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

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12
votes
1answer
256 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 = ...
0
votes
0answers
33 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
889 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 ...
8
votes
2answers
229 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
1answer
57 views

Function with infinite maxima and minima [closed]

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
36 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
68 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 ...
0
votes
1answer
64 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
8 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
10 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
163 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
4k 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 ...
4
votes
1answer
91 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 $a$, $b$, ...
1
vote
1answer
33 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
0
votes
0answers
14 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
49 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
57 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
53 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
92 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) $S(x)$.
9
votes
2answers
2k 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
33 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 ...
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
61 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
213 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
39 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
111 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} ...
46
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
51 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
28 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 $$ ...
3
votes
1answer
57 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
56 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
31 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
156 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
54 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
92 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
121 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
78 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 ...
2
votes
1answer
67 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 ...
0
votes
0answers
54 views

Prove that this expression involving $_2 F_1$ and Gamma functions is identically zero

While attempting an answer to the question Evaluation of $\displaystyle \int_{1}^{3}\left[\sqrt{1+(x-1)^3}+(x^2-1)^{\frac{1}{3}}\right]dx$, after a few manipulations I came across the following ...
1
vote
1answer
62 views

Meijer G-function limit for $z\rightarrow\infty$

I am trying to understand if the integral $$f(R)=\int_a^R\frac{K_1(r)dr}{r}$$ has a finite limit for $R\rightarrow\infty$. With Wolfram Mathematica I got the following primitive: $$\frac{1}{4} ...
0
votes
2answers
1k views

Find intersection of linear and logarithmic lines

I have equations for two lines, one of which is linear and the other is logarithmic, ie: $$y = m_1 x + c_1$$ $$y = m_2 \cdot \ln(x) + c_2$$ ..and I need to find out where (if at all) these lines ...
1
vote
0answers
62 views

How To Prove The following equation?

The equation arised in the paper:Exact and asympototic representations of the sound field in a stratified ocean.That is the equation(3.12) for solving the problem $$\Delta ...
2
votes
1answer
71 views

A limit involving the Hurwitz zeta function

I want to show that $$ \lim_{s \to 1} \left( \zeta(s,a) - \frac{1}{s-1} \right) = - \psi(a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function. The only ...
0
votes
1answer
84 views

What is the following expression equal to?

What is the following expression equal to? $$z^{\alpha } \left(\, _2F_1\left(1,-\alpha ;1-\alpha ;\frac{1}{z}\right)+\, _2F_1(1,\alpha ;\alpha +1;z)-1\right)$$ The derivative of it with respect to z ...
2
votes
1answer
122 views

Re-Expressing the Digamma

I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...
8
votes
4answers
329 views

Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$

Do you think we can express the closed form of the integral below in a very nice and short way? As you already know, your opinions weighs much to me, so I need them! Calculate in closed-form ...
1
vote
1answer
35 views

gegenbauer polynomial

Usually, Gegenbuaer polynomial is denoted by $C^{(\lambda )}_{n}(x)$ with $\lambda >-1/2$. My question: is it possible to generalize Gegenbuaer polynomial for $Re(\lambda)>-1/2, \lambda \in ...
5
votes
2answers
615 views

Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled "On question 330 of Professor Sanjana" when I got confused regarding a proposition which I am unable to answer. The proposition is: $ \displaystyle ...
3
votes
0answers
53 views

Airy transform of gaussian on half-line: $\int_{0}^\infty dx\, e^{-x^2}\text{Ai}(y-x)$

Background. The Airy transform of $f$ is defined as $$\int_{-\infty}^\infty dx\, f(x)\,\text{Ai}(y-x)\;.$$ $\text{Ai}$ denotes Airy function, $$\text{Ai}(x)=\frac{1}{\pi}\int_{-\infty}^\infty ...