# Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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### Anger-Weber function for an integer value of the order

The Anger-Weber function is defined by $$A_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t$$ where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$. I am not able to numerically ...
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### Numerical evaluation of the Schläfli integral

I'm trying to numerically evaluate $$S_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t$$ where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$. This integral is a part of the ...
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### Series expansion involving Kummer and Tricomi functions analogy

Good day to everyone. I've got in a pickle while toying around with some transformations. It is well-known that the bivariate confluent hypergeometric function $\Phi_2(\cdot)$ can be expanded in the ...
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I was evaluating: $$I:=\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}\,dx$$ This is what I did, how can my answer be simplified if correct. The following is my work: $$I=2\int_{0}^{\infty}\frac{\sin(x)}{e^... 0 votes 0 answers 60 views ### Analytic Form of \int dx \frac{e^{ - m\sqrt{x^2+b^2}}}{\sqrt{x^2 + b^2}}e^{ikx } Does anyone know how to get an analytic expression of this?$$\int^L_{-L} dx \frac{e^{- m\sqrt{x^2+b^2}}}{\sqrt{x^2 + b^2}}e^{ikx} $$where k = \dfrac{n\pi}{L}, m, and b are real parameters. or ... 4 votes 2 answers 266 views ### Integral over a product of polynomial, exponential and Bessel function In a physics textbook I'm working through I found an interesting integral identity which I want to prove: \begin{equation} \int_0^\infty t^{\nu +1} J_\nu(\beta t) e^{-\alpha t} \, dt = \frac{2\alpha (... 2 votes 0 answers 36 views ### Explicit example of an entire function with simple zeros at precisely the square roots of the positive half integers I'm looking for an entire function with the property that f(\sqrt{n+1/2}) = 0 for n=0,1,2,\dots, all of which are simple zeros and f has no other zeros. I know that such functions exist and can ... 1 vote 1 answer 102 views ### D-dimensional Fourier transform of \exp{(-c|\vec{x}|^\alpha}) Prove that for \vec{x},\vec{\xi} \in \mathbb{R}^D, \mathbb{R} \ni c, \alpha = \mathrm{const.} the D-dimensional Fourier transform of  f(\vec{x}) = \exp{(-c|\vec{x}|^\alpha}) \begin{equation} \... 7 votes 6 answers 701 views ### A question on Beta function I need an asymptotic expansion/closed form for$$\sum_{k=1}^{\infty}\int_{0}^{\infty}(B(x+n+k,n+1))^2\ dx$$where B(m,n) is the Beta function and n\in\mathbb{N}. Denote$$I_n=\sum_{k=1}^{\infty}\...
Here are two formulas that involve Dottie number and Gamma function: $\Gamma \left( \frac{1}{2} + \frac{d}{\pi} \right) \Gamma \left( \frac{1}{2} - \frac{d}{\pi} \right) = \frac{\pi}{d}$ and \$\sqrt{\...