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

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11
votes
1answer
237 views

Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
1
vote
0answers
14 views

Show that $J_n(x)$ satisfies Bessel equation $ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $

Here is the definition of the Bessel function I am starting with a definition as an integral. $$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$ Essentially we have computed ...
27
votes
1answer
864 views

elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for ...
0
votes
1answer
26 views

The derivatives of Riemann xi function

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
3
votes
2answers
120 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
1
vote
0answers
35 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
1
vote
1answer
37 views

Function with infinite maxima and minima [on hold]

Can you please give an example of a function with an infinite number of maxima and minima occurring in any finite time interval? Edit: This question came to me as I was reading on the dirichlet ...
3
votes
1answer
31 views

A formula for length of representation of a number in a “base” without zeros

If you had 2 items the sequence would go like this: $$1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5, \ldots$$ This is $\lfloor\log_2(n+2)\rfloor$. What if I ...
1
vote
1answer
64 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
5
votes
1answer
525 views

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 ...
0
votes
1answer
47 views

Book recommendation on special functions

I am currently studying real analysis from rudin and really like the chapter on special functions. But Rudin does not give much knowledge about those topics. Reading the references I found book by ...
0
votes
0answers
6 views

Giving two examples of functions with some properties.

This is a question from a list. Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties: $f(x)=0 \Leftrightarrow 0\leq x\leq 1$; $g(x)=x$ if $|x|\leq 1$, ...
0
votes
1answer
8 views

Legendre functions - Derivation of the recursion relation

From the following: $$\sum_0^\infty [ n(n-1)a_nx^{n-2} - n(n-1)a_n x^n -2na_nx^n + l(l+1)a_n x^n ] = 0$$ (a) I'm trying to get to: $$\sum_0^\infty [ (n+2)(n+1)a_{n+2} - [n(n+1) + l(l+1)]a_n]x^n = ...
5
votes
0answers
154 views

Relationship between Dixonian elliptic functions and Borwein cubic theta functions

In this paper, it says that the three Borwein cubic theta functions obey the identity $a(q)^{3}=b(q)^{3}+c(q)^{3}$, which is strongly reminiscent of the identity that Dixonian elliptic functions obey ...
4
votes
2answers
3k views

Integrating Legendre Polynomials over half range

Solving for the potential of a conducting sphere with hemispheres at opposite potentials, (not using Green's function) I am stuck at this point: $$I_l = V_1 \int_0^1 P_l(x)dx+V_2 \int_{-1}^0 ...
2
votes
1answer
57 views

Integral with incomplete gamma function

I am trying to solve this integral: \begin{equation} \frac{1}{c^{b}}\int_{0}^{\infty} x^{n}\, e^{-a x}\, \gamma(b,c(-d+x)) \ \mathrm{d}x \end{equation} where, $n>0$ is an integer, and $a$, $b$, ...
0
votes
0answers
24 views

integration of product two lower incomplete gamma function and exponential [closed]

i need helping to find the value of this integration : $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$ where all parameters are positive. Can anyone help me how ...
1
vote
1answer
24 views

About Beta function $B(\alpha,r\alpha +1)‎\rightarrow‎ 0$

I want to show that $$B(\alpha,r\alpha +1)‎\rightarrow‎ 0$$ when $r‎\rightarrow‎ \infty$ and $0< \alpha <1$. with thanks
1
vote
1answer
36 views

integral include lower incomplete gamma square [closed]

I am trying to calculate the following integral: $$ \int_0^{\infty}e^{-\beta x}\{ \gamma(\alpha,\theta x)\}^{2} dx $$ where all parameters are positive. i need helping , thanks!
-1
votes
0answers
22 views

integral with two lower incomplete gamma

Can I get a step by step answer to this integration ? $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$
0
votes
0answers
7 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
0
votes
1answer
40 views

integral include lower incomplete gamma

I am trying to calculate the following integral: $$ \int_0^{\infty}e^{-\beta x}\gamma(\alpha,\theta x)dx $$ where all parameters are positive. Any help , Thanks!
1
vote
1answer
37 views

Integrals involving whittaker functions.

I want to compute the following integrals: $$ \int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy $$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind. I already know that ...
3
votes
1answer
48 views

Computing the limit of an integral

Consider the following integral $$ \int_{-\infty}^{\infty}f(t) K(\frac{a-t}{h})dt $$ where (1) $h>0$, $a \in \mathbb{R}$ (2) $f:\mathbb{R}\rightarrow[0,\infty)$ is such that ...
4
votes
1answer
86 views

Calculate (or estimate) $S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$.

Let $x\in\mathbb R$, $x>1$ and $$S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$$ where $\zeta(x)$ is the Riemann zeta function. Calculate (or estimate) $S(x)$.
1
vote
0answers
68 views

Limit of this monster function [closed]

Here's a function, and I am interested in finding its limit when $z\rightarrow \infty$. Any idea would be really appreciated. First thing I need to know is if this limit really exists. ...
9
votes
2answers
1k views

Are there well known lower bounds for the upper incomplete gamma function?

Let $a >0, b >0$, and $r \in \mathbb{R}$. I am trying to find a lower bound for the integral $$\int_a^\infty y^{-r} \exp\left( - b(y-a)^2\right) \,\mathrm dy.$$ After consulting the Wikipedia ...
1
vote
1answer
27 views

How to prove this limit of Airy Function.

I have no idea how to prove this limit $$\lim_{x\rightarrow \infty }\exp\left ( \frac{2x^{3/2}}{3} \right )\sqrt[4]{x}\mathrm{Ai}\left ( x \right )=\frac{1}{2\sqrt{\pi }}$$ where ...
3
votes
0answers
112 views

Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where ...
16
votes
1answer
375 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
0
votes
0answers
13 views

Behavior of $J/I$ w.r.t $m_1$, $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$

Let us define $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$. We assume that $m_1 \ge 0$, $k \ge 0$ and $k \le N$. Using the ...
0
votes
0answers
36 views

Is there a finite set comprising the solutions to indefinite integrals of common functions?

There are some integrals that are impossible to express in terms of elementary function, for example, $ \int \frac{e^x}{x} dx $ is only expressible as a "special" function $Ei(x)$, the exponential ...
1
vote
1answer
54 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
21
votes
1answer
193 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...
1
vote
0answers
28 views

Integration involving Error function

I am interested in the following integral $$\int_0^\infty x^n\text{Erf}[ax]j_m(bx)dx,$$ where Erf is the error function $j_n$ is the spherical Bessel function of first kind. Does the analytical ...
0
votes
0answers
97 views

How to verfy if the approximations of the complex error function have no poles?

I found an article published few days ago in arXiv:1601.01261 that shows a very simple Matlab code for computation of the complex error function (aka the Faddeeva function) defined as \begin{equation} ...
43
votes
1answer
2k views

Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?

Is it possible to simplify this expression? $$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$ Is there a systematic ...
3
votes
2answers
49 views

Indefinite Bessel integrals

I just ran into integrals of the Bessel type, but which are unfortunately indefinite integrals, such as $$ f(t)=\int \cos(\gamma\cos(\omega t))\cos(\omega t)\mathrm dt. $$ I'm conscious of the fact ...
0
votes
0answers
25 views

What the inverse function of $_2F_1(a,b;c;z)$

What the inverse function of the function $f(z)$ given by $$ f(z) = \, _2F_1(a,b;c;z), \quad \mid z \mid <1, $$ where is the Gauss hypergeometric function given by $$ ...
2
votes
1answer
42 views

Elliptical Integral that diverges at one point

I have to solve the following integral $$I=\int_{\lambda_1}^yd\lambda\frac{1}{1-\lambda}\sqrt{\frac{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_4)}{\lambda-\lambda_3}}$$ where ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
1
vote
1answer
28 views

Integral representation of the modified Bessel function involving $\sinh(t) \sinh(\alpha t)$

I've come across this peculiar integral representation for $K_\alpha(x)$: $\frac{\alpha}{x}K_\alpha(x) = \int_0^\infty dt \sinh(t) \sinh(\alpha t) e^{-x \cosh(t)}$ How does it come about? Are there ...
5
votes
2answers
147 views

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

How could one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt\,?$$ I have to apply special functions, so I thought that I have to use the change variable $$u=t^4,$$ but $$du=4t^3\,dt$$ and when ...
1
vote
3answers
50 views

How could I solve $\int_{-\infty}^{+\infty} x^2e^{-x^2}dx$ apply special function gamma

I try solve the integral $$\int_{-\infty}^{+\infty} t^2e^{-t^2}dt$$ I do not know but I think that I should apply $gamma\ function$, which is $$ \Gamma (x)=\int_{0}^{\infty} t^{x-1}e^{-t}dt$$ Like ...
0
votes
2answers
85 views

Evaluating an integral involving $(a-x)^ne^{-1/x}/x^2$

Let $a>0$ be a small parameter and consider a fixed integer $n\geq 0$. Is it true that $$ \int_0^a \frac{(a-x)^n e^{-1/x}}{x^2}\ dx=n! a^{2n}e^{-1/a}(1+O(a)) $$ as $a\to 0$? I have verified this ...
4
votes
2answers
96 views

Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for ...
2
votes
2answers
68 views

Integration of Associated Legendre Polynomial

I am interested in the following integral $$I=\int_{-1}^1P_\ell^2(x)P_n(x)\mathrm{d}x,$$ where $P_n(x)$ is Legendre Polynomial of $n$th order, and $P_\ell^2$ is Associated Legendre Polynomial. Any one ...
9
votes
0answers
169 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
2
votes
1answer
61 views

Calculating $I=\int_{-1}^1{\dfrac{1}{\sqrt{1-x}}P_n(x)} \, dx$ where $P_n$ is a Legendre Polynomial.

Calculating $$I=\int_{-1}^1{\dfrac{1}{\sqrt{1-x}}P_n(x)} \, dx$$ Where $P_n$ is a Legendre Polynomial. My progress: For any integral of the form: $$\int_{-1}^1{f(x)P_n(x)} \, dx$$ Usinng Rodrigues ...
28
votes
1answer
722 views

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 ...