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Bounds and inequalities for the gamma function

Table of contents: Real argument - Complex argument - Derivatives

Related topics: Gamma function

Real argument

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arg min*x(0,)Γ(x)[1.46163214496836234126265954233±4.281030]\mathop{\operatorname{arg\,min*}}\limits_{x \in \left(0, \infty\right)} \Gamma(x) \in \left[1.46163214496836234126265954233 \pm 4.28 \cdot 10^{-30}\right]
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minx(0,)Γ(x)[0.885603194410888700278815900583±4.121031]\mathop{\min}\limits_{x \in \left(0, \infty\right)} \Gamma(x) \in \left[0.885603194410888700278815900583 \pm 4.12 \cdot 10^{-31}\right]
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minx(0,)logΓ(x)[0.121486290535849608095514557178±3.091031]\mathop{\min}\limits_{x \in \left(0, \infty\right)} \log \Gamma(x) \in \left[-0.121486290535849608095514557178 \pm 3.09 \cdot 10^{-31}\right]
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Γ(x)>(2π)1/2xx1/2ex\Gamma(x) > {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x}
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Γ(x)<(2π)1/2xx1/2exexp ⁣(112x)\Gamma(x) < {\left(2 \pi\right)}^{1 / 2} {x}^{x - 1 / 2} {e}^{-x} \exp\!\left(\frac{1}{12 x}\right)
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logΓ(x)>(x12)log(x)x+log ⁣(2π)2+k=12nB2k2k(2k1)x2k1\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}}
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logΓ(x)<(x12)log(x)x+log ⁣(2π)2+k=12n+1B2k2k(2k1)x2k1\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}}

Complex argument

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Γ(z)>0\left|\Gamma(z)\right| > 0
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1Γ(z)<\left|\frac{1}{\Gamma(z)}\right| < \infty
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Γ(z)(2π)1/2zx1/2eπy/2exp ⁣(16z)   where z=x+yi\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
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Γ(z)(2π)1/2zz1/2ezexp ⁣(16z)\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)
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Γ(z)(2π)1/2zz1/2ezexp ⁣(16z)\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)
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Γ ⁣(yi)=πysinh ⁣(πy)\left|\Gamma\!\left(y i\right)\right| = \sqrt{\frac{\pi}{y \sinh\!\left(\pi y\right)}}
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Γ ⁣(12+yi)=πcosh ⁣(πy)\left|\Gamma\!\left(\frac{1}{2} + y i\right)\right| = \sqrt{\frac{\pi}{\cosh\!\left(\pi y\right)}}
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Γ ⁣(1+yi)=πysinh ⁣(πy)\left|\Gamma\!\left(1 + y i\right)\right| = \sqrt{\frac{\pi y}{\sinh\!\left(\pi y\right)}}
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Γ ⁣(x+yi)=Γ(x)k=0(1+y2(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}
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Γ ⁣(x+yi)Γ(x)\left|\Gamma\!\left(x + y i\right)\right| \le \left|\Gamma(x)\right|
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Γ ⁣(x+yi)<Γ ⁣(x+ti)\left|\Gamma\!\left(x + y i\right)\right| < \left|\Gamma\!\left(x + t i\right)\right|
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Γ ⁣(x+yi)Γ(x)cosh ⁣(πy)\left|\Gamma\!\left(x + y i\right)\right| \ge \frac{\Gamma(x)}{\sqrt{\cosh\!\left(\pi y\right)}}
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Γ ⁣(x+yi)Γ(x)eπy/2\left|\Gamma\!\left(x + y i\right)\right| \ge \Gamma(x) {e}^{-\pi \left|y\right| / 2}
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1Γ(z)eπR/2RR+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|

Derivatives

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1n![dndxn1Γ(x)]x=02n!\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma(x)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC