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Prove that $\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$

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#calculus #integration #special-functions #beta-function
435
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Evaluating $\int_0^\infty \sqrt{\frac{x}{e^x-1}}dx$ in terms of special functions

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#calculus #definite-integrals #improper-integrals #special-functions #closed-form
300
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asymptotic expansion for Bessel function $I_0(z)$ in terms of Gauss hypergeometric functions ${}_2F_1$

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#real-analysis #asymptotics #special-functions #bessel-functions #hypergeometric-function
635
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Simplify Infinite Series Involving Gamma Function $\Gamma$

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#sequences-and-series #power-series #special-functions #gamma-function
84
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Closed form for generating function of Riemann Xi function

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#special-functions #generating-functions #closed-form #riemann-zeta
286
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Can this series be expressed as a Hyper Geometric function

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#sequences-and-series #special-functions #hypergeometric-function
136
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Is $ d^m_xP_l(x) d^{m+1}_xP_{l+1}(x)- d^m_xP_{l+1}(x) d^{m+1}_xP_{l}(x)$ positive?

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#calculus #real-analysis #analysis #special-functions #legendre-polynomials
74
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Calculus function create

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#calculus #special-functions
3.9k
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What even *are* elliptic functions?

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#complex-analysis #special-functions #analytic-number-theory #elliptic-integrals #elliptic-functions
87
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Is there a simplified way to express the following function that is defined by two Gauss' hypergeometric functions?

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#real-analysis #special-functions #real-numbers #hypergeometric-function #algebra-precalculus
45
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Is there a power series solution for this ODE with non-recursive coefficients?

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#ordinary-differential-equations #power-series #special-functions #hypergeometric-function
325
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Real and imaginary parts of $\ln \Gamma(i b)$

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#complex-analysis #special-functions #gamma-function #digamma-function
61
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Simplifing double integral with known function(s)

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#definite-integrals #improper-integrals #special-functions #trigonometric-integrals
91
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Integration involving some special functions

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#definite-integrals #asymptotics #special-functions #closed-form #bessel-functions
310
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How to solve this integral using some special functions? $I=\int_a^b \frac{\sin u}{(c-u-\sin u)^{2/3}}du$

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#definite-integrals #special-functions #gamma-function #hypergeometric-function #elliptic-functions

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