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

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2
votes
0answers
27 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
1
vote
2answers
109 views

Summing Lerch Transcendents

The Lerch transcendent is given by $$ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}. $$ While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} ...
1
vote
0answers
53 views

integral from gradshteyn and ryzhik

I'm interested in evaluating the integral $$ \int_{a}^\infty e^{-x\cosh\alpha}\,K_{\nu}(x\sinh\alpha)\,\frac{dx}{x}, $$ where $a>0$ and $\nu$ is purely imaginary. Here $K$ denotes the MacDonald ...
2
votes
2answers
18 views

Lipschitz-like behaviour of quartic polynomials

I have observed the following phenomenon: Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to ...
0
votes
1answer
121 views

Solve $x^a = 1 - \exp(-x)$ for $x$

I would like to obtain a closed-form solution for the equation $x^a = 1 - \exp(-x)$, in which $x$ is the (real strictly positive) unknown and $a$ is a real positive parameter. So far, I have tried ...
0
votes
1answer
40 views

Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
4
votes
1answer
281 views

Showing that the Barnes G-function satisfies the functional equation $G(z+1) = \Gamma(z) G(z) $

Let $G(z)$ be the Barnes G-function. I want to use the infinite product representation $$ G(z+1)=(2\pi)^{z/2}\text{exp}\left(-\frac{z(z+1)}{2}- \frac{\gamma z^{2}}{2}\right)\, ...
25
votes
1answer
639 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 ...
1
vote
1answer
44 views

Integral that resembles an exponential integral

$$ I(y;c,\lambda) \equiv\int_{0}^\infty \frac{\lambda c}{x} \exp\left(-\lambda x\right)\exp\left(-\frac{c}{x}y\right)dx$$ where $c,\lambda>0$. Q: Can this integration be made in analytic form ...
1
vote
1answer
65 views

A differential equation I

Consider the second order differential equation \begin{align} 2 t^{3} y'' + (5 t^{2} - t) y' + (t^{2} - t + 1) y = 0 \end{align} with the conditions $y(0) = 0$ and $y'(0) = 1$. A solution is known in ...
6
votes
0answers
48 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
3
votes
3answers
261 views

Approximate Riemann zeta function

Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$. In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function. My question is: Is there a Functional equation for ...
3
votes
0answers
112 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
4
votes
0answers
38 views

Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
1
vote
1answer
261 views

asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
7
votes
1answer
152 views

Computing a double gamma-digamma-trigamma series

What are your thoughts on this series? $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k)^2 \Gamma (n) }{\Gamma (2 k+n)}((\psi ^{(0)}(n)-\psi ^{(0)}(2 k+n)) (\psi ^{(0)}(k)-\psi ^{(0)}(2 ...
6
votes
3answers
189 views

Why is the Gamma function off by 1 from the factorial? [duplicate]

Why didn't they define it as $$ \tilde \Gamma(x) = \int_0^\infty t^x e^{-t} \, dt ?$$ Then the definition would have two less characters than the standard definition of $\Gamma(x)$, and we would have ...
1
vote
3answers
36 views

Switching the order of summations of a certain function

I am looking to switch the order of the summations of the following function: $$ \lambda = -\sum_{c=1}^{n-1} \sum_{k=c}^n {k \choose c} \frac{(-1)^k}{k!} f^{k-c}U(-c,k-2c+1,-f)\phi(n,k) $$ I don't ...
0
votes
2answers
31 views

Logarithmic derivative of Polygamma functions

While studying Gamma function and related functions I noticed that its logarithmic derivative (the so-called Digamma function) is studied more than its "normal" derivative but on the other hand I ...
1
vote
0answers
75 views

Derivation of approximation of Error function

In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). ...
2
votes
1answer
72 views

Autocorrelated, discrete, bounded and symmetric random walk with no edge attraction

I need to move over a discrete set of linearly organized.. let's say "Japan steps" $S=\{0,\dots,c\}, c \in \mathbb{N}^*$. My current position is given by $d \in S$. On each time step, I need to draw ...
4
votes
0answers
76 views

Is there a name for this type of integral $\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$?

Given a polynomial of arbitrary degree, $P(x)$, on $[a,b]$ is there a name for this type of integral: $$\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$$
1
vote
1answer
100 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
1
vote
2answers
71 views

Looking for a function that fits a certain criteria

I am looking for a function that fits this description: $$ \frac{d^n}{dx^n}[f(x)] = n! f(x) $$ or $$ \frac{d^n}{dx^n}[f(x)] = (n-1)! f(x) $$ For all values of $n$, with this function i am looking to ...
5
votes
2answers
7k views

Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
0
votes
1answer
36 views

How to find the derivative of $|f(x)|$

The original question was to find domain of derivative of $y=|\arcsin(2x^2−1)|$. First method My attempt was to break $y$ into intervals ,i.e., where $\arcsin(2x^2−1)\geq 0$ and where ...
14
votes
3answers
248 views

Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form

What tools other than beta function you might like to use here? $$\sum _{k=1}^{\infty } \frac{\displaystyle \Gamma \left(\frac{k}{2}+1\right)}{\displaystyle k^2 \Gamma ...
3
votes
1answer
87 views

A possible dilogarithm identity?

I'm curious to find out if the sum can be expressed in some known constants. What do you think about that? Is it feasible? Have you met it before? $$2 ...
11
votes
0answers
124 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
0
votes
2answers
37 views

Other forms for the derivative of the Gamma function

When I searched for the derivative of the Gamma function I got something of the form: $$\Gamma'(x)=\Gamma(x) \psi(x)$$ But from the definition of the Digamma function to me it's like writing: ...
0
votes
0answers
29 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 ...
4
votes
1answer
58 views

$\int x J_0(k x)e^{-x^2/2}dx$ Bessel function decomposition of a gaussian

$$\int ^\infty _0 x J_0(k x)e^{-x^2/2}dx$$ The integral above corresponds to fourier transform in radial coordinates. The fourier transform of a 2D gaussian is still a 2D gaussian. So the integral ...
0
votes
2answers
51 views

Finding the inverse of the function $f(k, x) = k^{x}x.$

Recently, I have been looking at the function $f(x) = e^{x}x,$ where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ...
6
votes
1answer
115 views

Does $ \sum_{(m,n) \neq (0,0)} \frac{(-1)^{m+n}}{m^2 + n^2} $ have an exact value?

I am looking for an $\mathbb{Z}[i]$ analogue of the alternating harmonic series: $L(1,\chi)=\sum_{n=0}^\infty (-1)^n \frac{1}{n} = \frac{\pi}{4}$. If we try adding the reciprocals of the Gaussian ...
3
votes
1answer
141 views

A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
1
vote
1answer
43 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
0
votes
2answers
45 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
12
votes
1answer
150 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
12
votes
4answers
300 views

Evaluating $\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$

I was trying to find a closed form for $$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz = -2.454199511\cdots$$ where $\text{Li}_2(z)$ is the ...
0
votes
0answers
20 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
1
vote
0answers
40 views

An integral involving the Gamma function

By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} ...
0
votes
0answers
16 views

Laplace Transform of Kelvin functions

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? ...
8
votes
1answer
259 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
0
votes
1answer
22 views

Properties of greatest integer function

I am curious to know some properties of the floor functions, for instance, $\lfloor a \cdot x \rfloor$, $\lfloor a1\cdot x1+a2\cdot x2 \rfloor$, etc. Is there any book that contains such properties ?
9
votes
2answers
124 views

Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, ...
2
votes
1answer
229 views

How to prove this integral representation of Bessel function?

I'm trying to prove this integral representation: $$J_\nu = \frac{({x/2})^\nu}{\Gamma(\nu+1/2)\sqrt{\pi}}\int_{-1}^{1}(1-t^2)^{\nu-1/2}e^{itx}dt$$ I think countour integration would be useful, but ...
1
vote
0answers
31 views

Series involving Laguerre polynomials

Given the series \begin{align} S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a) \end{align} where $L_{m}(x)$ is the Laguerre polynomial. By using \begin{align} L_{n}(z) = ...
4
votes
1answer
56 views

Simplifying real part of hypergeometric function with complex parameters

I am looking for a simpler representation of the following hypergeometric function with complex parameters in terms of more basic functions and manifestly real parameters: ...
2
votes
0answers
71 views

Conway's box function generalized as a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
8
votes
4answers
152 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.