# 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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### Check this value of $\int_{0}^{x}\frac{t^m}{(x-t)^\alpha}dt$

I want to prove that: $$\int_{0}^{x}\frac{t^m}{(x-t)^\alpha}dt=\frac{\Gamma(1-\alpha)\Gamma(m+1)}{\Gamma(m-\alpha+2)}x^{m-\alpha+1}$$ where $m$ is a positive integer and $\alpha \in [0,1]$. I ...
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### Integrating a product of complicated exponential function and error function

I have a problem with the following integral $$\int_0^\infty\dfrac{{\rm e}^{-t-(x^2-a^2)/t}}{t}Erf\left(\frac{a}{\sqrt{t}}\right)\,{\rm d}t,$$ where $0\le a<x$ and Erf stands for the error ...
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### Functions that satisfy the identity $f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$

I am looking for function(s) which satisfy the following property: $$f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$$ I am not sure if there is any function ...
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### combinatorial identity involving fraction and product of bionomial coefficients

How can I prove the following identity for $i\geq 1$: $$\sum_{t=i}^{s-1} \frac{i}{t}\binom{2(s-t-1)}{s-t-1}\binom{2t-i-1}{t-1}= \binom{2s-i-2}{s-1}.$$ Perhaps I'll need to go to hypergeometric ...
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### The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental

Prove that the number: $$\sum_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$$ is transcendental. I don't have a direct proof but a round one. The series can be expressed in terms of $\vartheta_3$ ...
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### What is an example of a polynomial of degree as small as possible which meets this condition?

If $f$ is a function, what polynomial is a good approximation of order $n$ for $f$ near $x=0$? Here we say that $P$ is a good approximation of order $n$ for $f$ near $x=0$ when $E(x)$ approaches $0$ ...
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### Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx$ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
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### How can I scale a value when it is within a threshold?

I am not a mathematician so I'm not even sure of the correct language to describe this. I also don't know what appropriate tags are for this question so please amend as necessary. I am looking ...
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### Derivatives wrt order of MacDonald function

I'm looking for a closed-form expression for $$\left.\left[\frac{\partial^n}{\partial \nu^n}K_{\nu}(z)\right]\right|_{\nu=\pm\tfrac{1}{2}},\;\;n\ge1$$ where $K_{\nu}(z)$ denotes the MacDonald ...
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### Deriving Hermite polynomial derivative recurrence relation straight from differential equation.

I want to derive the derivative recurrence relation for the Hermite polynomials straight from the Hermite differential equation. That is, I want to go from left to right in the following sequence ...
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I have the following integral that I am trying to solve $$I= \int_0^\infty e^{-\beta x} x^{\mu-1} \tilde{\gamma}(\nu, \alpha x)^\xi dx$$ where $\beta \in \mathbb{R}^+$, $\nu \in \mathbb{R}^+$, $\xi ... 0answers 40 views ### About the domain of the Gamma function I started to read about the history of the Gamma Function. There are three places I liked most, The early history of the factorial function (p. 239 - 243) Leonhard Euler's Integral: An Historical ... 1answer 39 views ### Name of a particular improper integral I am curious if there is a particular name for this,$\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions? 4answers 272 views ### Solving$\ln{x}=\tan{x}$with infinitely many solutions Lets take$f(x)=\ln{x}$and$g(x)=\tan{x}$When$f(x)=g(x)$that is$\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to$x$but the two ... 1answer 27 views ### Asymptotic limit of the following integral? I am interested in the asymptotic limit of the following integral for$a\rightarrow\infty$, $$\int_0^1\mathop{\mathrm{d}x}J_2(ax)x^n,$$ where$n>-1$and$J_2(x)$is the Bessel function of first ... 0answers 293 views ### Calculate$\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$Prove that: $$I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2$$ Using ... 1answer 34 views ### Legendre Polynomial definite integral identity I'm doing a problem involving legendre polynomials and I got stuck in this integral: $$I_k=\int_{-1}^{1} x P_{2k+1}(x)dx$$ Update: Note that the function in the integral is even If$k=0$, then ... 0answers 26 views ### Differential equation with variable coefficients I saw this differential equation somewhere $$y''+xy=0$$ it was solved using the substitution $$y=x^\alpha u$$ where$\alpha$, is a constant. My question is how can we substitute for$y$with an ... 1answer 101 views ### Approximating hypergeometric function F(1,1+a,2+a,z) for z->1 in my studies a normalization constant for a pmf includes the hypergeometric function${}_2F_1(1,1+a,2+a,z)$The parameters are in the range$0.99<z<1$and$0<a<5$. I have tried some ... 2answers 662 views ### Definite integral involving modified bessel function of the first kind I would like to solve the following integral that is a variation of this one (Integral involving Modified Bessel Function of the First Kind). Namely, I have: $$\frac{1}{\sqrt{2\pi w^2}}\int_{-\infty}... 0answers 29 views ### Infinite sum of a product of hyperbolic functions, help!! Let g_{a,b}=\mathrm{csch}(n(a-b)) when a is different from b and 0 if a=b. n is a positive real. I am trying to compute the following sum \sum_{k=0}^{\infty}(2k-a)g_{0,k}... 2answers 44 views ### Does anyone know a function that can describe a harmonic series? I want to find a function that satisfies the following functional equation: F(z+1)=1/z+F(z) This is a generalization of harmonic series 1 + 1/2 + 1/3 + 1/4 + ...,... 0answers 16 views ### On Hyper-geometric function differential equation The hypergeometric function$$_2F_1(a,b;c+n:z) = \sum_{m=0}^\infty \frac{(a)_m(b)_m}{(c+n)_m}\frac{z^m}{m!}$$should satisfy the differential equation$$z(1-z)\frac{d^2u}{dz^2} + [c+n-(a+b+1)]\frac{... 0answers 26 views ### On Hyper-geometric Functions and its recurrence relation I research in generating functions of Hyper-geometric functions$_2F_1(a+n,b;c+n;x)$using Lie group theoretic method and so the recurrence relation is important in this method. I want recurrence ... 1answer 198 views ### Closed form double integral$ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$Is there a closed form expression for $$S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}}$$... 1answer 34 views ### Want to check that$\sum_{j=0}^{k-1}w^{ jm}=0$,$m\not\equiv 0 \pmod{k}$where$w=e^{2\pi i/k}$If$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$, then $$\sum_{n=0}^{\infty}a_{kn+m}x^{kn+m}=\frac{1}{k}\sum_{j=0}^{k-1}w^{-jm}f(w^j x) \tag{1},$$ where$w=e^{2\pi i/k}$is a primitive$k$th root of ... 2answers 43 views ### How to prove this gamma identity? How to prove this? $$2^n \ \Gamma(n+\frac{1}{2})\ =\ 1.3.5...(2n-1)\ \sqrt{\pi}$$ I tried rewriting the right-hand side as $$\frac{(2n-1)!}{2(n-1/2)}\ \sqrt{\pi}=\frac{\Gamma(2n)}{2\Gamma(n+1/2)}\sqrt{... 1answer 74 views ### Evaluating an integral by substitution and special functions [duplicate] How can I evaluate this integral?$$\int_{0}^{1} \frac{dx}{\sqrt{{1+x^4} }}$$I tried using the substitution x=\mathrm{e}^{-u} but I got nowhere. 3answers 152 views ### Hypergeometric function integral representation How to prove the following relation?$$ \, _2{F}_1(K,K;K+1;1-m) = \frac{\Gamma (K+1)}{\Gamma (K)} \int_0^{\infty } \frac{1}{(1+x) (m+x)^K} \, dx $$where _2{F}_1(.,.;.;.) is the hypergeometric ... 1answer 57 views ### Show some properties of the Digamma Function Let \psi(z) denote the Digamma function, \psi(z)=\frac{d}{dz}\ln \Gamma(z)=\frac{\Gamma'(z)}{\Gamma(z)}. I am meant to show the following properties of \psi: \psi is meromorphic in \mathbb{... 1answer 65 views ### Integral representation of Bessel function K_v(y) = \frac{1}{2} \int_{0}^{\infty} t^{v-1} \text{exp}(-\frac{1}{2}y(t+t^{-1}))\text{d}t. How does one find the following representation of the bessel function K_v(y):$$K_v(y) = \frac{1}{2} \int_{0}^{\infty} t^{v-1} \exp \left(-\frac{1}{2}y\left(t+t^{-1}\right) \right)\,\mathrm{d}t.$$I ... 1answer 116 views ### An elliptic integral? I ran into an integral a little while ago that looks like an elliptic integral of the first kind, however I am having trouble seeing how it can be put into the standard form. I've tried messing ... 5answers 87 views ### What are some functions that respect the following criteria? : f(1/x) = f(x) and \int_{0}^{+\infty} f(x) dx = 1 I'm looking for some functions that respect these six criteria: f is defined on [0 ; +\infty[ f is differentiable everywhere in [0 ; +\infty[ f(0) = 0 \lim\limits_{x \to +\infty} f(x) = 0... 4answers 64 views ### integrating this infinite gaussian integral How does one integrate \int_{-\infty}^{+\infty}x e^{-\lambda ( x-a )^2 }dx where \lambda is a positive constant. My integral tables are not returning anything useable. The best it return is ... 0answers 48 views ### Expansion of some singular kernel with the help of Bessel and Neumann spherical harmonic functions With the following notations: j_n: spherical Bessel functions, y_n: spherical Neumann function, P_n: Legendre polynomial, r, \rho, \theta, \lambda arbitrary complex, R=\sqrt{r^2+\rho^... 1answer 57 views ### Perturbation of the Upper Incomplete Gamma Function The Upper Incomplete Gamma function, for t \in \mathbb{R}, is defined as: $$\Gamma(α,β)=\int_{β}^{\infty}t^{α-1}e^{-t}dt$$ For the problem which I am studying it takes ... 2answers 559 views ### Approximation of \mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t} [duplicate] I am reading about the Riemann hypothesis, and the article mentioned the Li function:$$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$They said that this function can be approximated:$$\... 1answer 25 views ### Normalisation of Bessel functions I've done the integration by parts and obtained $$\frac{-1}{\alpha^2} \int z^2 J J'$$ but I have no idea how to use Bessel's equation to simplify this as it only appears to get far more complicated.... 1answer 62 views ### Solving differential equation$y''(x)+Q(x)y(x)=0$[closed] How to solve the following differential equation $$y''(x)+Q(x)y(x)=0$$ And how to find exact solution$y(x)$in terms of special functions? 0answers 20 views ### Division of half-integer order legendre functions of the second kind with different arguments I'm in search of a formula for:$\frac{Q_{n-\frac{1}{2}}(\chi_1)}{Q_{n-\frac{1}{2}}(\chi_2)}= ??$where I am hoping the result to be a function of$\frac{\chi_1}{\chi_2}\$. Does anyone know of such ...
I can not solve the following integral using the hypergeometric function: $$\int_a^b (\sin x)^{(1/n)}dx$$ Wolframalpha showed the following result. but I do not understand how Wolframalpha came ...