New posts in gamma-function

Calculating half derivative of $x$ using Cauchy's repeated integral formula

integration derivatives definite-integrals gamma-function fractional-calculus

Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed.

sequences-and-series solution-verification gamma-function hypergeometric-function arc-length

How to find $\mathrm{\sum_{n=1}^\infty E_{n}(n)= \lim_{k\to\infty}\int_1^\infty \frac{e^{kt+t}t^k-e^t}{e^{kt+2t}t^{k+1}-e^{kt+t}t^k}dt=.26929…}$?

integration sequences-and-series improper-integrals special-functions gamma-function

The limit $\lim_{r\to0}\frac1r\left(1-\binom{n}{r}^{-1}\right)$

limits binomial-coefficients gamma-function harmonic-numbers

Generating function for gamma function (or factorial)

combinatorics generating-functions factorial gamma-function

The derivative of $x!$ and its continuity

derivatives continuity factorial gamma-function

Euler gamma function differential

calculus gamma-function

Integral of $\ln(x)\operatorname{sech}(x)$

integration definite-integrals improper-integrals gamma-function hyperbolic-functions

Approximation of the Gamma function for small value

inequality reference-request gamma-function

Series expansion of incomplete gamma function ratio

asymptotics special-functions gamma-function

Proof that the gamma function is an extension of the factorial function

calculus factorial gamma-function

Solve $\sum_{n=1}^\inftyΓ(-n,-n)\mathop= \pi\left(\frac1e-1\right)i+ \sum_{n=1}^\infty \frac{(-1)^n \text{Ei}(n)}{n!}+\sum_{n=1}^\infty a_n･(-e)^n $

sequences-and-series complex-numbers gamma-function constants tetration

Fourier transform of $\Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) /\sqrt{ \cosh(p/2)}$

integration complex-analysis fourier-analysis asymptotics gamma-function

Prove known closed form for $\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx$

definite-integrals closed-form gamma-function bessel-functions elliptic-integrals

Proof that the factorial is nonelementary

factorial gamma-function elementary-functions

How to prove the formula for the Reciprocal Multifactorial constant?

sequences-and-series factorial gamma-function

Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$

definite-integrals closed-form gamma-function hypergeometric-function trigonometric-integrals

A gamma function inequality

inequality special-functions gamma-function

Evaluating this integral using the Gamma function

calculus integration definite-integrals improper-integrals gamma-function

Proving $\int_0^\infty\left(\frac{x^xe^{-x}}{\Gamma(x+1)}-\frac1{\sqrt{2\pi x}}\right)dx=-\frac13$

integration sequences-and-series definite-integrals gamma-function