Symbol: EllipticK — K ( m ) K(m) K ( m )
— Legendre complete elliptic integral of the first kind
Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind
Source code for this entry:
Entry(ID("e8ae42"),
SymbolDefinition(EllipticK, EllipticK(m), "Legendre complete elliptic integral of the first kind"))
Symbol: EllipticE — E ( m ) E(m) E ( m )
— Legendre complete elliptic integral of the second kind
Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind
Source code for this entry:
Entry(ID("723fd0"),
SymbolDefinition(EllipticE, EllipticE(m), "Legendre complete elliptic integral of the second kind"))
Symbol: EllipticPi — Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
— Legendre complete elliptic integral of the third kind
Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind
Source code for this entry:
Entry(ID("34482b"),
SymbolDefinition(EllipticPi, EllipticPi(n, m), "Legendre complete elliptic integral of the third kind"))
Symbol: IncompleteEllipticF — F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
— Legendre incomplete elliptic integral of the first kind
Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind
Source code for this entry:
Entry(ID("107140"),
SymbolDefinition(IncompleteEllipticF, IncompleteEllipticF(phi, m), "Legendre incomplete elliptic integral of the first kind"))
Symbol: IncompleteEllipticE — E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
— Legendre incomplete elliptic integral of the second kind
Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind
Source code for this entry:
Entry(ID("afdf5d"),
SymbolDefinition(IncompleteEllipticE, IncompleteEllipticE(phi, m), "Legendre incomplete elliptic integral of the second kind"))
Symbol: IncompleteEllipticPi — Π ( n , ϕ , m ) \Pi\!\left(n, \phi, m\right) Π ( n , ϕ , m )
— Legendre incomplete elliptic integral of the third kind
Definitions:
Fungrim symbol Notation Short description IncompleteEllipticPi Π ( n , ϕ , m ) \Pi\!\left(n, \phi, m\right) Π ( n , ϕ , m )
Legendre incomplete elliptic integral of the third kind
Source code for this entry:
Entry(ID("53b1e7"),
SymbolDefinition(IncompleteEllipticPi, IncompleteEllipticPi(n, phi, m), "Legendre incomplete elliptic integral of the third kind"))
Image: Plot of
K ( m ) K(m) K ( m )
on
m ∈ [ − 2 , 2 ] m \in \left[-2, 2\right] m ∈ [ − 2 , 2 ]
Big 🔍 Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval
Source code for this entry:
Entry(ID("89d93c"),
Image(Description("Plot of", EllipticK(m), "on", Element(m, ClosedInterval(-2, 2))), ImageSource("plot_elliptic_k")))
Image: Plot of
E ( m ) E(m) E ( m )
on
m ∈ [ − 2 , 2 ] m \in \left[-2, 2\right] m ∈ [ − 2 , 2 ]
Big 🔍 Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval
Source code for this entry:
Entry(ID("210213"),
Image(Description("Plot of", EllipticE(m), "on", Element(m, ClosedInterval(-2, 2))), ImageSource("plot_elliptic_e")))
Image: Plot of
F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
on
ϕ ∈ [ − 2 π , 2 π ] \phi \in \left[-2 \pi, 2 \pi\right] ϕ ∈ [ − 2 π , 2 π ]
Big 🔍 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("4704f9"),
Image(Description("Plot of", IncompleteEllipticF(phi, m), "on", Element(phi, ClosedInterval(Neg(Mul(2, Pi)), Mul(2, Pi)))), ImageSource("plot_incomplete_elliptic_f")))
Image: Plot of
E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
on
ϕ ∈ [ − 2 π , 2 π ] \phi \in \left[-2 \pi, 2 \pi\right] ϕ ∈ [ − 2 π , 2 π ]
Big 🔍 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("20d72c"),
Image(Description("Plot of", IncompleteEllipticE(phi, m), "on", Element(phi, ClosedInterval(Neg(Mul(2, Pi)), Mul(2, Pi)))), ImageSource("plot_incomplete_elliptic_e")))
K ( m ) = ∫ 0 π / 2 1 1 − m sin 2 ( x ) d x K(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx K ( m ) = ∫ 0 π / 2 1 − m sin 2 ( x ) 1 d x
Assumptions: m ∈ C ∖ [ 1 , ∞ ) m \in \mathbb{C} \setminus \left[1, \infty\right) m ∈ C ∖ [ 1 , ∞ )
TeX:
K(m) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
m \in \mathbb{C} \setminus \left[1, \infty\right) Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine Pi π \pi π
The constant pi (3.14...) CC C \mathbb{C} C
Complex numbers ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b )
Closed-open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("0455b3"),
Formula(Equal(EllipticK(m), Integral(Div(1, Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2))))), For(x, 0, Div(Pi, 2))))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity)))))
E ( m ) = ∫ 0 π / 2 1 − m sin 2 ( x ) d x E(m) = \int_{0}^{\pi / 2} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx E ( m ) = ∫ 0 π / 2 1 − m sin 2 ( x ) d x
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
E(m) = \int_{0}^{\pi / 2} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine Pi π \pi π
The constant pi (3.14...) CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("190843"),
Formula(Equal(EllipticE(m), Integral(Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2)))), For(x, 0, Div(Pi, 2))))),
Variables(m),
Assumptions(Element(m, CC)))
Π ( n , m ) = ∫ 0 π / 2 1 ( 1 − n sin 2 ( x ) ) 1 − m sin 2 ( x ) d x \Pi\!\left(n, m\right) = \int_{0}^{\pi / 2} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx Π ( n , m ) = ∫ 0 π / 2 ( 1 − n sin 2 ( x ) ) 1 − m sin 2 ( x ) 1 d x
Assumptions: n ∈ ( − ∞ , 1 ) and m ∈ ( − ∞ , 1 ) n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) n ∈ ( − ∞ , 1 ) a n d m ∈ ( − ∞ , 1 )
TeX:
\Pi\!\left(n, m\right) = \int_{0}^{\pi / 2} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...) OpenInterval ( a , b ) \left(a, b\right) ( a , b )
Open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("83a535"),
Formula(Equal(EllipticPi(n, m), Integral(Div(1, Mul(Sub(1, Mul(n, Pow(Sin(x), 2))), Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2)))))), For(x, 0, Div(Pi, 2))))),
Variables(n, m),
Assumptions(And(Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 1)))))
K ( m ) = ∫ 0 1 1 1 − x 2 1 − m x 2 d x K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx K ( m ) = ∫ 0 1 1 − x 2 1 − m x 2 1 d x
Assumptions: m ∈ C ∖ [ 1 , ∞ ) m \in \mathbb{C} \setminus \left[1, \infty\right) m ∈ C ∖ [ 1 , ∞ )
TeX:
K(m) = \int_{0}^{1} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
m \in \mathbb{C} \setminus \left[1, \infty\right) Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power CC C \mathbb{C} C
Complex numbers ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b )
Closed-open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("47dead"),
Formula(Equal(EllipticK(m), Integral(Div(1, Mul(Sqrt(Sub(1, Pow(x, 2))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, 1)))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity)))))
E ( m ) = ∫ 0 1 1 − m x 2 1 − x 2 d x E(m) = \int_{0}^{1} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx E ( m ) = ∫ 0 1 1 − x 2 1 − m x 2 d x
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
E(m) = \int_{0}^{1} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("fa8666"),
Formula(Equal(EllipticE(m), Integral(Div(Sqrt(Sub(1, Mul(m, Pow(x, 2)))), Sqrt(Sub(1, Pow(x, 2)))), For(x, 0, 1)))),
Variables(m),
Assumptions(Element(m, CC)))
Π ( n , m ) = ∫ 0 1 1 ( 1 − n x 2 ) 1 − x 2 1 − m x 2 d x \Pi\!\left(n, m\right) = \int_{0}^{1} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx Π ( n , m ) = ∫ 0 1 ( 1 − n x 2 ) 1 − x 2 1 − m x 2 1 d x
Assumptions: n ∈ ( − ∞ , 1 ) and m ∈ ( − ∞ , 1 ) n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) n ∈ ( − ∞ , 1 ) a n d m ∈ ( − ∞ , 1 )
TeX:
\Pi\!\left(n, m\right) = \int_{0}^{1} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Pow a b {a}^{b} a b
Power Sqrt z \sqrt{z} z
Principal square root OpenInterval ( a , b ) \left(a, b\right) ( a , b )
Open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("c10014"),
Formula(Equal(EllipticPi(n, m), Integral(Div(1, Mul(Mul(Sub(1, Mul(n, Pow(x, 2))), Sqrt(Sub(1, Pow(x, 2)))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, 1)))),
Variables(n, m),
Assumptions(And(Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 1)))))
K ( m ) = ∫ 1 ∞ 1 x 2 − 1 x 2 − m d x K(m) = \int_{1}^{\infty} \frac{1}{\sqrt{{x}^{2} - 1} \sqrt{{x}^{2} - m}} \, dx K ( m ) = ∫ 1 ∞ x 2 − 1 x 2 − m 1 d x
Assumptions: m ∈ C ∖ [ 1 , ∞ ) m \in \mathbb{C} \setminus \left[1, \infty\right) m ∈ C ∖ [ 1 , ∞ )
TeX:
K(m) = \int_{1}^{\infty} \frac{1}{\sqrt{{x}^{2} - 1} \sqrt{{x}^{2} - m}} \, dx
m \in \mathbb{C} \setminus \left[1, \infty\right) Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Infinity ∞ \infty ∞
Positive infinity CC C \mathbb{C} C
Complex numbers ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b )
Closed-open interval
Source code for this entry:
Entry(ID("7cd257"),
Formula(Equal(EllipticK(m), Integral(Div(1, Mul(Sqrt(Sub(Pow(x, 2), 1)), Sqrt(Sub(Pow(x, 2), m)))), For(x, 1, Infinity)))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity)))))
F ( ϕ , m ) = ∫ 0 ϕ 1 1 − m sin 2 ( x ) d x F\!\left(\phi, m\right) = \int_{0}^{\phi} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx F ( ϕ , m ) = ∫ 0 ϕ 1 − m sin 2 ( x ) 1 d x
Assumptions: ϕ ∈ [ − π 2 , π 2 ] and m ∈ C ∖ [ 1 , ∞ ) \phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right) ϕ ∈ [ 2 − π , 2 π ] a n d m ∈ C ∖ [ 1 , ∞ )
TeX:
F\!\left(\phi, m\right) = \int_{0}^{\phi} \frac{1}{\sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...) CC C \mathbb{C} C
Complex numbers ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b )
Closed-open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("81fb10"),
Formula(Equal(IncompleteEllipticF(phi, m), Integral(Div(1, Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2))))), For(x, 0, phi)))),
Variables(phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))
E ( ϕ , m ) = ∫ 0 ϕ 1 − m sin 2 ( x ) d x E\!\left(\phi, m\right) = \int_{0}^{\phi} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx E ( ϕ , m ) = ∫ 0 ϕ 1 − m sin 2 ( x ) d x
Assumptions: ϕ ∈ [ − π 2 , π 2 ] and m ∈ C \phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} ϕ ∈ [ 2 − π , 2 π ] a n d m ∈ C
TeX:
E\!\left(\phi, m\right) = \int_{0}^{\phi} \sqrt{1 - m \sin^{2}\!\left(x\right)} \, dx
\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...) CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("2ff7e7"),
Formula(Equal(IncompleteEllipticE(phi, m), Integral(Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2)))), For(x, 0, phi)))),
Variables(phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, CC))))
Π ( n , ϕ , m ) = ∫ 0 ϕ 1 ( 1 − n sin 2 ( x ) ) 1 − m sin 2 ( x ) d x \Pi\!\left(n, \phi, m\right) = \int_{0}^{\phi} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx Π ( n , ϕ , m ) = ∫ 0 ϕ ( 1 − n sin 2 ( x ) ) 1 − m sin 2 ( x ) 1 d x
Assumptions: ϕ ∈ [ − π 2 , π 2 ] and n ∈ ( − ∞ , 1 ) and m ∈ ( − ∞ , 1 ) \phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) ϕ ∈ [ 2 − π , 2 π ] a n d n ∈ ( − ∞ , 1 ) a n d m ∈ ( − ∞ , 1 )
TeX:
\Pi\!\left(n, \phi, m\right) = \int_{0}^{\phi} \frac{1}{\left(1 - n \sin^{2}\!\left(x\right)\right) \sqrt{1 - m \sin^{2}\!\left(x\right)}} \, dx
\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticPi Π ( n , ϕ , m ) \Pi\!\left(n, \phi, m\right) Π ( n , ϕ , m )
Legendre incomplete elliptic integral of the third kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine Sqrt z \sqrt{z} z
Principal square root ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...) OpenInterval ( a , b ) \left(a, b\right) ( a , b )
Open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("60f858"),
Formula(Equal(IncompleteEllipticPi(n, phi, m), Integral(Div(1, Mul(Sub(1, Mul(n, Pow(Sin(x), 2))), Sqrt(Sub(1, Mul(m, Pow(Sin(x), 2)))))), For(x, 0, phi)))),
Variables(n, phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 1)))))
F ( ϕ , m ) = ∫ 0 sin ( ϕ ) 1 1 − x 2 1 − m x 2 d x F\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx F ( ϕ , m ) = ∫ 0 sin ( ϕ ) 1 − x 2 1 − m x 2 1 d x
Assumptions: ϕ ∈ [ − π 2 , π 2 ] and m ∈ C ∖ [ 1 , ∞ ) \phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right) ϕ ∈ [ 2 − π , 2 π ] a n d m ∈ C ∖ [ 1 , ∞ )
TeX:
F\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \setminus \left[1, \infty\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...) CC C \mathbb{C} C
Complex numbers ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b )
Closed-open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("33ee4a"),
Formula(Equal(IncompleteEllipticF(phi, m), Integral(Div(1, Mul(Sqrt(Sub(1, Pow(x, 2))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, Sin(phi))))),
Variables(phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, SetMinus(CC, ClosedOpenInterval(1, Infinity))))))
E ( ϕ , m ) = ∫ 0 sin ( ϕ ) 1 − m x 2 1 − x 2 d x E\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx E ( ϕ , m ) = ∫ 0 sin ( ϕ ) 1 − x 2 1 − m x 2 d x
Assumptions: ϕ ∈ [ − π 2 , π 2 ] and m ∈ C \phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} ϕ ∈ [ 2 − π , 2 π ] a n d m ∈ C
TeX:
E\!\left(\phi, m\right) = \int_{0}^{\sin(\phi)} \frac{\sqrt{1 - m {x}^{2}}}{\sqrt{1 - {x}^{2}}} \, dx
\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Sin sin ( z ) \sin(z) sin ( z )
Sine ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...) CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("5e869b"),
Formula(Equal(IncompleteEllipticE(phi, m), Integral(Div(Sqrt(Sub(1, Mul(m, Pow(x, 2)))), Sqrt(Sub(1, Pow(x, 2)))), For(x, 0, Sin(phi))))),
Variables(phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(m, CC))))
Π ( n , ϕ , m ) = ∫ 0 sin ( ϕ ) 1 ( 1 − n x 2 ) 1 − x 2 1 − m x 2 d x \Pi\!\left(n, \phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx Π ( n , ϕ , m ) = ∫ 0 sin ( ϕ ) ( 1 − n x 2 ) 1 − x 2 1 − m x 2 1 d x
Assumptions: ϕ ∈ [ − π 2 , π 2 ] and n ∈ ( − ∞ , 1 ) and m ∈ ( − ∞ , 1 ) \phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) ϕ ∈ [ 2 − π , 2 π ] a n d n ∈ ( − ∞ , 1 ) a n d m ∈ ( − ∞ , 1 )
TeX:
\Pi\!\left(n, \phi, m\right) = \int_{0}^{\sin(\phi)} \frac{1}{\left(1 - n {x}^{2}\right) \sqrt{1 - {x}^{2}} \sqrt{1 - m {x}^{2}}} \, dx
\phi \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \;\mathbin{\operatorname{and}}\; n \in \left(-\infty, 1\right) \;\mathbin{\operatorname{and}}\; m \in \left(-\infty, 1\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticPi Π ( n , ϕ , m ) \Pi\!\left(n, \phi, m\right) Π ( n , ϕ , m )
Legendre incomplete elliptic integral of the third kind Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Integral Pow a b {a}^{b} a b
Power Sqrt z \sqrt{z} z
Principal square root Sin sin ( z ) \sin(z) sin ( z )
Sine ClosedInterval [ a , b ] \left[a, b\right] [ a , b ]
Closed interval Pi π \pi π
The constant pi (3.14...) OpenInterval ( a , b ) \left(a, b\right) ( a , b )
Open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("06223c"),
Formula(Equal(IncompleteEllipticPi(n, phi, m), Integral(Div(1, Mul(Mul(Sub(1, Mul(n, Pow(x, 2))), Sqrt(Sub(1, Pow(x, 2)))), Sqrt(Sub(1, Mul(m, Pow(x, 2)))))), For(x, 0, Sin(phi))))),
Variables(n, phi, m),
Assumptions(And(Element(phi, ClosedInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Element(n, OpenInterval(Neg(Infinity), 1)), Element(m, OpenInterval(Neg(Infinity), 1)))))
K ( 0 ) = π 2 K(0) = \frac{\pi}{2} K ( 0 ) = 2 π
TeX:
K(0) = \frac{\pi}{2} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("bb4501"),
Formula(Equal(EllipticK(0), Div(Pi, 2))))
E ( 0 ) = π 2 E(0) = \frac{\pi}{2} E ( 0 ) = 2 π
TeX:
E(0) = \frac{\pi}{2} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("1d62a7"),
Formula(Equal(EllipticE(0), Div(Pi, 2))))
K ( 1 ) = ∞ K(1) = \infty K ( 1 ) = ∞
TeX:
K(1) = \infty Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("45b157"),
Formula(Equal(EllipticK(1), Infinity)))
TeX:
E(1) = 1 Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind
Source code for this entry:
Entry(ID("958a3f"),
Formula(Equal(EllipticE(1), 1)))
K ( − 1 ) = ( Γ ( 1 4 ) ) 2 4 2 π K(-1) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} K ( − 1 ) = 4 2 π ( Γ ( 4 1 ) ) 2
TeX:
K(-1) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("afb22a"),
Formula(Equal(EllipticK(-1), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi)))))))
K ( 1 2 ) = ( Γ ( 1 4 ) ) 2 4 π K\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} K ( 2 1 ) = 4 π ( Γ ( 4 1 ) ) 2
TeX:
K\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("cc22bf"),
Formula(Equal(EllipticK(Div(1, 2)), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Pi))))))
K ( 2 ) = ( Γ ( 1 4 ) ) 2 4 2 π ( 1 − i ) K(2) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right) K ( 2 ) = 4 2 π ( Γ ( 4 1 ) ) 2 ( 1 − i )
TeX:
K(2) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} \left(1 - i\right) Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...) ConstI i i i
Imaginary unit
Source code for this entry:
Entry(ID("630eca"),
Formula(Equal(EllipticK(2), Mul(Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi)))), Sub(1, ConstI)))))
E ( − 1 ) = 2 ( ( Γ ( 1 4 ) ) 2 8 π + π 3 / 2 ( Γ ( 1 4 ) ) 2 ) E(-1) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right) E ( − 1 ) = 2 ( 8 π ( Γ ( 4 1 ) ) 2 + ( Γ ( 4 1 ) ) 2 π 3 / 2 )
TeX:
E(-1) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right) Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("9f3474"),
Formula(Equal(EllipticE(-1), Mul(Sqrt(2), Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(8, Sqrt(Pi))), Div(Pow(Pi, Div(3, 2)), Pow(Gamma(Div(1, 4)), 2)))))))
E ( 1 2 ) = ( Γ ( 1 4 ) ) 2 8 π + π 3 / 2 ( Γ ( 1 4 ) ) 2 E\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} E ( 2 1 ) = 8 π ( Γ ( 4 1 ) ) 2 + ( Γ ( 4 1 ) ) 2 π 3 / 2
TeX:
E\!\left(\frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("3b272e"),
Formula(Equal(EllipticE(Div(1, 2)), Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(8, Sqrt(Pi))), Div(Pow(Pi, Div(3, 2)), Pow(Gamma(Div(1, 4)), 2))))))
E ( 2 ) = 2 π 3 / 2 ( Γ ( 1 4 ) ) 2 ( 1 + i ) E(2) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right) E ( 2 ) = ( Γ ( 4 1 ) ) 2 2 π 3 / 2 ( 1 + i )
TeX:
E(2) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} \left(1 + i\right) Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Pi π \pi π
The constant pi (3.14...) Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function ConstI i i i
Imaginary unit
Source code for this entry:
Entry(ID("5d2c01"),
Formula(Equal(EllipticE(2), Mul(Div(Mul(Sqrt(2), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)), Add(1, ConstI)))))
K ( ( 3 − 2 2 ) 2 ) = ( 2 + 2 ) ( Γ ( 1 4 ) ) 2 16 π K\!\left({\left(3 - 2 \sqrt{2}\right)}^{2}\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}} K ( ( 3 − 2 2 ) 2 ) = 1 6 π ( 2 + 2 ) ( Γ ( 4 1 ) ) 2
TeX:
K\!\left({\left(3 - 2 \sqrt{2}\right)}^{2}\right) = \frac{\left(2 + \sqrt{2}\right) {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{16 \sqrt{\pi}} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pow a b {a}^{b} a b
Power Sqrt z \sqrt{z} z
Principal square root Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("2991b5"),
Formula(Equal(EllipticK(Pow(Sub(3, Mul(2, Sqrt(2))), 2)), Div(Mul(Add(2, Sqrt(2)), Pow(Gamma(Div(1, 4)), 2)), Mul(16, Sqrt(Pi))))))
K ( 4 − 3 2 8 ) = ( Γ ( 1 4 ) ) 2 4 ⋅ 2 1 / 4 π K\!\left(\frac{4 - 3 \sqrt{2}}{8}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \cdot {2}^{1 / 4} \sqrt{\pi}} K ( 8 4 − 3 2 ) = 4 ⋅ 2 1 / 4 π ( Γ ( 4 1 ) ) 2
TeX:
K\!\left(\frac{4 - 3 \sqrt{2}}{8}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \cdot {2}^{1 / 4} \sqrt{\pi}} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("4b040d"),
Formula(Equal(EllipticK(Div(Sub(4, Mul(3, Sqrt(2))), 8)), Div(Pow(Gamma(Div(1, 4)), 2), Mul(Mul(4, Pow(2, Div(1, 4))), Sqrt(Pi))))))
K ( 1 + 3 i 2 ) = e i π / 12 ⋅ 3 1 / 4 ( Γ ( 1 3 ) ) 3 2 7 / 3 π K\!\left(\frac{1 + \sqrt{3} i}{2}\right) = \frac{{e}^{i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi} K ( 2 1 + 3 i ) = 2 7 / 3 π e i π / 1 2 ⋅ 3 1 / 4 ( Γ ( 3 1 ) ) 3
TeX:
K\!\left(\frac{1 + \sqrt{3} i}{2}\right) = \frac{{e}^{i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Sqrt z \sqrt{z} z
Principal square root ConstI i i i
Imaginary unit Exp e z {e}^{z} e z
Exponential function Pi π \pi π
The constant pi (3.14...) Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function
Source code for this entry:
Entry(ID("0abbe1"),
Formula(Equal(EllipticK(Div(Add(1, Mul(Sqrt(3), ConstI)), 2)), Div(Mul(Mul(Exp(Div(Mul(ConstI, Pi), 12)), Pow(3, Div(1, 4))), Pow(Gamma(Div(1, 3)), 3)), Mul(Pow(2, Div(7, 3)), Pi)))))
K ( 1 − 3 i 2 ) = e − i π / 12 ⋅ 3 1 / 4 ( Γ ( 1 3 ) ) 3 2 7 / 3 π K\!\left(\frac{1 - \sqrt{3} i}{2}\right) = \frac{{e}^{-i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi} K ( 2 1 − 3 i ) = 2 7 / 3 π e − i π / 1 2 ⋅ 3 1 / 4 ( Γ ( 3 1 ) ) 3
TeX:
K\!\left(\frac{1 - \sqrt{3} i}{2}\right) = \frac{{e}^{-i \pi / 12} \cdot {3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{7 / 3} \pi} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Sqrt z \sqrt{z} z
Principal square root ConstI i i i
Imaginary unit Exp e z {e}^{z} e z
Exponential function Pi π \pi π
The constant pi (3.14...) Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function
Source code for this entry:
Entry(ID("175b7a"),
Formula(Equal(EllipticK(Div(Sub(1, Mul(Sqrt(3), ConstI)), 2)), Div(Mul(Mul(Exp(Neg(Div(Mul(ConstI, Pi), 12))), Pow(3, Div(1, 4))), Pow(Gamma(Div(1, 3)), 3)), Mul(Pow(2, Div(7, 3)), Pi)))))
K ( 4 3 − 7 ) = 3 + 2 3 ( Γ ( 1 3 ) ) 3 2 10 / 3 π K\!\left(4 \sqrt{3} - 7\right) = \frac{\sqrt{3 + 2 \sqrt{3}} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{10 / 3} \pi} K ( 4 3 − 7 ) = 2 1 0 / 3 π 3 + 2 3 ( Γ ( 3 1 ) ) 3
TeX:
K\!\left(4 \sqrt{3} - 7\right) = \frac{\sqrt{3 + 2 \sqrt{3}} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{{2}^{10 / 3} \pi} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("b95ffa"),
Formula(Equal(EllipticK(Sub(Mul(4, Sqrt(3)), 7)), Div(Mul(Sqrt(Add(3, Mul(2, Sqrt(3)))), Pow(Gamma(Div(1, 3)), 3)), Mul(Pow(2, Div(10, 3)), Pi)))))
K ( 1 2 − 3 4 ) = 3 1 / 4 ( Γ ( 1 3 ) ) 3 4 ⋅ 2 1 / 3 π K\!\left(\frac{1}{2} - \frac{\sqrt{3}}{4}\right) = \frac{{3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{4 \cdot {2}^{1 / 3} \pi} K ( 2 1 − 4 3 ) = 4 ⋅ 2 1 / 3 π 3 1 / 4 ( Γ ( 3 1 ) ) 3
TeX:
K\!\left(\frac{1}{2} - \frac{\sqrt{3}}{4}\right) = \frac{{3}^{1 / 4} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3}}{4 \cdot {2}^{1 / 3} \pi} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("40a376"),
Formula(Equal(EllipticK(Sub(Div(1, 2), Div(Sqrt(3), 4))), Div(Mul(Pow(3, Div(1, 4)), Pow(Gamma(Div(1, 3)), 3)), Mul(Mul(4, Pow(2, Div(1, 3))), Pi)))))
Π ( 0 , 0 ) = π 2 \Pi\!\left(0, 0\right) = \frac{\pi}{2} Π ( 0 , 0 ) = 2 π
TeX:
\Pi\!\left(0, 0\right) = \frac{\pi}{2} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("618a54"),
Formula(Equal(EllipticPi(0, 0), Div(Pi, 2))))
Π ( 0 , 1 ) = ∞ \Pi\!\left(0, 1\right) = \infty Π ( 0 , 1 ) = ∞
TeX:
\Pi\!\left(0, 1\right) = \infty Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("18e226"),
Formula(Equal(EllipticPi(0, 1), Infinity)))
Π ( 1 , 0 ) = ∞ ~ \Pi\!\left(1, 0\right) = {\tilde \infty} Π ( 1 , 0 ) = ∞ ~
TeX:
\Pi\!\left(1, 0\right) = {\tilde \infty} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind UnsignedInfinity ∞ ~ {\tilde \infty} ∞ ~
Unsigned infinity
Source code for this entry:
Entry(ID("061c49"),
Formula(Equal(EllipticPi(1, 0), UnsignedInfinity)))
Π ( 0 , 1 2 ) = ( Γ ( 1 4 ) ) 2 4 π \Pi\!\left(0, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} Π ( 0 , 2 1 ) = 4 π ( Γ ( 4 1 ) ) 2
TeX:
\Pi\!\left(0, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("3c4979"),
Formula(Equal(EllipticPi(0, Div(1, 2)), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Pi))))))
Π ( 1 2 , 0 ) = π 2 2 \Pi\!\left(\frac{1}{2}, 0\right) = \frac{\pi \sqrt{2}}{2} Π ( 2 1 , 0 ) = 2 π 2
TeX:
\Pi\!\left(\frac{1}{2}, 0\right) = \frac{\pi \sqrt{2}}{2} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Pi π \pi π
The constant pi (3.14...) Sqrt z \sqrt{z} z
Principal square root
Source code for this entry:
Entry(ID("124d02"),
Formula(Equal(EllipticPi(Div(1, 2), 0), Div(Mul(Pi, Sqrt(2)), 2))))
Π ( 1 2 , 1 2 ) = ( Γ ( 1 4 ) ) 2 4 π + 2 π 3 / 2 ( Γ ( 1 4 ) ) 2 \Pi\!\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} + \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} Π ( 2 1 , 2 1 ) = 4 π ( Γ ( 4 1 ) ) 2 + ( Γ ( 4 1 ) ) 2 2 π 3 / 2
TeX:
\Pi\!\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{\pi}} + \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("9b0385"),
Formula(Equal(EllipticPi(Div(1, 2), Div(1, 2)), Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Pi))), Div(Mul(2, Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2))))))
Π ( 1 , m ) = ∞ ~ \Pi\!\left(1, m\right) = {\tilde \infty} Π ( 1 , m ) = ∞ ~
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
\Pi\!\left(1, m\right) = {\tilde \infty}
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind UnsignedInfinity ∞ ~ {\tilde \infty} ∞ ~
Unsigned infinity CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("ce4df4"),
Formula(Equal(EllipticPi(1, m), UnsignedInfinity)),
Variables(m),
Assumptions(Element(m, CC)))
Π ( n , 1 ) = { ( 1 − n ) − 1 ∞ , n ≠ 1 ∞ ~ , n = 1 \Pi\!\left(n, 1\right) = \begin{cases} {\left(1 - n\right)}^{-1} \infty, & n \ne 1\\{\tilde \infty}, & n = 1\\ \end{cases} Π ( n , 1 ) = { ( 1 − n ) − 1 ∞ , ∞ ~ , n = 1 n = 1
Assumptions: n ∈ C n \in \mathbb{C} n ∈ C
TeX:
\Pi\!\left(n, 1\right) = \begin{cases} {\left(1 - n\right)}^{-1} \infty, & n \ne 1\\{\tilde \infty}, & n = 1\\ \end{cases}
n \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Pow a b {a}^{b} a b
Power Infinity ∞ \infty ∞
Positive infinity UnsignedInfinity ∞ ~ {\tilde \infty} ∞ ~
Unsigned infinity CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("e9c797"),
Formula(Equal(EllipticPi(n, 1), Cases(Tuple(Mul(Pow(Sub(1, n), -1), Infinity), NotEqual(n, 1)), Tuple(UnsignedInfinity, Equal(n, 1))))),
Variables(n),
Assumptions(Element(n, CC)))
Π ( n , 0 ) = π 2 1 − n \Pi\!\left(n, 0\right) = \frac{\pi}{2 \sqrt{1 - n}} Π ( n , 0 ) = 2 1 − n π
Assumptions: n ∈ C n \in \mathbb{C} n ∈ C
TeX:
\Pi\!\left(n, 0\right) = \frac{\pi}{2 \sqrt{1 - n}}
n \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind Pi π \pi π
The constant pi (3.14...) Sqrt z \sqrt{z} z
Principal square root CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("5d8804"),
Formula(Equal(EllipticPi(n, 0), Div(Pi, Mul(2, Sqrt(Sub(1, n)))))),
Variables(n),
Assumptions(Element(n, CC)))
Π ( 0 , m ) = K ( m ) \Pi\!\left(0, m\right) = K(m) Π ( 0 , m ) = K ( m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
\Pi\!\left(0, m\right) = K(m)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("dd67fb"),
Formula(Equal(EllipticPi(0, m), EllipticK(m))),
Variables(m),
Assumptions(Element(m, CC)))
Π ( m , m ) = E ( m ) 1 − m \Pi\!\left(m, m\right) = \frac{E(m)}{1 - m} Π ( m , m ) = 1 − m E ( m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
\Pi\!\left(m, m\right) = \frac{E(m)}{1 - m}
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("9227bf"),
Formula(Equal(EllipticPi(m, m), Div(EllipticE(m), Sub(1, m)))),
Variables(m),
Assumptions(Element(m, CC)))
F ( 0 , 0 ) = 0 F\!\left(0, 0\right) = 0 F ( 0 , 0 ) = 0
TeX:
F\!\left(0, 0\right) = 0 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind
Source code for this entry:
Entry(ID("ba1965"),
Formula(Equal(IncompleteEllipticF(0, 0), 0)))
F ( 0 , m ) = 0 F\!\left(0, m\right) = 0 F ( 0 , m ) = 0
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
F\!\left(0, m\right) = 0
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("4268fc"),
Formula(Equal(IncompleteEllipticF(0, m), 0)),
Variables(m),
Assumptions(Element(m, CC)))
F ( ϕ , 0 ) = ϕ F\!\left(\phi, 0\right) = \phi F ( ϕ , 0 ) = ϕ
Assumptions: ϕ ∈ C \phi \in \mathbb{C} ϕ ∈ C
TeX:
F\!\left(\phi, 0\right) = \phi
\phi \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("d2adb6"),
Formula(Equal(IncompleteEllipticF(phi, 0), phi)),
Variables(phi),
Assumptions(Element(phi, CC)))
F ( π 2 , m ) = K ( m ) F\!\left(\frac{\pi}{2}, m\right) = K(m) F ( 2 π , m ) = K ( m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
F\!\left(\frac{\pi}{2}, m\right) = K(m)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("0b8fd6"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 2), m), EllipticK(m))),
Variables(m),
Assumptions(Element(m, CC)))
F ( − π 2 , m ) = − K ( m ) F\!\left(\frac{-\pi}{2}, m\right) = -K(m) F ( 2 − π , m ) = − K ( m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
F\!\left(\frac{-\pi}{2}, m\right) = -K(m)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("81f7db"),
Formula(Equal(IncompleteEllipticF(Div(Neg(Pi), 2), m), Neg(EllipticK(m)))),
Variables(m),
Assumptions(Element(m, CC)))
F ( π k 2 , m ) = k K ( m ) F\!\left(\frac{\pi k}{2}, m\right) = k K(m) F ( 2 π k , m ) = k K ( m )
Assumptions: m ∈ C and k ∈ Z and ( k ≠ 0 or m ≠ 1 ) m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \left(k \ne 0 \;\mathbin{\operatorname{or}}\; m \ne 1\right) m ∈ C a n d k ∈ Z a n d ( k = 0 o r m = 1 )
TeX:
F\!\left(\frac{\pi k}{2}, m\right) = k K(m)
m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \left(k \ne 0 \;\mathbin{\operatorname{or}}\; m \ne 1\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers ZZ Z \mathbb{Z} Z
Integers
Source code for this entry:
Entry(ID("afabeb"),
Formula(Equal(IncompleteEllipticF(Div(Mul(Pi, k), 2), m), Mul(k, EllipticK(m)))),
Variables(m, k),
Assumptions(And(Element(m, CC), Element(k, ZZ), Or(NotEqual(k, 0), NotEqual(m, 1)))))
F ( π 2 , 0 ) = π 2 F\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2} F ( 2 π , 0 ) = 2 π
TeX:
F\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("c0ad12"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 2), 0), Div(Pi, 2))))
F ( π 2 , − 1 ) = ( Γ ( 1 4 ) ) 2 4 2 π F\!\left(\frac{\pi}{2}, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} F ( 2 π , − 1 ) = 4 2 π ( Γ ( 4 1 ) ) 2
TeX:
F\!\left(\frac{\pi}{2}, -1\right) = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{4 \sqrt{2 \pi}} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function Sqrt z \sqrt{z} z
Principal square root
Source code for this entry:
Entry(ID("ace837"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 2), -1), Div(Pow(Gamma(Div(1, 4)), 2), Mul(4, Sqrt(Mul(2, Pi)))))))
F ( π 2 , 1 ) = ∞ F\!\left(\frac{\pi}{2}, 1\right) = \infty F ( 2 π , 1 ) = ∞
TeX:
F\!\left(\frac{\pi}{2}, 1\right) = \infty Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("16612f"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 2), 1), Infinity)))
F ( − π 2 , 1 ) = − ∞ F\!\left(\frac{-\pi}{2}, 1\right) = -\infty F ( 2 − π , 1 ) = − ∞
TeX:
F\!\left(\frac{-\pi}{2}, 1\right) = -\infty Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("04c829"),
Formula(Equal(IncompleteEllipticF(Div(Neg(Pi), 2), 1), Neg(Infinity))))
F ( π 3 , 1 ) = log ( 2 + 3 ) F\!\left(\frac{\pi}{3}, 1\right) = \log\!\left(2 + \sqrt{3}\right) F ( 3 π , 1 ) = log ( 2 + 3 )
TeX:
F\!\left(\frac{\pi}{3}, 1\right) = \log\!\left(2 + \sqrt{3}\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Log log ( z ) \log(z) log ( z )
Natural logarithm Sqrt z \sqrt{z} z
Principal square root
Source code for this entry:
Entry(ID("c584c3"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 3), 1), Log(Add(2, Sqrt(3))))))
F ( π 4 , 1 ) = log ( 1 + 2 ) F\!\left(\frac{\pi}{4}, 1\right) = \log\!\left(1 + \sqrt{2}\right) F ( 4 π , 1 ) = log ( 1 + 2 )
TeX:
F\!\left(\frac{\pi}{4}, 1\right) = \log\!\left(1 + \sqrt{2}\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Log log ( z ) \log(z) log ( z )
Natural logarithm Sqrt z \sqrt{z} z
Principal square root
Source code for this entry:
Entry(ID("f5d489"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 4), 1), Log(Add(1, Sqrt(2))))))
F ( π 6 , 1 ) = log ( 3 ) 2 F\!\left(\frac{\pi}{6}, 1\right) = \frac{\log(3)}{2} F ( 6 π , 1 ) = 2 log ( 3 )
TeX:
F\!\left(\frac{\pi}{6}, 1\right) = \frac{\log(3)}{2} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Log log ( z ) \log(z) log ( z )
Natural logarithm
Source code for this entry:
Entry(ID("a91f8d"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 6), 1), Div(Log(3), 2))))
F ( ϕ , 1 ) = { log ( 1 + sin ( ϕ ) cos ( ϕ ) ) , − π 2 ≤ Re ( ϕ ) ≤ π 2 and ϕ ∉ { − π 2 , π 2 } sgn ( ϕ ) ∞ , ϕ ∈ { − π 2 , π 2 } ∞ ~ , otherwise F\!\left(\phi, 1\right) = \begin{cases} \log\!\left(\frac{1 + \sin(\phi)}{\cos(\phi)}\right), & \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2} \;\mathbin{\operatorname{and}}\; \phi \notin \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\\operatorname{sgn}(\phi) \infty, & \phi \in \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases} F ( ϕ , 1 ) = ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ log ( cos ( ϕ ) 1 + sin ( ϕ ) ) , s g n ( ϕ ) ∞ , ∞ ~ , 2 − π ≤ R e ( ϕ ) ≤ 2 π a n d ϕ ∈ / { 2 − π , 2 π } ϕ ∈ { 2 − π , 2 π } otherwise
Assumptions: ϕ ∈ C \phi \in \mathbb{C} ϕ ∈ C
TeX:
F\!\left(\phi, 1\right) = \begin{cases} \log\!\left(\frac{1 + \sin(\phi)}{\cos(\phi)}\right), & \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2} \;\mathbin{\operatorname{and}}\; \phi \notin \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\\operatorname{sgn}(\phi) \infty, & \phi \in \left\{\frac{-\pi}{2}, \frac{\pi}{2}\right\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}
\phi \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Log log ( z ) \log(z) log ( z )
Natural logarithm Sin sin ( z ) \sin(z) sin ( z )
Sine Cos cos ( z ) \cos(z) cos ( z )
Cosine Pi π \pi π
The constant pi (3.14...) Re Re ( z ) \operatorname{Re}(z) R e ( z )
Real part Sign sgn ( z ) \operatorname{sgn}(z) s g n ( z )
Sign function Infinity ∞ \infty ∞
Positive infinity UnsignedInfinity ∞ ~ {\tilde \infty} ∞ ~
Unsigned infinity CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("b7cfb3"),
Formula(Equal(IncompleteEllipticF(phi, 1), Cases(Tuple(Log(Div(Add(1, Sin(phi)), Cos(phi))), And(LessEqual(Div(Neg(Pi), 2), Re(phi), Div(Pi, 2)), NotElement(phi, Set(Div(Neg(Pi), 2), Div(Pi, 2))))), Tuple(Mul(Sign(phi), Infinity), Element(phi, Set(Div(Neg(Pi), 2), Div(Pi, 2)))), Tuple(UnsignedInfinity, Otherwise)))),
Variables(phi),
Assumptions(Element(phi, CC)))
F ( π 4 , 2 ) = 2 ( Γ ( 1 4 ) ) 2 8 π F\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} F ( 4 π , 2 ) = 8 π 2 ( Γ ( 4 1 ) ) 2
TeX:
F\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function
Source code for this entry:
Entry(ID("8b4be6"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 4), 2), Div(Mul(Sqrt(2), Pow(Gamma(Div(1, 4)), 2)), Mul(8, Sqrt(Pi))))))
F ( π 6 , 4 ) = K ( 1 4 ) 2 F\!\left(\frac{\pi}{6}, 4\right) = \frac{K\!\left(\frac{1}{4}\right)}{2} F ( 6 π , 4 ) = 2 K ( 4 1 )
TeX:
F\!\left(\frac{\pi}{6}, 4\right) = \frac{K\!\left(\frac{1}{4}\right)}{2} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind
Source code for this entry:
Entry(ID("aac129"),
Formula(Equal(IncompleteEllipticF(Div(Pi, 6), 4), Div(EllipticK(Div(1, 4)), 2))))
F ( asin ( 1 m ) , m ) = K ( 1 m ) m F\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \frac{K\!\left(\frac{1}{m}\right)}{\sqrt{m}} F ( a s i n ( m 1 ) , m ) = m K ( m 1 )
Assumptions: m ∈ C ∖ { 0 } m \in \mathbb{C} \setminus \left\{0\right\} m ∈ C ∖ { 0 }
TeX:
F\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \frac{K\!\left(\frac{1}{m}\right)}{\sqrt{m}}
m \in \mathbb{C} \setminus \left\{0\right\} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Sqrt z \sqrt{z} z
Principal square root EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("087a7c"),
Formula(Equal(IncompleteEllipticF(Asin(Div(1, Sqrt(m))), m), Div(EllipticK(Div(1, m)), Sqrt(m)))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, Set(0)))))
E ( 0 , 0 ) = 0 E\!\left(0, 0\right) = 0 E ( 0 , 0 ) = 0
TeX:
E\!\left(0, 0\right) = 0 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind
Source code for this entry:
Entry(ID("a6c07e"),
Formula(Equal(IncompleteEllipticE(0, 0), 0)))
E ( 0 , m ) = 0 E\!\left(0, m\right) = 0 E ( 0 , m ) = 0
TeX:
E\!\left(0, m\right) = 0 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind
Source code for this entry:
Entry(ID("be3e09"),
Formula(Equal(IncompleteEllipticE(0, m), 0)))
E ( ϕ , 0 ) = ϕ E\!\left(\phi, 0\right) = \phi E ( ϕ , 0 ) = ϕ
Assumptions: ϕ ∈ C \phi \in \mathbb{C} ϕ ∈ C
TeX:
E\!\left(\phi, 0\right) = \phi
\phi \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("efc7a4"),
Formula(Equal(IncompleteEllipticE(phi, 0), phi)),
Variables(phi),
Assumptions(Element(phi, CC)))
E ( π 2 , m ) = E ( m ) E\!\left(\frac{\pi}{2}, m\right) = E(m) E ( 2 π , m ) = E ( m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
E\!\left(\frac{\pi}{2}, m\right) = E(m)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("1b881e"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 2), m), EllipticE(m))),
Variables(m),
Assumptions(Element(m, CC)))
E ( − π 2 , m ) = − E ( m ) E\!\left(\frac{-\pi}{2}, m\right) = -E(m) E ( 2 − π , m ) = − E ( m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
E\!\left(\frac{-\pi}{2}, m\right) = -E(m)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("2ef763"),
Formula(Equal(IncompleteEllipticE(Div(Neg(Pi), 2), m), Neg(EllipticE(m)))),
Variables(m),
Assumptions(Element(m, CC)))
E ( π k 2 , m ) = k E ( m ) E\!\left(\frac{\pi k}{2}, m\right) = k E(m) E ( 2 π k , m ) = k E ( m )
Assumptions: m ∈ C and k ∈ Z m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} m ∈ C a n d k ∈ Z
TeX:
E\!\left(\frac{\pi k}{2}, m\right) = k E(m)
m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers ZZ Z \mathbb{Z} Z
Integers
Source code for this entry:
Entry(ID("a14442"),
Formula(Equal(IncompleteEllipticE(Div(Mul(Pi, k), 2), m), Mul(k, EllipticE(m)))),
Variables(m, k),
Assumptions(And(Element(m, CC), Element(k, ZZ))))
E ( ϕ , 1 ) = sin ( ϕ ) E\!\left(\phi, 1\right) = \sin(\phi) E ( ϕ , 1 ) = sin ( ϕ )
Assumptions: ϕ ∈ C and ( Re ( ϕ ) ∈ [ − π 2 , π 2 ) or ϕ = π 2 ) \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}(\phi) \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right) \;\mathbin{\operatorname{or}}\; \phi = \frac{\pi}{2}\right) ϕ ∈ C a n d ( R e ( ϕ ) ∈ [ 2 − π , 2 π ) o r ϕ = 2 π )
TeX:
E\!\left(\phi, 1\right) = \sin(\phi)
\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}(\phi) \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right) \;\mathbin{\operatorname{or}}\; \phi = \frac{\pi}{2}\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Sin sin ( z ) \sin(z) sin ( z )
Sine CC C \mathbb{C} C
Complex numbers Re Re ( z ) \operatorname{Re}(z) R e ( z )
Real part ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b )
Closed-open interval Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("75e141"),
Formula(Equal(IncompleteEllipticE(phi, 1), Sin(phi))),
Variables(phi),
Assumptions(And(Element(phi, CC), Or(Element(Re(phi), ClosedOpenInterval(Div(Neg(Pi), 2), Div(Pi, 2))), Equal(phi, Div(Pi, 2))))))
E ( ϕ , 1 ) = ( − 1 ) ⌊ Re ( ϕ ) / π + 1 / 2 ⌋ sin ( ϕ ) + 2 ⌊ Re ( ϕ ) π + 1 2 ⌋ E\!\left(\phi, 1\right) = {\left(-1\right)}^{\left\lfloor \operatorname{Re}(\phi) / \pi + 1 / 2 \right\rfloor} \sin(\phi) + 2 \left\lfloor \frac{\operatorname{Re}(\phi)}{\pi} + \frac{1}{2} \right\rfloor E ( ϕ , 1 ) = ( − 1 ) ⌊ R e ( ϕ ) / π + 1 / 2 ⌋ sin ( ϕ ) + 2 ⌊ π R e ( ϕ ) + 2 1 ⌋
Assumptions: ϕ ∈ C \phi \in \mathbb{C} ϕ ∈ C
TeX:
E\!\left(\phi, 1\right) = {\left(-1\right)}^{\left\lfloor \operatorname{Re}(\phi) / \pi + 1 / 2 \right\rfloor} \sin(\phi) + 2 \left\lfloor \frac{\operatorname{Re}(\phi)}{\pi} + \frac{1}{2} \right\rfloor
\phi \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pow a b {a}^{b} a b
Power Re Re ( z ) \operatorname{Re}(z) R e ( z )
Real part Pi π \pi π
The constant pi (3.14...) Sin sin ( z ) \sin(z) sin ( z )
Sine CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("f35a37"),
Formula(Equal(IncompleteEllipticE(phi, 1), Add(Mul(Pow(-1, Floor(Add(Div(Re(phi), Pi), Div(1, 2)))), Sin(phi)), Mul(2, Floor(Add(Div(Re(phi), Pi), Div(1, 2))))))),
Variables(phi),
Assumptions(Element(phi, CC)))
E ( π 2 , 0 ) = π 2 E\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2} E ( 2 π , 0 ) = 2 π
TeX:
E\!\left(\frac{\pi}{2}, 0\right) = \frac{\pi}{2} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("51a946"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 2), 0), Div(Pi, 2))))
E ( π 2 , − 1 ) = 2 ( ( Γ ( 1 4 ) ) 2 8 π + π 3 / 2 ( Γ ( 1 4 ) ) 2 ) E\!\left(\frac{\pi}{2}, -1\right) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right) E ( 2 π , − 1 ) = 2 ( 8 π ( Γ ( 4 1 ) ) 2 + ( Γ ( 4 1 ) ) 2 π 3 / 2 )
TeX:
E\!\left(\frac{\pi}{2}, -1\right) = \sqrt{2} \left(\frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}{8 \sqrt{\pi}} + \frac{{\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function
Source code for this entry:
Entry(ID("2573ba"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 2), -1), Mul(Sqrt(2), Add(Div(Pow(Gamma(Div(1, 4)), 2), Mul(8, Sqrt(Pi))), Div(Pow(Pi, Div(3, 2)), Pow(Gamma(Div(1, 4)), 2)))))))
E ( π 2 , 1 ) = 1 E\!\left(\frac{\pi}{2}, 1\right) = 1 E ( 2 π , 1 ) = 1
TeX:
E\!\left(\frac{\pi}{2}, 1\right) = 1 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("b62aae"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 2), 1), 1)))
E ( − π 2 , 1 ) = − 1 E\!\left(\frac{-\pi}{2}, 1\right) = -1 E ( 2 − π , 1 ) = − 1
TeX:
E\!\left(\frac{-\pi}{2}, 1\right) = -1 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("dec0d2"),
Formula(Equal(IncompleteEllipticE(Div(Neg(Pi), 2), 1), -1)))
E ( π k 2 , 1 ) = k E\!\left(\frac{\pi k}{2}, 1\right) = k E ( 2 π k , 1 ) = k
Assumptions: k ∈ Z k \in \mathbb{Z} k ∈ Z
TeX:
E\!\left(\frac{\pi k}{2}, 1\right) = k
k \in \mathbb{Z} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) ZZ Z \mathbb{Z} Z
Integers
Source code for this entry:
Entry(ID("2245df"),
Formula(Equal(IncompleteEllipticE(Div(Mul(Pi, k), 2), 1), k)),
Variables(k),
Assumptions(Element(k, ZZ)))
E ( π 3 , 1 ) = 3 2 E\!\left(\frac{\pi}{3}, 1\right) = \frac{\sqrt{3}}{2} E ( 3 π , 1 ) = 2 3
TeX:
E\!\left(\frac{\pi}{3}, 1\right) = \frac{\sqrt{3}}{2} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) Sqrt z \sqrt{z} z
Principal square root
Source code for this entry:
Entry(ID("3aed02"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 3), 1), Div(Sqrt(3), 2))))
E ( π 6 , 1 ) = 1 2 E\!\left(\frac{\pi}{6}, 1\right) = \frac{1}{2} E ( 6 π , 1 ) = 2 1
TeX:
E\!\left(\frac{\pi}{6}, 1\right) = \frac{1}{2} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...)
Source code for this entry:
Entry(ID("d88dd1"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 6), 1), Div(1, 2))))
E ( π 4 , 2 ) = 2 π 3 / 2 ( Γ ( 1 4 ) ) 2 E\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} E ( 4 π , 2 ) = ( Γ ( 4 1 ) ) 2 2 π 3 / 2
TeX:
E\!\left(\frac{\pi}{4}, 2\right) = \frac{\sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) Sqrt z \sqrt{z} z
Principal square root Pow a b {a}^{b} a b
Power Gamma Γ ( z ) \Gamma(z) Γ ( z )
Gamma function
Source code for this entry:
Entry(ID("4dabda"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 4), 2), Div(Mul(Sqrt(2), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))
E ( π 6 , 4 ) = 2 E ( 1 4 ) − 3 2 K ( 1 4 ) E\!\left(\frac{\pi}{6}, 4\right) = 2 E\!\left(\frac{1}{4}\right) - \frac{3}{2} K\!\left(\frac{1}{4}\right) E ( 6 π , 4 ) = 2 E ( 4 1 ) − 2 3 K ( 4 1 )
TeX:
E\!\left(\frac{\pi}{6}, 4\right) = 2 E\!\left(\frac{1}{4}\right) - \frac{3}{2} K\!\left(\frac{1}{4}\right) Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind
Source code for this entry:
Entry(ID("eba27c"),
Formula(Equal(IncompleteEllipticE(Div(Pi, 6), 4), Sub(Mul(2, EllipticE(Div(1, 4))), Mul(Div(3, 2), EllipticK(Div(1, 4)))))))
E ( asin ( 1 m ) , m ) = m ( E ( 1 m ) − ( 1 − 1 m ) K ( 1 m ) ) E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right) E ( a s i n ( m 1 ) , m ) = m ( E ( m 1 ) − ( 1 − m 1 ) K ( m 1 ) )
Assumptions: m ∈ C ∖ { 0 , 1 } m \in \mathbb{C} \setminus \left\{0, 1\right\} m ∈ C ∖ { 0 , 1 }
TeX:
E\!\left(\operatorname{asin}\!\left(\frac{1}{\sqrt{m}}\right), m\right) = \sqrt{m} \left(E\!\left(\frac{1}{m}\right) - \left(1 - \frac{1}{m}\right) K\!\left(\frac{1}{m}\right)\right)
m \in \mathbb{C} \setminus \left\{0, 1\right\} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Sqrt z \sqrt{z} z
Principal square root EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("f0bcb5"),
Formula(Equal(IncompleteEllipticE(Asin(Div(1, Sqrt(m))), m), Mul(Sqrt(m), Sub(EllipticE(Div(1, m)), Mul(Sub(1, Div(1, m)), EllipticK(Div(1, m))))))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, Set(0, 1)))))
K ( m ‾ ) = K ( m ) ‾ K\!\left(\overline{m}\right) = \overline{K(m)} K ( m ) = K ( m )
Assumptions: m ∈ C ∖ ( 1 , ∞ ) m \in \mathbb{C} \setminus \left(1, \infty\right) m ∈ C ∖ ( 1 , ∞ )
TeX:
K\!\left(\overline{m}\right) = \overline{K(m)}
m \in \mathbb{C} \setminus \left(1, \infty\right) Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Conjugate z ‾ \overline{z} z
Complex conjugate CC C \mathbb{C} C
Complex numbers OpenInterval ( a , b ) \left(a, b\right) ( a , b )
Open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("713966"),
Formula(Equal(EllipticK(Conjugate(m)), Conjugate(EllipticK(m)))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, OpenInterval(1, Infinity)))))
E ( m ‾ ) = E ( m ) ‾ E\!\left(\overline{m}\right) = \overline{E(m)} E ( m ) = E ( m )
Assumptions: m ∈ C ∖ ( 1 , ∞ ) m \in \mathbb{C} \setminus \left(1, \infty\right) m ∈ C ∖ ( 1 , ∞ )
TeX:
E\!\left(\overline{m}\right) = \overline{E(m)}
m \in \mathbb{C} \setminus \left(1, \infty\right) Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Conjugate z ‾ \overline{z} z
Complex conjugate CC C \mathbb{C} C
Complex numbers OpenInterval ( a , b ) \left(a, b\right) ( a , b )
Open interval Infinity ∞ \infty ∞
Positive infinity
Source code for this entry:
Entry(ID("8e5c81"),
Formula(Equal(EllipticE(Conjugate(m)), Conjugate(EllipticE(m)))),
Variables(m),
Assumptions(Element(m, SetMinus(CC, OpenInterval(1, Infinity)))))
F ( − ϕ , m ) = − F ( ϕ , m ) F\!\left(-\phi, m\right) = -F\!\left(\phi, m\right) F ( − ϕ , m ) = − F ( ϕ , m )
Assumptions: ϕ ∈ C and m ∈ C \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} ϕ ∈ C a n d m ∈ C
TeX:
F\!\left(-\phi, m\right) = -F\!\left(\phi, m\right)
\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("b0eb37"),
Formula(Equal(IncompleteEllipticF(Neg(phi), m), Neg(IncompleteEllipticF(phi, m)))),
Variables(phi, m),
Assumptions(And(Element(phi, CC), Element(m, CC))))
E ( − ϕ , m ) = − E ( ϕ , m ) E\!\left(-\phi, m\right) = -E\!\left(\phi, m\right) E ( − ϕ , m ) = − E ( ϕ , m )
Assumptions: ϕ ∈ C and m ∈ C \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} ϕ ∈ C a n d m ∈ C
TeX:
E\!\left(-\phi, m\right) = -E\!\left(\phi, m\right)
\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("aa1b8e"),
Formula(Equal(IncompleteEllipticE(Neg(phi), m), Neg(IncompleteEllipticE(phi, m)))),
Variables(phi, m),
Assumptions(And(Element(phi, CC), Element(m, CC))))
Π ( n , − ϕ , m ) = − Π ( n , ϕ , m ) \Pi\!\left(n, -\phi, m\right) = -\Pi\!\left(n, \phi, m\right) Π ( n , − ϕ , m ) = − Π ( n , ϕ , m )
Assumptions: n ∈ C and ϕ ∈ C and m ∈ C n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} n ∈ C a n d ϕ ∈ C a n d m ∈ C
TeX:
\Pi\!\left(n, -\phi, m\right) = -\Pi\!\left(n, \phi, m\right)
n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticPi Π ( n , ϕ , m ) \Pi\!\left(n, \phi, m\right) Π ( n , ϕ , m )
Legendre incomplete elliptic integral of the third kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("255d81"),
Formula(Equal(IncompleteEllipticPi(n, Neg(phi), m), Neg(IncompleteEllipticPi(n, phi, m)))),
Variables(n, phi, m),
Assumptions(And(Element(n, CC), Element(phi, CC), Element(m, CC))))
F ( ϕ + k π , m ) = F ( ϕ , m ) + 2 k K ( m ) F\!\left(\phi + k \pi, m\right) = F\!\left(\phi, m\right) + 2 k K(m) F ( ϕ + k π , m ) = F ( ϕ , m ) + 2 k K ( m )
Assumptions: ϕ ∈ C and m ∈ C and k ∈ Z and m ≠ 1 \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \ne 1 ϕ ∈ C a n d m ∈ C a n d k ∈ Z a n d m = 1
TeX:
F\!\left(\phi + k \pi, m\right) = F\!\left(\phi, m\right) + 2 k K(m)
\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \ne 1 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticF F ( ϕ , m ) F\!\left(\phi, m\right) F ( ϕ , m )
Legendre incomplete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers ZZ Z \mathbb{Z} Z
Integers
Source code for this entry:
Entry(ID("685126"),
Formula(Equal(IncompleteEllipticF(Add(phi, Mul(k, Pi)), m), Add(IncompleteEllipticF(phi, m), Mul(Mul(2, k), EllipticK(m))))),
Variables(phi, m, k),
Assumptions(And(Element(phi, CC), Element(m, CC), Element(k, ZZ), NotEqual(m, 1))))
E ( ϕ + k π , m ) = E ( ϕ , m ) + 2 k E ( m ) E\!\left(\phi + k \pi, m\right) = E\!\left(\phi, m\right) + 2 k E(m) E ( ϕ + k π , m ) = E ( ϕ , m ) + 2 k E ( m )
Assumptions: ϕ ∈ C and m ∈ C and k ∈ Z \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} ϕ ∈ C a n d m ∈ C a n d k ∈ Z
TeX:
E\!\left(\phi + k \pi, m\right) = E\!\left(\phi, m\right) + 2 k E(m)
\phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} Definitions:
Fungrim symbol Notation Short description IncompleteEllipticE E ( ϕ , m ) E\!\left(\phi, m\right) E ( ϕ , m )
Legendre incomplete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers ZZ Z \mathbb{Z} Z
Integers
Source code for this entry:
Entry(ID("c28288"),
Formula(Equal(IncompleteEllipticE(Add(phi, Mul(k, Pi)), m), Add(IncompleteEllipticE(phi, m), Mul(Mul(2, k), EllipticE(m))))),
Variables(phi, m, k),
Assumptions(And(Element(phi, CC), Element(m, CC), Element(k, ZZ))))
Π ( n , ϕ + k π , m ) = Π ( n , ϕ , m ) + 2 k Π ( n , m ) \Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right) Π ( n , ϕ + k π , m ) = Π ( n , ϕ , m ) + 2 k Π ( n , m )
Assumptions: n ∈ C and ϕ ∈ C and m ∈ C and k ∈ Z and n ≠ 1 and m ≠ 1 n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 1 \;\mathbin{\operatorname{and}}\; m \ne 1 n ∈ C a n d ϕ ∈ C a n d m ∈ C a n d k ∈ Z a n d n = 1 a n d m = 1
TeX:
\Pi\!\left(n, \phi + k \pi, m\right) = \Pi\!\left(n, \phi, m\right) + 2 k \Pi\!\left(n, m\right)
n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 1 \;\mathbin{\operatorname{and}}\; m \ne 1 Definitions:
Fungrim symbol Notation Short description IncompleteEllipticPi Π ( n , ϕ , m ) \Pi\!\left(n, \phi, m\right) Π ( n , ϕ , m )
Legendre incomplete elliptic integral of the third kind Pi π \pi π
The constant pi (3.14...) EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind CC C \mathbb{C} C
Complex numbers ZZ Z \mathbb{Z} Z
Integers
Source code for this entry:
Entry(ID("5f84d9"),
Formula(Equal(IncompleteEllipticPi(n, Add(phi, Mul(k, Pi)), m), Add(IncompleteEllipticPi(n, phi, m), Mul(Mul(2, k), EllipticPi(n, m))))),
Variables(n, phi, m, k),
Assumptions(And(Element(n, CC), Element(phi, CC), Element(m, CC), Element(k, ZZ), NotEqual(n, 1), NotEqual(m, 1))))
K ( m ) = π 2 2 F 1 ( 1 2 , 1 2 , 1 , m ) K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, m\right) K ( m ) = 2 π 2 F 1 ( 2 1 , 2 1 , 1 , m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, m\right)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Hypergeometric2F1 2 F 1 ( a , b , c , z ) \,{}_2F_1\!\left(a, b, c, z\right) 2 F 1 ( a , b , c , z )
Gauss hypergeometric function CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("b760d1"),
Formula(Equal(EllipticK(m), Mul(Div(Pi, 2), Hypergeometric2F1(Div(1, 2), Div(1, 2), 1, m)))),
Variables(m),
Assumptions(Element(m, CC)))
E ( m ) = π 2 2 F 1 ( − 1 2 , 1 2 , 1 , m ) E(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}, 1, m\right) E ( m ) = 2 π 2 F 1 ( − 2 1 , 2 1 , 1 , m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
E(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}, 1, m\right)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind Pi π \pi π
The constant pi (3.14...) Hypergeometric2F1 2 F 1 ( a , b , c , z ) \,{}_2F_1\!\left(a, b, c, z\right) 2 F 1 ( a , b , c , z )
Gauss hypergeometric function CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("16d2e1"),
Formula(Equal(EllipticE(m), Mul(Div(Pi, 2), Hypergeometric2F1(Neg(Div(1, 2)), Div(1, 2), 1, m)))),
Variables(m),
Assumptions(Element(m, CC)))
2 E ( m ) − K ( m ) = π 2 2 F 1 ( − 1 2 , 3 2 , 1 , m ) 2 E(m) - K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{3}{2}, 1, m\right) 2 E ( m ) − K ( m ) = 2 π 2 F 1 ( − 2 1 , 2 3 , 1 , m )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
2 E(m) - K(m) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{3}{2}, 1, m\right)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) Hypergeometric2F1 2 F 1 ( a , b , c , z ) \,{}_2F_1\!\left(a, b, c, z\right) 2 F 1 ( a , b , c , z )
Gauss hypergeometric function CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("752619"),
Formula(Equal(Sub(Mul(2, EllipticE(m)), EllipticK(m)), Mul(Div(Pi, 2), Hypergeometric2F1(Neg(Div(1, 2)), Div(3, 2), 1, m)))),
Variables(m),
Assumptions(Element(m, CC)))
K ( m ) = π 2 agm ( 1 , 1 − m ) K(m) = \frac{\pi}{2 \operatorname{agm}\!\left(1, \sqrt{1 - m}\right)} K ( m ) = 2 a g m ( 1 , 1 − m ) π
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
K(m) = \frac{\pi}{2 \operatorname{agm}\!\left(1, \sqrt{1 - m}\right)}
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind Pi π \pi π
The constant pi (3.14...) AGM agm ( a , b ) \operatorname{agm}\!\left(a, b\right) a g m ( a , b )
Arithmetic-geometric mean Sqrt z \sqrt{z} z
Principal square root CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("e15f43"),
Formula(Equal(EllipticK(m), Div(Pi, Mul(2, AGM(1, Sqrt(Sub(1, m))))))),
Variables(m),
Assumptions(Element(m, CC)))
K ( m ) = R F ( 0 , 1 − m , 1 ) K(m) = R_F\!\left(0, 1 - m, 1\right) K ( m ) = R F ( 0 , 1 − m , 1 )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
K(m) = R_F\!\left(0, 1 - m, 1\right)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticK K ( m ) K(m) K ( m )
Legendre complete elliptic integral of the first kind CarlsonRF R F ( x , y , z ) R_F\!\left(x, y, z\right) R F ( x , y , z )
Carlson symmetric elliptic integral of the first kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("0cc11f"),
Formula(Equal(EllipticK(m), CarlsonRF(0, Sub(1, m), 1))),
Variables(m),
Assumptions(Element(m, CC)))
E ( m ) = 2 R G ( 0 , 1 − m , 1 ) E(m) = 2 R_G\!\left(0, 1 - m, 1\right) E ( m ) = 2 R G ( 0 , 1 − m , 1 )
Assumptions: m ∈ C m \in \mathbb{C} m ∈ C
TeX:
E(m) = 2 R_G\!\left(0, 1 - m, 1\right)
m \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description EllipticE E ( m ) E(m) E ( m )
Legendre complete elliptic integral of the second kind CarlsonRG R G ( x , y , z ) R_G\!\left(x, y, z\right) R G ( x , y , z )
Carlson symmetric elliptic integral of the second kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("6520e7"),
Formula(Equal(EllipticE(m), Mul(2, CarlsonRG(0, Sub(1, m), 1)))),
Variables(m),
Assumptions(Element(m, CC)))
Π ( n , m ) = R F ( 0 , 1 − m , 1 ) + n 3 R J ( 0 , 1 − m , 1 , 1 − n ) \Pi\!\left(n, m\right) = R_F\!\left(0, 1 - m, 1\right) + \frac{n}{3} R_J\!\left(0, 1 - m, 1, 1 - n\right) Π ( n , m ) = R F ( 0 , 1 − m , 1 ) + 3 n R J ( 0 , 1 − m , 1 , 1 − n )
Assumptions: n ∈ C and m ∈ C and m ≠ 1 n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \ne 1 n ∈ C a n d m ∈ C a n d m = 1
TeX:
\Pi\!\left(n, m\right) = R_F\!\left(0, 1 - m, 1\right) + \frac{n}{3} R_J\!\left(0, 1 - m, 1, 1 - n\right)
n \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \ne 1 Definitions:
Fungrim symbol Notation Short description EllipticPi Π ( n , m ) \Pi\!\left(n, m\right) Π ( n , m )
Legendre complete elliptic integral of the third kind CarlsonRF R F ( x , y , z ) R_F\!\left(x, y, z\right) R F ( x , y , z )
Carlson symmetric elliptic integral of the first kind CarlsonRJ R J ( x , y , z , w ) R_J\!\left(x, y, z, w\right) R J ( x , y , z , w )
Carlson symmetric elliptic integral of the third kind CC C \mathbb{C} C
Complex numbers
Source code for this entry:
Entry(ID("9ccaef"),
Formula(Equal(EllipticPi(n, m), Add(CarlsonRF(0, Sub(1, m), 1), Mul(Div(n, 3), CarlsonRJ(0, Sub(1, m), 1, Sub(1, n)))))),
Variables(n, m),
Assumptions(And(Element(n, CC), Element(m, CC), NotEqual(m, 1))))