Table of contents: Definitions - Illustrations - Integral representations - Connection formulas - Symmetry - Scale invariance - Domain - Representation of other functions - Specific values - Functional equations - Derivatives and differential equations - Series representations - Bounds and inequalities
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Entry(ID("5cd377"), SymbolDefinition(CarlsonRF, CarlsonRF(x, y, z), "Carlson symmetric elliptic integral of the first kind"))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Entry(ID("8f7c2a"), SymbolDefinition(CarlsonRG, CarlsonRG(x, y, z), "Carlson symmetric elliptic integral of the second kind"))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
Entry(ID("bac745"), SymbolDefinition(CarlsonRJ, CarlsonRJ(x, y, z, w), "Carlson symmetric elliptic integral of the third kind"))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Entry(ID("663d75"), SymbolDefinition(CarlsonRD, CarlsonRD(x, y, z), "Degenerate Carlson symmetric elliptic integral of the third kind"))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
Entry(ID("132ec5"), SymbolDefinition(CarlsonRC, CarlsonRC(x, y), "Degenerate Carlson symmetric elliptic integral of the first kind"))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
ClosedInterval | [a,b] | Closed interval |
Entry(ID("b0921b"), Image(Description("Plot of", CarlsonRF(x, y, 1), "on", Element(x, ClosedInterval(0, 4)), "for", Element(y, Set(Decimal("0.1"), 1, 10))), ImageSource("plot_carlson_rf")))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
ClosedInterval | [a,b] | Closed interval |
Entry(ID("6ae152"), Image(Description("Plot of", CarlsonRG(x, y, 1), "on", Element(x, ClosedInterval(0, 4)), "for", Element(y, Set(Decimal("0.1"), 1, 10))), ImageSource("plot_carlson_rg")))
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
ClosedInterval | [a,b] | Closed interval |
ConstI | i | Imaginary unit |
Im | Im(z) | Imaginary part |
Re | Re(z) | Real part |
Abs | ∣z∣ | Absolute value |
Entry(ID("cc234c"), Image(Description("X-ray of", CarlsonRF(z, 1, 2), "on", Element(z, Add(ClosedInterval(-4, 4), Mul(ClosedInterval(-4, 4), ConstI)))), ImageSource("xray_carlson_rf")), Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))
R_F\!\left(x, y, z\right) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Infinity | ∞ | Positive infinity |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Entry(ID("9357b9"), Formula(Equal(CarlsonRF(x, y, z), Mul(Div(1, 2), Integral(Div(1, Mul(Mul(Sqrt(Add(t, x)), Sqrt(Add(t, y))), Sqrt(Add(t, z)))), For(t, 0, Infinity))))), Variables(x, y, z), Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
R_G\!\left(x, y, z\right) = \frac{1}{4} \int_{0}^{\infty} \frac{t}{\sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) \, dt x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Infinity | ∞ | Positive infinity |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Entry(ID("dab889"), Formula(Equal(CarlsonRG(x, y, z), Mul(Div(1, 4), Integral(Mul(Div(t, Mul(Mul(Sqrt(Add(t, x)), Sqrt(Add(t, y))), Sqrt(Add(t, z)))), Add(Add(Div(x, Add(t, x)), Div(y, Add(t, y))), Div(z, Add(t, z)))), For(t, 0, Infinity))))), Variables(x, y, z), Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenInterval(Neg(Infinity), 0))))))
R_J\!\left(x, y, z, w\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + w\right) \sqrt{t + x} \sqrt{t + y} \sqrt{t + z}} \, dt x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Infinity | ∞ | Positive infinity |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
OpenClosedInterval | (a,b] | Open-closed interval |
Entry(ID("02a8d7"), Formula(Equal(CarlsonRJ(x, y, z, w), Mul(Div(3, 2), Integral(Div(1, Mul(Mul(Mul(Add(t, w), Sqrt(Add(t, x))), Sqrt(Add(t, y))), Sqrt(Add(t, z)))), For(t, 0, Infinity))))), Variables(x, y, z, w), Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(w, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
R_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Integral | ∫abf(x)dx | Integral |
Pow | ab | Power |
Infinity | ∞ | Positive infinity |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
OpenClosedInterval | (a,b] | Open-closed interval |
Entry(ID("944a14"), Formula(Equal(CarlsonRD(x, y, z), Mul(Div(3, 2), Integral(Div(1, Mul(Mul(Add(t, x), Add(t, y)), Pow(Add(t, z), Div(3, 2)))), For(t, 0, Infinity))))), Variables(x, y, z), Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Or(NotEqual(x, 0), NotEqual(y, 0)))))
R_C\!\left(x, y\right) = \frac{1}{2} \int_{0}^{\infty} \frac{1}{\left(t + y\right) \sqrt{t + x}} \, dt x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right]
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Infinity | ∞ | Positive infinity |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
OpenClosedInterval | (a,b] | Open-closed interval |
Entry(ID("f3b8dc"), Formula(Equal(CarlsonRC(x, y), Mul(Div(1, 2), Integral(Div(1, Mul(Add(t, y), Sqrt(Add(t, x)))), For(t, 0, Infinity))))), Variables(x, y), Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))
R_F\!\left(0, y, z\right) = \int_{0}^{\pi / 2} \frac{1}{\sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)}} \, d\theta y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Pow | ab | Power |
Cos | cos(z) | Cosine |
Sin | sin(z) | Sine |
Pi | π | The constant pi (3.14...) |
CC | C | Complex numbers |
Re | Re(z) | Real part |
Entry(ID("da16db"), Formula(Equal(CarlsonRF(0, y, z), Integral(Div(1, Sqrt(Add(Mul(y, Pow(Cos(theta), 2)), Mul(z, Pow(Sin(theta), 2))))), For(theta, 0, Div(Pi, 2))))), Variables(y, z), Assumptions(And(Element(y, CC), Element(z, CC), Greater(Re(y), 0), Greater(Re(z), 0))))
R_G\!\left(0, y, z\right) = \frac{1}{2} \int_{0}^{\pi / 2} \sqrt{y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)} \, d\theta y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Pow | ab | Power |
Cos | cos(z) | Cosine |
Sin | sin(z) | Sine |
Pi | π | The constant pi (3.14...) |
CC | C | Complex numbers |
Re | Re(z) | Real part |
Entry(ID("7fbbe8"), Formula(Equal(CarlsonRG(0, y, z), Mul(Div(1, 2), Integral(Sqrt(Add(Mul(y, Pow(Cos(theta), 2)), Mul(z, Pow(Sin(theta), 2)))), For(theta, 0, Div(Pi, 2)))))), Variables(y, z), Assumptions(And(Element(y, CC), Element(z, CC), GreaterEqual(Re(y), 0), GreaterEqual(Re(z), 0))))
R_D\!\left(0, y, z\right) = 3 \int_{0}^{\pi / 2} \frac{\sin^{2}\!\left(\theta\right)}{{\left(y \cos^{2}\!\left(\theta\right) + z \sin^{2}\!\left(\theta\right)\right)}^{3 / 2}} \, d\theta y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Integral | ∫abf(x)dx | Integral |
Pow | ab | Power |
Sin | sin(z) | Sine |
Cos | cos(z) | Cosine |
Pi | π | The constant pi (3.14...) |
CC | C | Complex numbers |
Re | Re(z) | Real part |
Entry(ID("9a0bc8"), Formula(Equal(CarlsonRD(0, y, z), Mul(3, Integral(Div(Pow(Sin(theta), 2), Pow(Add(Mul(y, Pow(Cos(theta), 2)), Mul(z, Pow(Sin(theta), 2))), Div(3, 2))), For(theta, 0, Div(Pi, 2)))))), Variables(y, z), Assumptions(And(Element(y, CC), Element(z, CC), Greater(Re(y), 0), Greater(Re(z), 0))))
R_F\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{\sin(\theta)}{\sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)}} \, d\theta \, d\phi x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Pi | π | The constant pi (3.14...) |
Integral | ∫abf(x)dx | Integral |
Sin | sin(z) | Sine |
Sqrt | z | Principal square root |
Pow | ab | Power |
Cos | cos(z) | Cosine |
CC | C | Complex numbers |
Re | Re(z) | Real part |
Entry(ID("8f0a91"), Formula(Equal(CarlsonRF(x, y, z), Mul(Div(1, Mul(4, Pi)), Integral(Integral(Div(Sin(theta), Sqrt(Add(Add(Mul(Mul(x, Pow(Sin(theta), 2)), Pow(Cos(phi), 2)), Mul(Mul(y, Pow(Sin(theta), 2)), Pow(Sin(phi), 2))), Mul(z, Pow(Cos(theta), 2))))), For(theta, 0, Pi)), For(phi, 0, Mul(2, Pi)))))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), GreaterEqual(Re(x), 0), GreaterEqual(Re(y), 0), GreaterEqual(Re(z), 0), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
R_G\!\left(x, y, z\right) = \frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \sqrt{x \sin^{2}\!\left(\theta\right) \cos^{2}\!\left(\phi\right) + y \sin^{2}\!\left(\theta\right) \sin^{2}\!\left(\phi\right) + z \cos^{2}\!\left(\theta\right)} \sin(\theta) \, d\theta \, d\phi x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(x) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(y) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Pi | π | The constant pi (3.14...) |
Integral | ∫abf(x)dx | Integral |
Sqrt | z | Principal square root |
Pow | ab | Power |
Sin | sin(z) | Sine |
Cos | cos(z) | Cosine |
CC | C | Complex numbers |
Re | Re(z) | Real part |
Entry(ID("0d8639"), Formula(Equal(CarlsonRG(x, y, z), Mul(Div(1, Mul(4, Pi)), Integral(Integral(Mul(Sqrt(Add(Add(Mul(Mul(x, Pow(Sin(theta), 2)), Pow(Cos(phi), 2)), Mul(Mul(y, Pow(Sin(theta), 2)), Pow(Sin(phi), 2))), Mul(z, Pow(Cos(theta), 2)))), Sin(theta)), For(theta, 0, Pi)), For(phi, 0, Mul(2, Pi)))))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), GreaterEqual(Re(x), 0), GreaterEqual(Re(y), 0), GreaterEqual(Re(z), 0))))
R_C\!\left(x, y\right) = R_F\!\left(x, y, y\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
CC | C | Complex numbers |
Entry(ID("61f98d"), Formula(Equal(CarlsonRC(x, y), CarlsonRF(x, y, y))), Variables(x, y), Assumptions(And(Element(x, CC), Element(y, CC))))
R_D\!\left(x, y, z\right) = R_J\!\left(x, y, z, z\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Entry(ID("409873"), Formula(Equal(CarlsonRD(x, y, z), CarlsonRJ(x, y, z, z))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))
2 R_G\!\left(x, y, z\right) = z R_F\!\left(x, y, z\right) - \frac{\left(x - z\right) \left(y - z\right)}{3} R_D\!\left(x, y, z\right) + \frac{\sqrt{x} \sqrt{y}}{\sqrt{z}} x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right) \;\mathbin{\operatorname{and}}\; z \ne 0
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Sqrt | z | Principal square root |
CC | C | Complex numbers |
Entry(ID("7609c8"), Formula(Equal(Mul(2, CarlsonRG(x, y, z)), Add(Sub(Mul(z, CarlsonRF(x, y, z)), Mul(Div(Mul(Sub(x, z), Sub(y, z)), 3), CarlsonRD(x, y, z))), Div(Mul(Sqrt(x), Sqrt(y)), Sqrt(z))))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(NotEqual(x, 0), NotEqual(y, 0)), NotEqual(z, 0))))
R_F\!\left(x, y, z\right) = R_F\!\left(x, z, y\right) = R_F\!\left(y, x, z\right) = R_F\!\left(y, z, x\right) = R_F\!\left(z, x, y\right) = R_F\!\left(z, y, x\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
CC | C | Complex numbers |
Entry(ID("f29729"), Formula(Equal(CarlsonRF(x, y, z), CarlsonRF(x, z, y), CarlsonRF(y, x, z), CarlsonRF(y, z, x), CarlsonRF(z, x, y), CarlsonRF(z, y, x))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))
R_G\!\left(x, y, z\right) = R_G\!\left(x, z, y\right) = R_G\!\left(y, x, z\right) = R_G\!\left(y, z, x\right) = R_G\!\left(z, x, y\right) = R_G\!\left(z, y, x\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
CC | C | Complex numbers |
Entry(ID("b478a1"), Formula(Equal(CarlsonRG(x, y, z), CarlsonRG(x, z, y), CarlsonRG(y, x, z), CarlsonRG(y, z, x), CarlsonRG(z, x, y), CarlsonRG(z, y, x))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))
R_J\!\left(x, y, z, w\right) = R_J\!\left(x, z, y, w\right) = R_J\!\left(y, x, z, w\right) = R_J\!\left(y, z, x, w\right) = R_J\!\left(z, x, y, w\right) = R_J\!\left(z, y, x, w\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Entry(ID("655a2b"), Formula(Equal(CarlsonRJ(x, y, z, w), CarlsonRJ(x, z, y, w), CarlsonRJ(y, x, z, w), CarlsonRJ(y, z, x, w), CarlsonRJ(z, x, y, w), CarlsonRJ(z, y, x, w))), Variables(x, y, z, w), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, CC))))
R_D\!\left(x, y, z\right) = R_D\!\left(y, x, z\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Entry(ID("1e8061"), Formula(Equal(CarlsonRD(x, y, z), CarlsonRD(y, x, z))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))
R_F\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-1 / 2} R_F\!\left(x, y, z\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Pow | ab | Power |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Infinity | ∞ | Positive infinity |
Entry(ID("7a168a"), Formula(Equal(CarlsonRF(Mul(lamda, x), Mul(lamda, y), Mul(lamda, z)), Mul(Pow(lamda, Neg(Div(1, 2))), CarlsonRF(x, y, z)))), Variables(x, y, z, lamda), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(lamda, OpenInterval(0, Infinity)))))
R_G\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{1 / 2} R_G\!\left(x, y, z\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Pow | ab | Power |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Infinity | ∞ | Positive infinity |
Entry(ID("f9ca94"), Formula(Equal(CarlsonRG(Mul(lamda, x), Mul(lamda, y), Mul(lamda, z)), Mul(Pow(lamda, Div(1, 2)), CarlsonRG(x, y, z)))), Variables(x, y, z, lamda), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(lamda, OpenInterval(0, Infinity)))))
R_J\!\left(\lambda x, \lambda y, \lambda z, \lambda w\right) = {\lambda}^{-3 / 2} R_J\!\left(x, y, z, w\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
Pow | ab | Power |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Infinity | ∞ | Positive infinity |
Entry(ID("4e21c7"), Formula(Equal(CarlsonRJ(Mul(lamda, x), Mul(lamda, y), Mul(lamda, z), Mul(lamda, w)), Mul(Pow(lamda, Neg(Div(3, 2))), CarlsonRJ(x, y, z, w)))), Variables(x, y, z, w, lamda), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, CC), Element(lamda, OpenInterval(0, Infinity)))))
R_D\!\left(\lambda x, \lambda y, \lambda z\right) = {\lambda}^{-3 / 2} R_D\!\left(x, y, z\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Pow | ab | Power |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Infinity | ∞ | Positive infinity |
Entry(ID("197a91"), Formula(Equal(CarlsonRD(Mul(lamda, x), Mul(lamda, y), Mul(lamda, z)), Mul(Pow(lamda, Neg(Div(3, 2))), CarlsonRD(x, y, z)))), Variables(x, y, z, lamda), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(lamda, OpenInterval(0, Infinity)))))
R_C\!\left(\lambda x, \lambda y\right) = {\lambda}^{-1 / 2} R_C\!\left(x, y\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \lambda \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
Pow | ab | Power |
CC | C | Complex numbers |
OpenInterval | (a,b) | Open interval |
Infinity | ∞ | Positive infinity |
Entry(ID("a839d5"), Formula(Equal(CarlsonRC(Mul(lamda, x), Mul(lamda, y)), Mul(Pow(lamda, Neg(Div(1, 2))), CarlsonRC(x, y)))), Variables(x, y, lamda), Assumptions(And(Element(x, CC), Element(y, CC), Element(lamda, OpenInterval(0, Infinity)))))
\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_F\!\left(x, y, z\right) \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CC | C | Complex numbers |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Entry(ID("655f6b"), Formula(Implies(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0)))), Element(CarlsonRF(x, y, z), CC))), Variables(x, y, z))
\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}\right) \;\implies\; R_G\!\left(x, y, z\right) \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CC | C | Complex numbers |
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Entry(ID("c90834"), Formula(Implies(And(Element(x, CC), Element(y, CC), Element(z, CC)), Element(CarlsonRG(x, y, z), CC))), Variables(x, y, z))
\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_J\!\left(x, y, z, w\right) \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CC | C | Complex numbers |
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
Entry(ID("8bac89"), Formula(Implies(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, SetMinus(CC, Set(0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0)))), Element(CarlsonRJ(x, y, z, w), CC))), Variables(x, y, z, w))
\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)\right) \;\implies\; R_D\!\left(x, y, z\right) \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CC | C | Complex numbers |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Entry(ID("ba7b32"), Formula(Implies(And(Element(x, CC), Element(y, CC), Element(z, SetMinus(CC, Set(0))), Or(NotEqual(x, 0), NotEqual(y, 0))), Element(CarlsonRD(x, y, z), CC))), Variables(x, y, z))
\left(x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\}\right) \;\implies\; R_C\!\left(x, y\right) \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CC | C | Complex numbers |
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
Entry(ID("7aa9be"), Formula(Implies(And(Element(x, CC), Element(y, SetMinus(CC, Set(0)))), Element(CarlsonRC(x, y), CC))), Variables(x, y))
\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_F\!\left(x, y, z\right) \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
ClosedOpenInterval | [a,b) | Closed-open interval |
Infinity | ∞ | Positive infinity |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
OpenInterval | (a,b) | Open interval |
Entry(ID("cc4cd8"), Formula(Implies(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, ClosedOpenInterval(0, Infinity)), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0)))), Element(CarlsonRF(x, y, z), OpenInterval(0, Infinity)))), Variables(x, y, z))
\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right)\right) \;\implies\; R_G\!\left(x, y, z\right) \in \left[0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
ClosedOpenInterval | [a,b) | Closed-open interval |
Infinity | ∞ | Positive infinity |
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Entry(ID("9c9173"), Formula(Implies(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, ClosedOpenInterval(0, Infinity))), Element(CarlsonRG(x, y, z), ClosedOpenInterval(0, Infinity)))), Variables(x, y, z))
\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; w \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)\right) \;\implies\; R_J\!\left(x, y, z, w\right) \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
ClosedOpenInterval | [a,b) | Closed-open interval |
Infinity | ∞ | Positive infinity |
OpenInterval | (a,b) | Open interval |
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
Entry(ID("671fcb"), Formula(Implies(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, ClosedOpenInterval(0, Infinity)), Element(w, OpenInterval(0, Infinity)), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0)))), Element(CarlsonRJ(x, y, z, w), OpenInterval(0, Infinity)))), Variables(x, y, z, w))
\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)\right) \;\implies\; R_D\!\left(x, y, z\right) \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
ClosedOpenInterval | [a,b) | Closed-open interval |
Infinity | ∞ | Positive infinity |
OpenInterval | (a,b) | Open interval |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Entry(ID("8236ff"), Formula(Implies(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, OpenInterval(0, Infinity)), Or(NotEqual(x, 0), NotEqual(y, 0))), Element(CarlsonRD(x, y, z), OpenInterval(0, Infinity)))), Variables(x, y, z))
\left(x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left(0, \infty\right)\right) \;\implies\; R_C\!\left(x, y\right) \in \left(0, \infty\right)
Fungrim symbol | Notation | Short description |
---|---|---|
ClosedOpenInterval | [a,b) | Closed-open interval |
Infinity | ∞ | Positive infinity |
OpenInterval | (a,b) | Open interval |
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
Entry(ID("da33ce"), Formula(Implies(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, OpenInterval(0, Infinity))), Element(CarlsonRC(x, y), OpenInterval(0, Infinity)))), Variables(x, y))
f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_F\!\left(\alpha, y, z\right), \alpha \mapsto R_F\!\left(x, \alpha, z\right), \alpha \mapsto R_F\!\left(x, y, \alpha\right)\right\} x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
IsHolomorphic | f(z) is holomorphic at z=c | Holomorphic predicate |
CC | C | Complex numbers |
OpenClosedInterval | (a,b] | Open-closed interval |
Infinity | ∞ | Positive infinity |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Entry(ID("0ba30f"), Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRF(alpha, y, z)), Fun(alpha, CarlsonRF(x, alpha, z)), Fun(alpha, CarlsonRF(x, y, alpha)))))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_G\!\left(\alpha, y, z\right), \alpha \mapsto R_G\!\left(x, \alpha, z\right), \alpha \mapsto R_G\!\left(x, y, \alpha\right)\right\} x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
IsHolomorphic | f(z) is holomorphic at z=c | Holomorphic predicate |
CC | C | Complex numbers |
OpenClosedInterval | (a,b] | Open-closed interval |
Infinity | ∞ | Positive infinity |
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
Entry(ID("c56825"), Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRG(alpha, y, z)), Fun(alpha, CarlsonRG(x, alpha, z)), Fun(alpha, CarlsonRG(x, y, alpha)))))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))
f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_J\!\left(\alpha, y, z, w\right), \alpha \mapsto R_J\!\left(x, \alpha, z, w\right), \alpha \mapsto R_J\!\left(x, y, \alpha, w\right), \alpha \mapsto R_J\!\left(x, y, w, \alpha\right)\right\} x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
IsHolomorphic | f(z) is holomorphic at z=c | Holomorphic predicate |
CC | C | Complex numbers |
OpenClosedInterval | (a,b] | Open-closed interval |
Infinity | ∞ | Positive infinity |
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
Entry(ID("583c27"), Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRJ(alpha, y, z, w)), Fun(alpha, CarlsonRJ(x, alpha, z, w)), Fun(alpha, CarlsonRJ(x, y, alpha, w)), Fun(alpha, CarlsonRJ(x, y, w, alpha)))))), Variables(x, y, z, w), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, SetMinus(CC, Set(0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_D\!\left(\alpha, y, z\right), \alpha \mapsto R_D\!\left(x, \alpha, z\right), \alpha \mapsto R_D\!\left(x, y, \alpha\right)\right\} x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)
Fungrim symbol | Notation | Short description |
---|---|---|
IsHolomorphic | f(z) is holomorphic at z=c | Holomorphic predicate |
CC | C | Complex numbers |
OpenClosedInterval | (a,b] | Open-closed interval |
Infinity | ∞ | Positive infinity |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
Entry(ID("114f9e"), Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRD(alpha, y, z)), Fun(alpha, CarlsonRD(x, alpha, z)), Fun(alpha, CarlsonRD(x, y, alpha)))))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, SetMinus(CC, Set(0))), Or(NotEqual(x, 0), NotEqual(y, 0)))))
f(\alpha) \text{ is holomorphic on } \alpha \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\text{ for all } f \in \left\{\alpha \mapsto R_C\!\left(\alpha, y\right), \alpha \mapsto R_C\!\left(x, \alpha\right)\right\} x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\}
Fungrim symbol | Notation | Short description |
---|---|---|
IsHolomorphic | f(z) is holomorphic at z=c | Holomorphic predicate |
CC | C | Complex numbers |
OpenClosedInterval | (a,b] | Open-closed interval |
Infinity | ∞ | Positive infinity |
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
Entry(ID("73cf98"), Formula(All(IsHolomorphic(f(alpha), ForElement(alpha, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))), ForElement(f, Set(Fun(alpha, CarlsonRC(alpha, y)), Fun(alpha, CarlsonRC(x, alpha)))))), Variables(x, y), Assumptions(And(Element(x, CC), Element(y, SetMinus(CC, Set(0))))))
R_F\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_F\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
RightLimit | limx→a+f(x) | Limiting value, from the right |
ConstI | i | Imaginary unit |
CC | C | Complex numbers |
Entry(ID("0d3186"), Formula(Equal(CarlsonRF(x, y, z), RightLimit(CarlsonRF(Add(x, Mul(epsilon, ConstI)), Add(y, Mul(epsilon, ConstI)), Add(z, Mul(epsilon, ConstI))), For(epsilon, 0)))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
R_G\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_G\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
RightLimit | limx→a+f(x) | Limiting value, from the right |
ConstI | i | Imaginary unit |
CC | C | Complex numbers |
Entry(ID("f9b773"), Formula(Equal(CarlsonRG(x, y, z), RightLimit(CarlsonRG(Add(x, Mul(epsilon, ConstI)), Add(y, Mul(epsilon, ConstI)), Add(z, Mul(epsilon, ConstI))), For(epsilon, 0)))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC))))
R_J\!\left(x, y, z, w\right) = \lim_{\varepsilon \to {0}^{+}} R_J\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i, w + \varepsilon i\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(\left(x \ne 0 \;\mathbin{\operatorname{and}}\; y \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(x \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right) \;\mathbin{\operatorname{or}}\; \left(y \ne 0 \;\mathbin{\operatorname{and}}\; z \ne 0\right)\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
RightLimit | limx→a+f(x) | Limiting value, from the right |
ConstI | i | Imaginary unit |
CC | C | Complex numbers |
Entry(ID("b8ca70"), Formula(Equal(CarlsonRJ(x, y, z, w), RightLimit(CarlsonRJ(Add(x, Mul(epsilon, ConstI)), Add(y, Mul(epsilon, ConstI)), Add(z, Mul(epsilon, ConstI)), Add(w, Mul(epsilon, ConstI))), For(epsilon, 0)))), Variables(x, y, z, w), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, CC), Element(w, SetMinus(CC, Set(0))), Or(And(NotEqual(x, 0), NotEqual(y, 0)), And(NotEqual(x, 0), NotEqual(z, 0)), And(NotEqual(y, 0), NotEqual(z, 0))))))
R_D\!\left(x, y, z\right) = \lim_{\varepsilon \to {0}^{+}} R_D\!\left(x + \varepsilon i, y + \varepsilon i, z + \varepsilon i\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
RightLimit | limx→a+f(x) | Limiting value, from the right |
ConstI | i | Imaginary unit |
CC | C | Complex numbers |
Entry(ID("9673f7"), Formula(Equal(CarlsonRD(x, y, z), RightLimit(CarlsonRD(Add(x, Mul(epsilon, ConstI)), Add(y, Mul(epsilon, ConstI)), Add(z, Mul(epsilon, ConstI))), For(epsilon, 0)))), Variables(x, y, z), Assumptions(And(Element(x, CC), Element(y, CC), Element(z, SetMinus(CC, Set(0))), Or(NotEqual(x, 0), NotEqual(y, 0)))))
R_C\!\left(x, y\right) = \lim_{\varepsilon \to {0}^{+}} R_C\!\left(x + \varepsilon i, y + \varepsilon i\right) x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\}
Fungrim symbol | Notation | Short description |
---|---|---|
CarlsonRC | RC(x,y) | Degenerate Carlson symmetric elliptic integral of the first kind |
RightLimit | limx→a+f(x) | Limiting value, from the right |
ConstI | i | Imaginary unit |
CC | C | Complex numbers |
Entry(ID("6923d5"), Formula(Equal(CarlsonRC(x, y), RightLimit(CarlsonRC(Add(x, Mul(epsilon, ConstI)), Add(y, Mul(epsilon, ConstI))), For(epsilon, 0)))), Variables(x, y), Assumptions(And(Element(x, CC), Element(y, SetMinus(CC, Set(0))))))
Related topics: Legendre elliptic integrals
K(m) = R_F\!\left(0, 1 - m, 1\right) m \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
EllipticK | K(m) | Legendre complete elliptic integral of the first kind |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
CC | C | Complex numbers |
Entry(ID("0cc11f"), Formula(Equal(EllipticK(m), CarlsonRF(0, Sub(1, m), 1))), Variables(m), Assumptions(Element(m, CC)))
E(m) = 2 R_G\!\left(0, 1 - m, 1\right) m \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
EllipticE | E(m) | Legendre complete elliptic integral of the second kind |
CarlsonRG | RG(x,y,z) | Carlson symmetric elliptic integral of the second kind |
CC | C | Complex numbers |
Entry(ID("6520e7"), Formula(Equal(EllipticE(m), Mul(2, CarlsonRG(0, Sub(1, m), 1)))), Variables(m), Assumptions(Element(m, CC)))
\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
Fungrim symbol | Notation | Short description |
---|---|---|
EllipticPi | Π(n,m) | Legendre complete elliptic integral of the third kind |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
CarlsonRJ | RJ(x,y,z,w) | Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
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))))
E(m) = \frac{1 - m}{3} \left(R_D\!\left(0, 1 - m, 1\right) + R_D\!\left(0, 1, 1 - m\right)\right) m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \ne 1
Fungrim symbol | Notation | Short description |
---|---|---|
EllipticE | E(m) | Legendre complete elliptic integral of the second kind |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Entry(ID("41cf8e"), Formula(Equal(EllipticE(m), Mul(Div(Sub(1, m), 3), Add(CarlsonRD(0, Sub(1, m), 1), CarlsonRD(0, 1, Sub(1, m)))))), Variables(m), Assumptions(And(Element(m, CC), NotEqual(m, 1))))
K(m) - E(m) = \frac{m}{3} R_D\!\left(0, 1 - m, 1\right) m \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
EllipticK | K(m) | Legendre complete elliptic integral of the first kind |
EllipticE | E(m) | Legendre complete elliptic integral of the second kind |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Entry(ID("94f646"), Formula(Equal(Sub(EllipticK(m), EllipticE(m)), Mul(Div(m, 3), CarlsonRD(0, Sub(1, m), 1)))), Variables(m), Assumptions(Element(m, CC)))
E(m) - \left(1 - m\right) K(m) = \frac{m \left(1 - m\right)}{3} R_D\!\left(0, 1, 1 - m\right) m \in \mathbb{C}
Fungrim symbol | Notation | Short description |
---|---|---|
EllipticE | E(m) | Legendre complete elliptic integral of the second kind |
EllipticK | K(m) | Legendre complete elliptic integral of the first kind |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Entry(ID("55d23d"), Formula(Equal(Sub(EllipticE(m), Mul(Sub(1, m), EllipticK(m))), Mul(Div(Mul(m, Sub(1, m)), 3), CarlsonRD(0, 1, Sub(1, m))))), Variables(m), Assumptions(Element(m, CC)))
F\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2}
Fungrim symbol | Notation | Short description |
---|---|---|
IncompleteEllipticF | F(ϕ,m) | Legendre incomplete elliptic integral of the first kind |
Sin | sin(z) | Sine |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Pow | ab | Power |
Cos | cos(z) | Cosine |
CC | C | Complex numbers |
Pi | π | The constant pi (3.14...) |
Re | Re(z) | Real part |
Entry(ID("e2445d"), Formula(Equal(IncompleteEllipticF(phi, m), Mul(Sin(phi), CarlsonRF(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1)))), Variables(phi, m), Assumptions(And(Element(phi, CC), Element(m, CC), LessEqual(Div(Neg(Pi), 2), Re(phi), Div(Pi, 2)))))
E\!\left(\phi, m\right) = \sin(\phi) R_F\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) - \frac{1}{3} m \sin^{3}\!\left(\phi\right) R_D\!\left(\cos^{2}\!\left(\phi\right), 1 - m \sin^{2}\!\left(\phi\right), 1\right) \phi \in \mathbb{C} \;\mathbin{\operatorname{and}}\; m \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \frac{-\pi}{2} \le \operatorname{Re}(\phi) \le \frac{\pi}{2}
Fungrim symbol | Notation | Short description |
---|---|---|
IncompleteEllipticE | E(ϕ,m) | Legendre incomplete elliptic integral of the second kind |
Sin | sin(z) | Sine |
CarlsonRF | RF(x,y,z) | Carlson symmetric elliptic integral of the first kind |
Pow | ab | Power |
Cos | cos(z) | Cosine |
CarlsonRD | RD(x,y,z) | Degenerate Carlson symmetric elliptic integral of the third kind |
CC | C | Complex numbers |
Pi | π | The constant pi (3.14...) |
Re | Re(z) | Real part |
Entry(ID("f48f54"), Formula(Equal(IncompleteEllipticE(phi, m), Sub(Mul(Sin(phi), CarlsonRF(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1)), Mul(Mul(Mul(Div(1, 3), m), Pow(Sin(phi), 3)), CarlsonRD(Pow(Cos(phi), 2), Sub(1, Mul(m, Pow(Sin(phi), 2))), 1))))), Variables(phi, m), Assumptions(And(Element(phi, CC), Element(m, CC), LessEqual(Div(Neg(Pi), 2), Re(phi), Div(Pi, 2)))))