# Tagged Questions

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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### Definite Integration of Hypergeometric function combined with two algebraic function?

Please help me to solve the integral: $$\int_0^{-w}\dfrac{{_2F_1(1,k,3/2;\phi/B)}}{\sqrt{\phi}\sqrt{-\phi-\omega}}d\phi$$ I have solved this in Mathematica.but I am not able to way a general result ...
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### Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
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### Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
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### Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: $$Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt$$ I am trying to calculate the derivative of $Γ$ with respect ...
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### Solving the definite integral $\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$

I need to solve this definite integral: $$\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$$ where $A$ is a real positive constant and $\psi\in[0,2\pi]$. I know that for $\psi=2\pi$ the ...
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### finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
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### Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
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### Solve the nth zero of a function. [closed]

Say I have a mystery continuous function, could be anything. f(x) Assuming we don't know the distribution of the zeros of the function, Is there a known way to solve the nth zero (hits the x-axis)? ...
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### Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
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### How to come up with the gamma function?

It always puzzles me, how the Gamma function's inventor came up with its definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
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### The sum $\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}$? [closed]

I'm interested in evaluating the following sum : $\displaystyle{\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}}$ where $x>0$. The existence of a closed form would be great but is perhaps too ...
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### Definite integral involving algebraic, exponential, and product of two Meijer's G function

I am having trouble with calculating the following integral: I = \int_{0}^{\infty}x\exp({-\beta x})\large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} \...
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### Problem on series expansion and Bessel functions

One way to define Bessel functions is $$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n.$$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
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### 2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
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### Integral of Bessel function multiplied with sine

I need advice on how to solve the following integral: $$\int_0^\infty J_0(bx) \sin(ax) dx$$ I've seen it referenced, e.g. here on MathSE, so I know the solution is $(a^2-b^2)^{-1/2}$ for $a>b$ ...
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### Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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### Integration of physicists' Hermite polynomial with exponential

I am trying to prove the lhs of the following equation is equal to rhs. \begin{align*} \int_{-\infty}^\infty H_n(x)e^{-x^2/2}\,\mathrm{d}x = \begin{cases} \frac{2\pi n!}{(n/2)!},& \text{if } ...