arg min* x ∈ ( 0 , ∞ ) Γ ( x ) ∈ [ 1.46163214496836234126265954233 ± 4.28 ⋅ 1 0 − 30 ] \mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right)} \Gamma(x) \in \left[1.46163214496836234126265954233 \pm 4.28 \cdot 10^{-30}\right] x ∈ ( 0 , ∞ ) a r g m i n * Γ ( x ) ∈ [ 1 . 4 6 1 6 3 2 1 4 4 9 6 8 3 6 2 3 4 1 2 6 2 6 5 9 5 4 2 3 3 ± 4 . 2 8 ⋅ 1 0 − 3 0 ] TeX:
\mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right)} \Gamma(x) \in \left[1.46163214496836234126265954233 \pm 4.28 \cdot 10^{-30}\right] Definitions:
Fungrim symbol Notation Short description ArgMinUnique arg min* x ∈ S f ( x ) \mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x) x ∈ S a r g m i n * f ( x ) Unique location of minimum value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("1bbbc7"),
Formula(Element(ArgMinUnique(Gamma(x), ForElement(x, OpenInterval(0, Infinity))), RealBall(Decimal("1.46163214496836234126265954233"), Decimal("4.28e-30"))))) min x ∈ ( 0 , ∞ ) Γ ( x ) ∈ [ 0.885603194410888700278815900583 ± 4.12 ⋅ 1 0 − 31 ] \mathop{\min}\limits_{x \in \left(0, \infty\right)} \Gamma(x) \in \left[0.885603194410888700278815900583 \pm 4.12 \cdot 10^{-31}\right] x ∈ ( 0 , ∞ ) min Γ ( x ) ∈ [ 0 . 8 8 5 6 0 3 1 9 4 4 1 0 8 8 8 7 0 0 2 7 8 8 1 5 9 0 0 5 8 3 ± 4 . 1 2 ⋅ 1 0 − 3 1 ] TeX:
\mathop{\min}\limits_{x \in \left(0, \infty\right)} \Gamma(x) \in \left[0.885603194410888700278815900583 \pm 4.12 \cdot 10^{-31}\right] Definitions:
Fungrim symbol Notation Short description Minimum min x ∈ S f ( x ) \mathop{\min}\limits_{x \in S} f(x) x ∈ S min f ( x ) Minimum value of a set or function Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("e010c9"),
Formula(Element(Minimum(Gamma(x), ForElement(x, OpenInterval(0, Infinity))), RealBall(Decimal("0.885603194410888700278815900583"), Decimal("4.12e-31"))))) min x ∈ ( 0 , ∞ ) log Γ ( x ) ∈ [ − 0.121486290535849608095514557178 ± 3.09 ⋅ 1 0 − 31 ] \mathop{\min}\limits_{x \in \left(0, \infty\right)} \log \Gamma(x) \in \left[-0.121486290535849608095514557178 \pm 3.09 \cdot 10^{-31}\right] x ∈ ( 0 , ∞ ) min log Γ ( x ) ∈ [ − 0 . 1 2 1 4 8 6 2 9 0 5 3 5 8 4 9 6 0 8 0 9 5 5 1 4 5 5 7 1 7 8 ± 3 . 0 9 ⋅ 1 0 − 3 1 ] TeX:
\mathop{\min}\limits_{x \in \left(0, \infty\right)} \log \Gamma(x) \in \left[-0.121486290535849608095514557178 \pm 3.09 \cdot 10^{-31}\right] Definitions:
Fungrim symbol Notation Short description Minimum min x ∈ S f ( x ) \mathop{\min}\limits_{x \in S} f(x) x ∈ S min f ( x ) Minimum value of a set or function LogGamma log Γ ( z ) \log \Gamma(z) log Γ ( z ) Logarithmic gamma function OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("b05f2b"),
Formula(Element(Minimum(LogGamma(x), ForElement(x, OpenInterval(0, Infinity))), RealBall(Decimal("-0.121486290535849608095514557178"), Decimal("3.09e-31"))))) Γ ( x ) > ( 2 π ) 1 / 2 x x − 1 / 2 e − x \Gamma(x) > {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} Γ ( x ) > ( 2 π ) 1 / 2 x x − 1 / 2 e − x Assumptions: x ∈ ( 0 , ∞ ) x \in \left(0, \infty\right) x ∈ ( 0 , ∞ ) TeX:
\Gamma(x) > {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x}
x \in \left(0, \infty\right) Definitions:
Fungrim symbol Notation Short description Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) Exp e z {e}^{z} e z Exponential function OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("2a47d7"),
Formula(Greater(Gamma(x), Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Pow(x, Sub(x, Div(1, 2)))), Exp(Neg(x))))),
Variables(x),
Assumptions(Element(x, OpenInterval(0, Infinity)))) Γ ( x ) < ( 2 π ) 1 / 2 x x − 1 / 2 e − x exp ( 1 12 x ) \Gamma(x) < {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right) Γ ( x ) < ( 2 π ) 1 / 2 x x − 1 / 2 e − x exp ( 1 2 x 1 ) Assumptions: x ∈ ( 0 , ∞ ) x \in \left(0, \infty\right) x ∈ ( 0 , ∞ ) TeX:
\Gamma(x) < {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right)
x \in \left(0, \infty\right) Definitions:
Fungrim symbol Notation Short description Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) Exp e z {e}^{z} e z Exponential function OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("a0ca3e"),
Formula(Less(Gamma(x), Mul(Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Pow(x, Sub(x, Div(1, 2)))), Exp(Neg(x))), Exp(Div(1, Mul(12, x)))))),
Variables(x),
Assumptions(Element(x, OpenInterval(0, Infinity)))) log Γ ( x ) > ( x − 1 2 ) log ( x ) − x + log ( 2 π ) 2 + ∑ k = 1 2 n B 2 k 2 k ( 2 k − 1 ) x 2 k − 1 \log \Gamma(x) > \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {x}^{2 k - 1}} log Γ ( x ) > ( x − 2 1 ) log ( x ) − x + 2 log ( 2 π ) + k = 1 ∑ 2 n 2 k ( 2 k − 1 ) x 2 k − 1 B 2 k Assumptions: x ∈ ( 0 , ∞ ) and n ∈ Z ≥ 0 x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} x ∈ ( 0 , ∞ ) a n d n ∈ Z ≥ 0 References:
H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66(217), pp. 373-389. Theorem 8. TeX:
\log \Gamma(x) > \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {x}^{2 k - 1}}
x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description LogGamma log Γ ( z ) \log \Gamma(z) log Γ ( z ) Logarithmic gamma function Log log ( z ) \log(z) log ( z ) Natural logarithm Pi π \pi π The constant pi (3.14...) Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum BernoulliB B n B_{n} B n Bernoulli number Pow a b {a}^{b} a b Power OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("2398a1"),
Formula(Greater(LogGamma(x), Add(Add(Sub(Mul(Sub(x, Div(1, 2)), Log(x)), x), Div(Log(Mul(2, Pi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(x, Sub(Mul(2, k), 1)))), For(k, 1, Mul(2, n)))))),
Variables(x, n),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(n, ZZGreaterEqual(0)))),
References("H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66(217), pp. 373-389. Theorem 8.")) log Γ ( x ) < ( x − 1 2 ) log ( x ) − x + log ( 2 π ) 2 + ∑ k = 1 2 n + 1 B 2 k 2 k ( 2 k − 1 ) x 2 k − 1 \log \Gamma(x) < \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n + 1} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {x}^{2 k - 1}} log Γ ( x ) < ( x − 2 1 ) log ( x ) − x + 2 log ( 2 π ) + k = 1 ∑ 2 n + 1 2 k ( 2 k − 1 ) x 2 k − 1 B 2 k Assumptions: x ∈ ( 0 , ∞ ) and n ∈ Z ≥ 0 x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} x ∈ ( 0 , ∞ ) a n d n ∈ Z ≥ 0 References:
H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66(217), pp. 373-389. Theorem 8. TeX:
\log \Gamma(x) < \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n + 1} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {x}^{2 k - 1}}
x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description LogGamma log Γ ( z ) \log \Gamma(z) log Γ ( z ) Logarithmic gamma function Log log ( z ) \log(z) log ( z ) Natural logarithm Pi π \pi π The constant pi (3.14...) Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum BernoulliB B n B_{n} B n Bernoulli number Pow a b {a}^{b} a b Power OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("99a9c6"),
Formula(Less(LogGamma(x), Add(Add(Sub(Mul(Sub(x, Div(1, 2)), Log(x)), x), Div(Log(Mul(2, Pi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(x, Sub(Mul(2, k), 1)))), For(k, 1, Add(Mul(2, n), 1)))))),
Variables(x, n),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(n, ZZGreaterEqual(0)))),
References("H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66(217), pp. 373-389. Theorem 8.")) ∣ Γ ( z ) ∣ > 0 \left|\Gamma(z)\right| > 0 ∣ Γ ( z ) ∣ > 0 Assumptions: z ∈ C z \in \mathbb{C} z ∈ C TeX:
\left|\Gamma(z)\right| > 0
z \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function CC C \mathbb{C} C Complex numbers
Source code for this entry:
Entry(ID("f50ec9"),
Formula(Greater(Abs(Gamma(z)), 0)),
Variables(z),
Assumptions(Element(z, CC))) ∣ 1 Γ ( z ) ∣ < ∞ \left|\frac{1}{\Gamma(z)}\right| < \infty ∣ ∣ ∣ ∣ Γ ( z ) 1 ∣ ∣ ∣ ∣ < ∞ Assumptions: z ∈ C z \in \mathbb{C} z ∈ C TeX:
\left|\frac{1}{\Gamma(z)}\right| < \infty
z \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Infinity ∞ \infty ∞ Positive infinity CC C \mathbb{C} C Complex numbers
Source code for this entry:
Entry(ID("143002"),
Formula(Less(Abs(Div(1, Gamma(z))), Infinity)),
Variables(z),
Assumptions(Element(z, CC))) ∣ Γ ( z ) ∣ ≤ ( 2 π ) 1 / 2 ∣ z ∣ x − 1 / 2 e − π ∣ y ∣ / 2 exp ( 1 6 ∣ z ∣ ) where z = x + y i \left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} {\left|z\right|}^{x - 1 / 2} {e}^{-\pi \left|y\right| / 2} \exp\!\left(\frac{1}{6 \left|z\right|}\right)\; \text{ where } z = x + y i ∣ Γ ( z ) ∣ ≤ ( 2 π ) 1 / 2 ∣ z ∣ x − 1 / 2 e − π ∣ y ∣ / 2 exp ( 6 ∣ z ∣ 1 ) where z = x + y i Assumptions: x ∈ [ 0 , ∞ ) and y ∈ R and x + y i ≠ 0 x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \ne 0 x ∈ [ 0 , ∞ ) a n d y ∈ R a n d x + y i = 0 References:
R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.19), p. 34. TeX:
\left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} {\left|z\right|}^{x - 1 / 2} {e}^{-\pi \left|y\right| / 2} \exp\!\left(\frac{1}{6 \left|z\right|}\right)\; \text{ where } z = x + y i
x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \ne 0 Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) Exp e z {e}^{z} e z Exponential function ConstI i i i Imaginary unit ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b ) Closed-open interval Infinity ∞ \infty ∞ Positive infinity RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("b7fec0"),
Formula(Where(LessEqual(Abs(Gamma(z)), Mul(Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Pow(Abs(z), Sub(x, Div(1, 2)))), Exp(Neg(Div(Mul(Pi, Abs(y)), 2)))), Exp(Div(1, Mul(6, Abs(z)))))), Equal(z, Add(x, Mul(y, ConstI))))),
Variables(x, y),
Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, RR), NotEqual(Add(x, Mul(y, ConstI)), 0))),
References("R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.19), p. 34.")) ∣ Γ ( z ) ∣ ≤ ( 2 π ) 1 / 2 ∣ z z − 1 / 2 e − z ∣ exp ( 1 6 ∣ z ∣ ) \left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(\frac{1}{6 \left|z\right|}\right) ∣ Γ ( z ) ∣ ≤ ( 2 π ) 1 / 2 ∣ ∣ ∣ z z − 1 / 2 e − z ∣ ∣ ∣ exp ( 6 ∣ z ∣ 1 ) Assumptions: z ∈ C and Re ( z ) ≥ 0 and z ≠ 0 z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; z \ne 0 z ∈ C a n d R e ( z ) ≥ 0 a n d z = 0 References:
R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34. TeX:
\left|\Gamma(z)\right| \le {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(\frac{1}{6 \left|z\right|}\right)
z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; z \ne 0 Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) Exp e z {e}^{z} e z Exponential function CC C \mathbb{C} C Complex numbers Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part
Source code for this entry:
Entry(ID("80f7dc"),
Formula(LessEqual(Abs(Gamma(z)), Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Abs(Mul(Pow(z, Sub(z, Div(1, 2))), Exp(Neg(z))))), Exp(Div(1, Mul(6, Abs(z))))))),
Variables(z),
Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0), NotEqual(z, 0))),
References("R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34.")) ∣ Γ ( z ) ∣ ≥ ( 2 π ) 1 / 2 ∣ z z − 1 / 2 e − z ∣ exp ( − 1 6 ∣ z ∣ ) \left|\Gamma(z)\right| \ge {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(-\frac{1}{6 \left|z\right|}\right) ∣ Γ ( z ) ∣ ≥ ( 2 π ) 1 / 2 ∣ ∣ ∣ z z − 1 / 2 e − z ∣ ∣ ∣ exp ( − 6 ∣ z ∣ 1 ) Assumptions: z ∈ C and Re ( z ) ≥ 0 and z ≠ 0 z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; z \ne 0 z ∈ C a n d R e ( z ) ≥ 0 a n d z = 0 TeX:
\left|\Gamma(z)\right| \ge {\left(2 \pi\right)}^{1 / 2} \left|{z}^{z - 1 / 2} {e}^{-z}\right| \exp\!\left(-\frac{1}{6 \left|z\right|}\right)
z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0 \;\mathbin{\operatorname{and}}\; z \ne 0 Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Pow a b {a}^{b} a b Power Pi π \pi π The constant pi (3.14...) Exp e z {e}^{z} e z Exponential function CC C \mathbb{C} C Complex numbers Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part
Source code for this entry:
Entry(ID("931d89"),
Formula(GreaterEqual(Abs(Gamma(z)), Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Abs(Mul(Pow(z, Sub(z, Div(1, 2))), Exp(Neg(z))))), Exp(Neg(Div(1, Mul(6, Abs(z)))))))),
Variables(z),
Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0), NotEqual(z, 0)))) ∣ Γ ( y i ) ∣ = π y sinh ( π y ) \left|\Gamma\!\left(y i\right)\right| = \sqrt{\frac{\pi}{y \sinh\!\left(\pi y\right)}} ∣ Γ ( y i ) ∣ = y sinh ( π y ) π Assumptions: y ∈ R ∖ { 0 } y \in \mathbb{R} \setminus \left\{0\right\} y ∈ R ∖ { 0 } TeX:
\left|\Gamma\!\left(y i\right)\right| = \sqrt{\frac{\pi}{y \sinh\!\left(\pi y\right)}}
y \in \mathbb{R} \setminus \left\{0\right\} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("1976db"),
Formula(Equal(Abs(Gamma(Mul(y, ConstI))), Sqrt(Div(Pi, Mul(y, Sinh(Mul(Pi, y))))))),
Variables(y),
Assumptions(Element(y, SetMinus(RR, Set(0))))) ∣ Γ ( 1 2 + y i ) ∣ = π cosh ( π y ) \left|\Gamma\!\left(\frac{1}{2} + y i\right)\right| = \sqrt{\frac{\pi}{\cosh\!\left(\pi y\right)}} ∣ ∣ ∣ ∣ Γ ( 2 1 + y i ) ∣ ∣ ∣ ∣ = cosh ( π y ) π Assumptions: y ∈ R y \in \mathbb{R} y ∈ R TeX:
\left|\Gamma\!\left(\frac{1}{2} + y i\right)\right| = \sqrt{\frac{\pi}{\cosh\!\left(\pi y\right)}}
y \in \mathbb{R} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("c7b921"),
Formula(Equal(Abs(Gamma(Add(Div(1, 2), Mul(y, ConstI)))), Sqrt(Div(Pi, Cosh(Mul(Pi, y)))))),
Variables(y),
Assumptions(Element(y, RR))) ∣ Γ ( 1 + y i ) ∣ = π y sinh ( π y ) \left|\Gamma\!\left(1 + y i\right)\right| = \sqrt{\frac{\pi y}{\sinh\!\left(\pi y\right)}} ∣ Γ ( 1 + y i ) ∣ = sinh ( π y ) π y Assumptions: y ∈ R ∖ { 0 } y \in \mathbb{R} \setminus \left\{0\right\} y ∈ R ∖ { 0 } TeX:
\left|\Gamma\!\left(1 + y i\right)\right| = \sqrt{\frac{\pi y}{\sinh\!\left(\pi y\right)}}
y \in \mathbb{R} \setminus \left\{0\right\} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("94db60"),
Formula(Equal(Abs(Gamma(Add(1, Mul(y, ConstI)))), Sqrt(Div(Mul(Pi, y), Sinh(Mul(Pi, y)))))),
Variables(y),
Assumptions(Element(y, SetMinus(RR, Set(0))))) ∣ Γ ( x + y i ) ∣ = ∣ Γ ( x ) ∣ ∏ k = 0 ∞ ( 1 + y 2 ( x + k ) 2 ) − 1 / 2 \left|\Gamma\!\left(x + y i\right)\right| = \left|\Gamma(x)\right| \prod_{k=0}^{\infty} {\left(1 + \frac{{y}^{2}}{{\left(x + k\right)}^{2}}\right)}^{-1 / 2} ∣ Γ ( x + y i ) ∣ = ∣ Γ ( x ) ∣ k = 0 ∏ ∞ ( 1 + ( x + k ) 2 y 2 ) − 1 / 2 Assumptions: x ∈ R and y ∈ R and x + y i ∉ { 0 , − 1 , … } x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \{0, -1, \ldots\} x ∈ R a n d y ∈ R a n d x + y i ∈ / { 0 , − 1 , … } References:
Abramowitz & Stegun 6.1.25 TeX:
\left|\Gamma\!\left(x + y i\right)\right| = \left|\Gamma(x)\right| \prod_{k=0}^{\infty} {\left(1 + \frac{{y}^{2}}{{\left(x + k\right)}^{2}}\right)}^{-1 / 2}
x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \{0, -1, \ldots\} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit Product ∏ n f ( n ) \prod_{n} f(n) ∏ n f ( n ) Product Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity RR R \mathbb{R} R Real numbers ZZLessEqual Z ≤ n \mathbb{Z}_{\le n} Z ≤ n Integers less than or equal to n
Source code for this entry:
Entry(ID("513a30"),
Formula(Equal(Abs(Gamma(Add(x, Mul(y, ConstI)))), Mul(Abs(Gamma(x)), Product(Pow(Add(1, Div(Pow(y, 2), Pow(Add(x, k), 2))), Neg(Div(1, 2))), For(k, 0, Infinity))))),
Variables(x, y),
Assumptions(And(Element(x, RR), Element(y, RR), NotElement(Add(x, Mul(y, ConstI)), ZZLessEqual(0)))),
References("Abramowitz & Stegun 6.1.25")) ∣ Γ ( x + y i ) ∣ ≤ ∣ Γ ( x ) ∣ \left|\Gamma\!\left(x + y i\right)\right| \le \left|\Gamma(x)\right| ∣ Γ ( x + y i ) ∣ ≤ ∣ Γ ( x ) ∣ Assumptions: x ∈ R and y ∈ R x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} x ∈ R a n d y ∈ R References:
B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-3. TeX:
\left|\Gamma\!\left(x + y i\right)\right| \le \left|\Gamma(x)\right|
x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("4a2ac8"),
Formula(LessEqual(Abs(Gamma(Add(x, Mul(y, ConstI)))), Abs(Gamma(x)))),
Variables(x, y),
Assumptions(And(Element(x, RR), Element(y, RR))),
References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-3.")) ∣ Γ ( x + y i ) ∣ < ∣ Γ ( x + t i ) ∣ \left|\Gamma\!\left(x + y i\right)\right| < \left|\Gamma\!\left(x + t i\right)\right| ∣ Γ ( x + y i ) ∣ < ∣ Γ ( x + t i ) ∣ Assumptions: x ∈ R and y ∈ R and t ∈ R and ∣ y ∣ > ∣ t ∣ x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; t \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|y\right| > \left|t\right| x ∈ R a n d y ∈ R a n d t ∈ R a n d ∣ y ∣ > ∣ t ∣ TeX:
\left|\Gamma\!\left(x + y i\right)\right| < \left|\Gamma\!\left(x + t i\right)\right|
x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; t \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|y\right| > \left|t\right| Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("dd5e3a"),
Formula(Less(Abs(Gamma(Add(x, Mul(y, ConstI)))), Abs(Gamma(Add(x, Mul(t, ConstI)))))),
Variables(x, y, t),
Assumptions(And(Element(x, RR), Element(y, RR), Element(t, RR), Greater(Abs(y), Abs(t))))) ∣ Γ ( x + y i ) ∣ ≥ Γ ( x ) cosh ( π y ) \left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma(x)}{\sqrt{\cosh\!\left(\pi y\right)}} ∣ Γ ( x + y i ) ∣ ≥ cosh ( π y ) Γ ( x ) Assumptions: x ∈ [ 1 2 , ∞ ) and y ∈ R x \in \left[\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} x ∈ [ 2 1 , ∞ ) a n d y ∈ R References:
B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4. TeX:
\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma(x)}{\sqrt{\cosh\!\left(\pi y\right)}}
x \in \left[\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit Sqrt z \sqrt{z} z Principal square root Pi π \pi π The constant pi (3.14...) ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b ) Closed-open interval Infinity ∞ \infty ∞ Positive infinity RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("e0b322"),
Formula(GreaterEqual(Abs(Gamma(Add(x, Mul(y, ConstI)))), Div(Gamma(x), Sqrt(Cosh(Mul(Pi, y)))))),
Variables(x, y),
Assumptions(And(Element(x, ClosedOpenInterval(Div(1, 2), Infinity)), Element(y, RR))),
References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4.")) ∣ Γ ( x + y i ) ∣ ≥ Γ ( x ) e − π ∣ y ∣ / 2 \left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma(x) {e}^{-\pi \left|y\right| / 2} ∣ Γ ( x + y i ) ∣ ≥ Γ ( x ) e − π ∣ y ∣ / 2 Assumptions: x ∈ [ 1 2 , ∞ ) and y ∈ R x \in \left[\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} x ∈ [ 2 1 , ∞ ) a n d y ∈ R References:
B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4. TeX:
\left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma(x) {e}^{-\pi \left|y\right| / 2}
x \in \left[\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function ConstI i i i Imaginary unit Exp e z {e}^{z} e z Exponential function Pi π \pi π The constant pi (3.14...) ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b ) Closed-open interval Infinity ∞ \infty ∞ Positive infinity RR R \mathbb{R} R Real numbers
Source code for this entry:
Entry(ID("7af1b9"),
Formula(GreaterEqual(Abs(Gamma(Add(x, Mul(y, ConstI)))), Mul(Gamma(x), Exp(Neg(Div(Mul(Pi, Abs(y)), 2)))))),
Variables(x, y),
Assumptions(And(Element(x, ClosedOpenInterval(Div(1, 2), Infinity)), Element(y, RR))),
References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Inequality 3.10-4.")) ∣ 1 Γ ( z ) ∣ ≤ e π R / 2 R R + 1 / 2 where R = ∣ z ∣ \left|\frac{1}{\Gamma(z)}\right| \le {e}^{\pi R / 2} {R}^{R + 1 / 2}\; \text{ where } R = \left|z\right| ∣ ∣ ∣ ∣ Γ ( z ) 1 ∣ ∣ ∣ ∣ ≤ e π R / 2 R R + 1 / 2 where R = ∣ z ∣ Assumptions: z ∈ C z \in \mathbb{C} z ∈ C TeX:
\left|\frac{1}{\Gamma(z)}\right| \le {e}^{\pi R / 2} {R}^{R + 1 / 2}\; \text{ where } R = \left|z\right|
z \in \mathbb{C} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function 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 CC C \mathbb{C} C Complex numbers
Source code for this entry:
Entry(ID("06260c"),
Formula(LessEqual(Abs(Div(1, Gamma(z))), Where(Mul(Exp(Div(Mul(Pi, R), 2)), Pow(R, Add(R, Div(1, 2)))), Equal(R, Abs(z))))),
Variables(z),
Assumptions(Element(z, CC))) ∣ 1 n ! [ d n d x n 1 Γ ( x ) ] x = 0 ∣ ≤ 2 n ! \left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma(x)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}} ∣ ∣ ∣ ∣ n ! 1 [ d x n d n Γ ( x ) 1 ] x = 0 ∣ ∣ ∣ ∣ ≤ n ! 2 Assumptions: n ∈ Z ≥ 0 n \in \mathbb{Z}_{\ge 0} n ∈ Z ≥ 0 References:
L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, https://arxiv.org/abs/1407.5983 (simplified version of (1.5)) TeX:
\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma(x)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}
n \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Factorial n ! n ! n ! Factorial ComplexDerivative d d z f ( z ) \frac{d}{d z}\, f\!\left(z\right) d z d f ( z ) Complex derivative Gamma Γ ( z ) \Gamma(z) Γ ( z ) Gamma function Sqrt z \sqrt{z} z Principal square root ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("cb5071"),
Formula(LessEqual(Abs(Mul(Div(1, Factorial(n)), ComplexDerivative(Div(1, Gamma(x)), For(x, 0, n)))), Div(2, Sqrt(Factorial(n))))),
Variables(n),
Assumptions(And(Element(n, ZZGreaterEqual(0)))),
References("L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, https://arxiv.org/abs/1407.5983 (simplified version of (1.5))"))