Gamma

The gamma function $\Gamma \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is the function initially defined for $\mathrm{Re}(z)>0$ by the integral by the formula $$\Gamma(x)=\displaystyle\int_0^{\infty} \xi^{x-1}e^{-\xi} \mathrm{d}\xi.$$ The analytic continuation of $\Gamma$ leads to a meromorphic function with poles at the negative integers.

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  Graph of $\Gamma$.

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  Domain coloring of $\Gamma$.

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  $ (1948).

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  Plot of $\Gamma$ and $\dfrac{1}{\Gamma}$ from Abramowitz&Stegun.

Properties

Theorem

The following formula holds: $$\Gamma(1)=1,$$ where $\Gamma$ denotes the gamma function.

Proof

Compute using the fundamental theorem of calculus, $$\begin{array}{ll} \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{0} e^{-\xi} \mathrm{d}\xi \\ &= \displaystyle\int_0^{\infty} e^{-\xi} \mathrm{d}\xi \\ &= \left[ -e^{-\xi} \right.\Bigg|_{0}^{\infty} \\ &= 1, \end{array}$$ as was to be shown. █

References

• 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 2.1

Theorem

The following formula holds: $$\Gamma(z+1)=z\Gamma(z),$$ where $\Gamma$ denotes gamma.

Proof

Use integration by parts to compute $$\begin{array}{ll} \Gamma(z+1) &= \displaystyle\int_0^{\infty} \xi^z e^{-\xi} \mathrm{d}\xi \\ &= -\xi^z e^{-\xi}\Bigg|_0^{\infty}- \displaystyle\int_0^{\infty} z \xi^{z-1} e^{-\xi} \mathrm{d}\xi \\ &= z\Gamma(z), \end{array}$$ as was to be shown. █

References

• 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 2.2
• 2010: Roelof Koekoek, Peter A. Lesky and René F. Swarttouw: Hypergeometric Orthogonal Polynomials and Their q-Analogues ... (previous) ... (next): $(1.2.2)$
• 2010: Richard Beals and Roderick Wong: Special functions, a graduate text ... (previous) ... (next): $(2.1.2)$

Theorem

If $n \in \{0,1,2,\ldots\}$, then $$\Gamma(n+1)=n!,$$ where $\Gamma$ denotes the gamma function and $n!$ denotes the factorial of $n$.

References

• 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 2.3

Proposition: The following formula holds: $$\Gamma(x)=2\displaystyle\int_0^{\infty} e^{-t^2}t^{2x-1}dt.$$

Proof: █

Proposition: The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} \cos^{2x-1}(\theta)\sin^{2y-1}(\theta) d\theta = \dfrac{\Gamma(x)\Gamma(y)}{2\Gamma(x+y)}.$$

Proof: █

Theorem: The following formula holds for $\mathrm{Re}(z)>0$: $$\Gamma(z)=\displaystyle\int_0^1 \log \left( \dfrac{1}{t} \right)^{z-1} dt,$$ where $\Gamma$ denotes the gamma function and $\log$ denotes the logarithm.

Proof: █

Proposition: The following formula holds: $$\Gamma(z)=\lim_{k \rightarrow \infty} \dfrac{k!k^z}{z(z+1)\ldots(z+k)}.$$

Proof: █

Gamma function Weierstrass product

Theorem

The following formula holds: $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ where $\Gamma$ denotes the gamma function and $\zeta$ denotes the Hurwitz zeta function.

References

Proposition: $\Gamma \left( \dfrac{1}{2} \right) = \sqrt{\pi}$.

Proof: █

Corollary: $\displaystyle\int_0^{\infty} e^{-t^2} dt = \dfrac{1}{2}\sqrt{\pi}$.

Theorem (Convexity): The gamma function is logarithmically convex.

Proof: █

Theorem (Legendre Duplication Formula): $$\Gamma(2x)=\dfrac{2^{2x-1}}{\sqrt{\pi}} \Gamma(x)\Gamma \left( x +\dfrac{1}{2} \right).$$

Proof: █

Proposition: If $z=0,-1,-2,\ldots$ then $\Gamma(z)=\infty$.

Proof: █

1. REDIRECT Gamma(z)Gamma(1-z)=pi/sin(pi z)

Theorem

The gamma function is the unique function $f$ such that $f(1)=1$, $f(x+1)=xf(x)$ for $x>0$, and $f$ is logarithmically convex.

References

Theorem: The following formula holds: $$\displaystyle\lim_{t \rightarrow \infty} \dfrac{\Gamma(t+\alpha)}{\Gamma(t)t^{\alpha}}=\displaystyle\lim_{t \rightarrow \infty} \dfrac{\Gamma(t)t^{\alpha}}{\Gamma(t+\alpha)}=1.$$

Proof: proof goes here █

Videos

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Gamma Function Of One-Half: Part 2
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See Also

Loggamma
Polygamma
Reciprocal gamma

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (next): §1.1 (1)

The Gamma Function by Emil Artin
The sine product formula and the gamma function
Leonhard Euler's Integral: A Historical Profile of the Gamma Function