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

learn more… | top users | synonyms

19
votes
0answers
532 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
0
votes
1answer
38 views

Improper integral involving sinc function and Pochhammer symbol

Can anyone please advise me how to integrate expressions of the form $\text{sinc}\,(x) / (1-x)_n$ along the real axis? Using a CAS, one could suggest that $$ n! \int_{-\infty}^\infty \frac{\sin \pi ...
1
vote
1answer
39 views

Asymptotics of a series involving cos integral functions

I'm looking for the asymptotic expansion( or value ) of the following function \begin{equation} F[y,t] = \sideset{}{'}\sum_{n \in \mathbb{Z}}\text{Ci}\big[\frac{n^2}{t}\big] - ...
1
vote
1answer
27 views

Are polylogarithms the simplest functions that decay exponentially in one limit, and grow polynomially in another limit?

I have a function $f(x)$ which is defined as the solution to a certain differential equation. The boundary conditions are that in the $x\rightarrow \infty$ limit, it should be asymptotically ...
0
votes
1answer
18 views

Mixing distribution.

Let $\theta$ in $[0,1]$ and defined for $u,v\in [0,1]$, $$C_\theta (u,v)=\begin{cases}\min(u.v),&&|v-u|\ge \theta\\ \max(u+v-1,0), && |u+v-1|\ge 1-\theta\\ (u+v-\theta)/2,&& ...
1
vote
1answer
31 views

Definition of elementary and special functions

This is a (perhaps) naive question, but one that I have been thinking about lately. Is it a true statement that all functions (elementary or special) can be defined as the solution to a particular ...
1
vote
0answers
28 views

Does asymptotic expansion of Whittaker function $W_{\lambda , \mu}(z)$ exist for $|\lambda| \to 0$?

Suppose Whittaker function $$ \tag 1 W_{\lambda , \mu}(z) $$ Does some asymptotic expansion exist for the case $|\lambda| \to 0$? I'm interested not in the case of $\lambda = 0$, but in the case of ...
3
votes
1answer
43 views

Asymptotics of this HyperGeometric Function

I have a function $$f(x)=x^{2m}\text{ }_2F_1\left(\frac{1}{2},-m;\frac{3}{2};-\frac{1}{x^2}\right)$$ where $x>0$. I am interested in asymptotics in the two extreme limits: $$\lim_{x\rightarrow 0} ...
1
vote
2answers
62 views

Closed-form solution to $\frac{\ln x}{x} = k$

What is the solution in $x$ to $$\frac{\ln x}{x} = k ?$$ I suspect this has something to do with the Lambert W function, since that's used in solutions of the form $x\ln(x) = k$, but the ...
0
votes
0answers
12 views

The Wronskian of parabolic cylinder function and the plane wave

Suppose equation $$ \tag 1 \ddot{y} + (t^2\theta (t - t_{i}) + p^2)y(t) = 0, \quad t \in (t_{0}, \infty) $$ (here $\theta (t - t_{i})$ is the step function) with initial condition $$ \tag 2 y(t \to ...
1
vote
1answer
38 views

The expression of the sum of infinite gaussian functions

Let $f(x|\mu,\sigma^2)$ be the gaussian function (normal distribution): $$f(x|\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}}e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$$ We know its integral over $\mathbb{R}$ is ...
1
vote
0answers
34 views

general procedure for contour integration of $\int_{0}^{\infty} \mathrm{Ai}(x)^{n} dx$

In Richard Crandall's On The Quantum Zeta Function, following eq. 4.11: $$ \int_{0}^{\infty}\mathrm{Ai}(x)^{2}dx=\frac{\Gamma(\tfrac{5}{6})}{2\pi^{5/6}12^{1/6}} $$ “again derivable by contour ...
0
votes
0answers
12 views

The domain of the Digamma function and its extension

First, we know that $\Gamma(x)>0$, for all $x>0$, so define $\psi(x)=\frac{\mathrm{d} }{\mathrm{d} x}\ln(\Gamma(x))=\Gamma'(x)/\Gamma(x)$, for all $x>0$. This is the Digamma function. It is ...
1
vote
0answers
79 views

Definite integral $\int_{0}^{\infty} e^{-a t} \log(t)\log(1+t)\,dt$

Is there a closed-form expression (possibly in terms of special functions) for the integral: $$ \int_{0}^{\infty} e^{-a t}\log(t)\log(1+t)\,dt, $$ where $a>0$?
0
votes
1answer
25 views

Integral of a power of the complementary error function

I would like to know if is it possible to calculate analytically the following integral: $$J=\int_0^{x_0}\operatorname{erfc}(x)^kdx$$ with $k=2,3,4,...N$ where $\operatorname{erfc}(x)$ is the ...
0
votes
2answers
31 views

Write function $F$, based on parameters?

Let $$U=x+y+z\, ,$$ $$V=xy+yz+zx\, ,$$ $$W=xyz\, .$$ and we have $F(U,V,W)=x^4+y^4+z^4$. My question it is, How to write function $F$, based on parameters $U$ , $V$ and $W$?
1
vote
1answer
37 views

Inequality for Gamma function

Prove that $$0<\frac{\Gamma(x+y)}{\Gamma(xy)-1}\leq3$$ for all $x>0,y>0, xy>2.$ And equality holds $x=y=2.$
6
votes
1answer
240 views

Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$

I've conjectured the following identity for $n\geq0$ integers: $$ \int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx = (-1)^{n+1}n! \cdot \left(-\operatorname{Ei}(1)+\sum_{k=1}^{n+1} ...
1
vote
1answer
25 views

The Wronskian of parabolic cylinder functions

Suppose I have the second order differential equation $$ y''(t) + (k^{2} + m^{2}t^2)y(t) = 0, \quad 0< t_{0} < t < \infty $$ The solution of this equation is parabolic cylinder functions, ...
1
vote
0answers
41 views

Where was the mistake

we know that $$\frac{\pi^2}{6}=\int_{0}^{\infty}\frac{t}{e^t-1}dt$$, we also know that $\frac{t}{e^t-1}$ is the generating function for the Bernoulli numbers i.e $ \frac{t}{e^t-1} ...
0
votes
0answers
13 views

Support width of the difference of two rect functions.

We let $\operatorname{rect}(t)\equiv \chi_{(-\frac 12, \frac 12)}(t) $ when $|t|\ne \frac 12$ and $\operatorname{rect}(t)=\frac 12$ if $|t|=\frac 12$. Let $\operatorname{rect}(t_1,t_2)\equiv 1$ when ...
0
votes
0answers
29 views

About asymptotic expansion of parabolic cylinder functions

Let's have the parabolic cylinder function $U(a,z)$. I'm interested in its asymptotics for large argument $z$. Here I've found it, but I'm a bit confuzed now because of expressions $(12.9.1)$ and ...
14
votes
1answer
195 views

Proving that $\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$

I am trying to prove that $$I=\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$$ where $\beta(s)$ is the Dirichlet Beta function and $G$ is the Catalan's constant. I managed ...
2
votes
0answers
52 views

How to evaluate following integral?

Suppose the integral $$ \tag 1 I = \int \limits_{-\pi}^{\pi}dx \int \limits_{-\pi}^{\pi}\frac{dy}{\tau - \cos (2x) -2\cos(x)\cos(y)}, \quad t > 3 $$ How to evaluate it in terms of elliptic ...
2
votes
1answer
25 views

$\operatorname{Re}(\operatorname{Li}_3(z))$ for real $z\geq1$ in terms of elementary functions?

According to the article by Wood, D. "The Computation of Polylogarithms. Technical Report 15-92*", listed in the references about the polylogarithm on the Wikipedia, there is a form in terms of ...
23
votes
4answers
746 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it known any ...
5
votes
1answer
138 views

Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
91
votes
0answers
2k views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
0
votes
1answer
45 views

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 ...
1
vote
0answers
70 views

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 ...
4
votes
1answer
259 views

Limit involving the inverse beta regularized function

Let $0<p<\frac{1}{2}$. I am looking for the limit: $$\lim_{t \to \infty} \left(\frac{t}{\frac{t}{I_{2 p}^{-1}\left(\frac{t}{2},\frac{1}{2}\right)}-2 \sqrt{t} \sqrt{\frac{1}{I_{2 ...
1
vote
0answers
35 views

How to make analytic continuation and compute imaginary part

Suppose I have the function $$ \tag 0 G(x) = g(x)K\left(k(x)\right), $$ where $$ k(x) = \frac{4\sqrt{x}}{(x-1)^{\frac{3}{2}}(x+3)^{\frac{1}{2}}}, $$ $$ g(x) = \frac{2}{\sqrt{x}}k(x), $$ and $K(x)$ is ...
7
votes
1answer
296 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
5
votes
0answers
48 views

Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...
6
votes
2answers
122 views

Integral of two logs and a power: $\int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du$

Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du.$$ The parameters are possibly complex, and satisfy ...
1
vote
2answers
33 views

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 ...
2
votes
1answer
50 views

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 ...
6
votes
1answer
155 views

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 $\vartheta_3$ ...
1
vote
1answer
31 views

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 $0$ ...
51
votes
6answers
5k views

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 ...
1
vote
0answers
18 views

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 ...
1
vote
0answers
34 views

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 ...
0
votes
1answer
44 views

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 ...
2
votes
0answers
61 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity ...
7
votes
1answer
150 views

How can I evaluate $\int_{0}^{1}\frac{(\arctan x)^2}{1+x^{2}}\ln\left ( 1+x^{2} \right )\mathrm{d}x$

How to calculate this relation? $$I=\int_{0}^{1}\frac{(\arctan x)^2}{1+x^{2}}\ln\left ( 1+x^{2} \right ...
2
votes
1answer
41 views

Hermite Polynomials: Rodrigues to Integral Representation

I would like to go from this representation of the Hermite polynomials: $$H_n(z)=(-1)^ne^{z^2}\frac{d^n}{dz^n}e^{-z^2} \tag{1}$$ to this representation ...
0
votes
1answer
50 views

Modified Bessel Function Integral representation proof $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{2}/4t}dt $

How do I proof the following integral representation for the Modified Bessel function of the second kind (Macdonald Function). ...
0
votes
0answers
29 views

Integral involving power of incomplete gamma function

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 ...
3
votes
0answers
39 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 ...
1
vote
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?