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

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8
votes
2answers
224 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
45 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
32 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
58 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
7 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
162 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 ...
3
votes
1answer
74 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
32 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
43 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
51 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
52 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
91 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
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
31 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
60 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
203 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
35 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
110 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} ...
45
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
50 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
27 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
48 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
30 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
153 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
51 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
91 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
116 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
75 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
66 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
60 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
61 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
120 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
316 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
32 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 ...
2
votes
1answer
129 views

Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. $\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$ Is there a closed form (free of polylogs and ...
1
vote
0answers
28 views

Estimate lower incomplete gamma function $\left|\gamma(n,z)-\gamma(n,-z)\right|$.

From Wikipedia, the upper incomplete gamma function is defined as: $$\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t ,\,\!$$ whereas the lower incomplete gamma function is defined as: ...
11
votes
3answers
210 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where ...