Bounds and inequalities for the gamma function

1bbbc7

\mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right)} \Gamma\!\left(x\right) \in \left[1.46163214496836234126265954233 \pm 4.28 \cdot 10^{-30}\right]

TeX:

\mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right)} \Gamma\!\left(x\right) \in \left[1.46163214496836234126265954233 \pm 4.28 \cdot 10^{-30}\right]

Definitions:

Fungrim symbol	Notation	Short description
`ArgMinUnique`	$\mathop{\operatorname{arg\,min*}}\limits_{P\left(x\right)} f\!\left(x\right)$	Unique location of minimum value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("1bbbc7"),
    Formula(Element(ArgMinUnique(GammaFunction(x), x, Element(x, OpenInterval(0, Infinity))), RealBall(Decimal("1.46163214496836234126265954233"), Decimal("4.28e-30")))))

e010c9

\mathop{\min}\limits_{x \in \left(0, \infty\right)} \Gamma\!\left(x\right) \in \left[0.885603194410888700278815900583 \pm 4.12 \cdot 10^{-31}\right]

TeX:

\mathop{\min}\limits_{x \in \left(0, \infty\right)} \Gamma\!\left(x\right) \in \left[0.885603194410888700278815900583 \pm 4.12 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("e010c9"),
    Formula(Element(Minimum(GammaFunction(x), x, Element(x, OpenInterval(0, Infinity))), RealBall(Decimal("0.885603194410888700278815900583"), Decimal("4.12e-31")))))

b05f2b

\mathop{\min}\limits_{x \in \left(0, \infty\right)} \log \Gamma\!\left(x\right) \in \left[-0.121486290535849608095514557178 \pm 3.09 \cdot 10^{-31}\right]

TeX:

\mathop{\min}\limits_{x \in \left(0, \infty\right)} \log \Gamma\!\left(x\right) \in \left[-0.121486290535849608095514557178 \pm 3.09 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function
`LogGamma`	$\log \Gamma\!\left(z\right)$	Logarithmic gamma function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("b05f2b"),
    Formula(Element(Minimum(LogGamma(x), x, Element(x, OpenInterval(0, Infinity))), RealBall(Decimal("-0.121486290535849608095514557178"), Decimal("3.09e-31")))))

2a47d7

\Gamma\!\left(x\right) \gt {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x}

Assumptions:

x \in \left(0, \infty\right)

TeX:

\Gamma\!\left(x\right) \gt {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x}

x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("2a47d7"),
    Formula(Greater(GammaFunction(x), Mul(Mul(Pow(Mul(2, ConstPi), Div(1, 2)), Pow(x, Sub(x, Div(1, 2)))), Exp(Neg(x))))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(0, Infinity))))

a0ca3e

\Gamma\!\left(x\right) \lt {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right)

Assumptions:

x \in \left(0, \infty\right)

TeX:

\Gamma\!\left(x\right) \lt {\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
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("a0ca3e"),
    Formula(Less(GammaFunction(x), Mul(Mul(Mul(Pow(Mul(2, ConstPi), 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))))

2398a1

\log \Gamma\!\left(x\right) \gt \left(x - \frac{1}{2}\right) \log\!\left(x\right) - 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}}

Assumptions:

x \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 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\!\left(x\right) \gt \left(x - \frac{1}{2}\right) \log\!\left(x\right) - 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`	$\log \Gamma\!\left(z\right)$	Logarithmic gamma function
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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, ConstPi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(x, Sub(Mul(2, k), 1)))), Tuple(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."))

99a9c6

\log \Gamma\!\left(x\right) \lt \left(x - \frac{1}{2}\right) \log\!\left(x\right) - 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}}

Assumptions:

x \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 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\!\left(x\right) \lt \left(x - \frac{1}{2}\right) \log\!\left(x\right) - 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`	$\log \Gamma\!\left(z\right)$	Logarithmic gamma function
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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, ConstPi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(x, Sub(Mul(2, k), 1)))), Tuple(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."))

f50ec9

\left|\Gamma\!\left(z\right)\right| \gt 0

Assumptions:

z \in \mathbb{C}

TeX:

\left|\Gamma\!\left(z\right)\right| \gt 0

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f50ec9"),
    Formula(Greater(Abs(GammaFunction(z)), 0)),
    Variables(z),
    Assumptions(Element(z, CC)))

143002

\left|\frac{1}{\Gamma\!\left(z\right)}\right| \lt \infty

Assumptions:

z \in \mathbb{C}

TeX:

\left|\frac{1}{\Gamma\!\left(z\right)}\right| \lt \infty

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("143002"),
    Formula(Less(Abs(Div(1, GammaFunction(z))), Infinity)),
    Variables(z),
    Assumptions(Element(z, CC)))

b7fec0

\left|\Gamma\!\left(z\right)\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

Assumptions:

x \in \left[0, \infty\right) \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x + y i \ne 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\!\left(z\right)\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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("b7fec0"),
    Formula(Where(LessEqual(Abs(GammaFunction(z)), Mul(Mul(Mul(Pow(Mul(2, ConstPi), Div(1, 2)), Pow(Abs(z), Sub(x, Div(1, 2)))), Exp(Neg(Div(Mul(ConstPi, 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), Unequal(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."))

80f7dc

\left|\Gamma\!\left(z\right)\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)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 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\!\left(z\right)\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}\!\left(z\right) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("80f7dc"),
    Formula(LessEqual(Abs(GammaFunction(z)), Mul(Mul(Pow(Mul(2, ConstPi), 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), Unequal(z, 0))),
    References("R. B. Paris and D. Kaminski (2001), Asymptotics of Mellin-Barnes integrals, Cambridge University Press. (2.1.18), p. 34."))

931d89

\left|\Gamma\!\left(z\right)\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)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

\left|\Gamma\!\left(z\right)\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}\!\left(z\right) \ge 0 \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("931d89"),
    Formula(GreaterEqual(Abs(GammaFunction(z)), Mul(Mul(Pow(Mul(2, ConstPi), 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), Unequal(z, 0))))

1976db

\left|\Gamma\!\left(y i\right)\right| = \sqrt{\frac{\pi}{y \sinh\!\left(\pi y\right)}}

Assumptions:

y \in \mathbb{R} \setminus \left\{0\right\}

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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("1976db"),
    Formula(Equal(Abs(GammaFunction(Mul(y, ConstI))), Sqrt(Div(ConstPi, Mul(y, Sinh(Mul(ConstPi, y))))))),
    Variables(y),
    Assumptions(Element(y, SetMinus(RR, Set(0)))))

c7b921

\left|\Gamma\!\left(\frac{1}{2} + y i\right)\right| = \sqrt{\frac{\pi}{\cosh\!\left(\pi y\right)}}

Assumptions:

y \in \mathbb{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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("c7b921"),
    Formula(Equal(Abs(GammaFunction(Add(Div(1, 2), Mul(y, ConstI)))), Sqrt(Div(ConstPi, Cosh(Mul(ConstPi, y)))))),
    Variables(y),
    Assumptions(Element(y, RR)))

94db60

\left|\Gamma\!\left(1 + y i\right)\right| = \sqrt{\frac{\pi y}{\sinh\!\left(\pi y\right)}}

Assumptions:

y \in \mathbb{R} \setminus \left\{0\right\}

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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("94db60"),
    Formula(Equal(Abs(GammaFunction(Add(1, Mul(y, ConstI)))), Sqrt(Div(Mul(ConstPi, y), Sinh(Mul(ConstPi, y)))))),
    Variables(y),
    Assumptions(Element(y, SetMinus(RR, Set(0)))))

513a30

\left|\Gamma\!\left(x + y i\right)\right| = \left|\Gamma\!\left(x\right)\right| \prod_{k=0}^{\infty} {\left(1 + \frac{{y}^{2}}{{\left(x + k\right)}^{2}}\right)}^{-1 / 2}

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x + y i \notin \{0, -1, \ldots\}

References:

Abramowitz & Stegun 6.1.25

TeX:

\left|\Gamma\!\left(x + y i\right)\right| = \left|\Gamma\!\left(x\right)\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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("513a30"),
    Formula(Equal(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Mul(Abs(GammaFunction(x)), Product(Pow(Add(1, Div(Pow(y, 2), Pow(Add(x, k), 2))), Neg(Div(1, 2))), Tuple(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"))

4a2ac8

\left|\Gamma\!\left(x + y i\right)\right| \le \left|\Gamma\!\left(x\right)\right|

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{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\!\left(x\right)\right|

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("4a2ac8"),
    Formula(LessEqual(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Abs(GammaFunction(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."))

dd5e3a

\left|\Gamma\!\left(x + y i\right)\right| \lt \left|\Gamma\!\left(x + t i\right)\right|

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, t \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left|y\right| \gt \left|t\right|

TeX:

\left|\Gamma\!\left(x + y i\right)\right| \lt \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| \gt \left|t\right|

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("dd5e3a"),
    Formula(Less(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Abs(GammaFunction(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)))))

e0b322

\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma\!\left(x\right)}{\sqrt{\cosh\!\left(\pi y\right)}}

Assumptions:

x \in \left[\frac{1}{2}, \infty\right) \,\mathbin{\operatorname{and}}\, y \in \mathbb{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\!\left(x\right)}{\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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("e0b322"),
    Formula(GreaterEqual(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Div(GammaFunction(x), Sqrt(Cosh(Mul(ConstPi, 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."))

7af1b9

\left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma\!\left(x\right) {e}^{-\pi \left|y\right| / 2}

Assumptions:

x \in \left[\frac{1}{2}, \infty\right) \,\mathbin{\operatorname{and}}\, y \in \mathbb{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\!\left(x\right) {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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("7af1b9"),
    Formula(GreaterEqual(Abs(GammaFunction(Add(x, Mul(y, ConstI)))), Mul(GammaFunction(x), Exp(Neg(Div(Mul(ConstPi, 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."))

06260c

\left|\frac{1}{\Gamma\!\left(z\right)}\right| \le {e}^{\pi R / 2} {R}^{R + 1 / 2}\; \text{ where } R = \left|z\right|

Assumptions:

z \in \mathbb{C}

TeX:

\left|\frac{1}{\Gamma\!\left(z\right)}\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`	$\left\|z\right\|$	Absolute value
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("06260c"),
    Formula(LessEqual(Abs(Div(1, GammaFunction(z))), Where(Mul(Exp(Div(Mul(ConstPi, R), 2)), Pow(R, Add(R, Div(1, 2)))), Equal(R, Abs(z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

cb5071

\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma\!\left(x\right)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}

Assumptions:

n \in \mathbb{Z}_{\ge 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\!\left(x\right)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Factorial`	$n !$	Factorial
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("cb5071"),
    Formula(LessEqual(Abs(Mul(Div(1, Factorial(n)), Derivative(Div(1, GammaFunction(x)), Tuple(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))"))

Real argument

Complex argument

Derivatives