New posts in special-functions

Understanding the Gamma Function

inequality special-functions gamma-function

The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$

analysis integration special-functions

Closed form solutions for a family of hypergeometric sums.

special-functions hypergeometric-function sequences-and-series

The ratio of jacobi theta functions

special-functions modular-forms continued-fractions q-series

Integral $T_n=\int_{0}^{\pi/2}x^{n}\ln(1+\tan x)\,dx$

calculus integration sequences-and-series definite-integrals special-functions

Erf squared approximation

special-functions error-function

Looking for a closed form for a ${}_4 F_3\left(\ldots,1\right)$

sequences-and-series special-functions closed-form hypergeometric-function elliptic-integrals

Proof that $Γ'(1) = -γ$?

calculus integration special-functions derivatives gamma-function

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

integration definite-integrals improper-integrals special-functions harmonic-numbers

On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

complex-analysis special-functions riemann-zeta gamma-function

Proving $\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx=\frac{3 \pi ^2}{160}$

integration definite-integrals special-functions closed-form

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

definite-integrals logarithms special-functions calculus closed-form

On $\mathrm{\sum\limits_{x=1}^\infty Ci(x)}$.

calculus sequences-and-series special-functions trigonometric-series constants

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$

ordinary-differential-equations derivatives special-functions proof-explanation bessel-functions

Solving $x^{x^x}=c$ with Lambert's W function

exponential-function special-functions lambert-w

Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$

complex-analysis summation special-functions big-list

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

calculus integration definite-integrals special-functions gamma-function

What are special functions for?

soft-question special-functions

Intuition why the volume and surface area of the unit sphere eventually decrease

special-functions intuition analytic-geometry

Are Complex Substitutions Legal in Integration?

complex-analysis integration special-functions improper-integrals fresnel-integrals