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

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0
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1answer
20 views

Approximation of a series containing Bessel functions

I have this series: $$\displaystyle S=\sum_{k=0}^N\left(J_k(x)-J_k(y)\right)$$ where: $J_k(\dot{})$ is the Bessel function of order $k$ with $x\in\mathbb{R}$ and $y\in\mathbb{R}$. I have to calculate ...
1
vote
0answers
37 views

Can my wrong derivation of the Gamma function be fixed?

I found the following simple but wrong derivation of the Gamma function: We start from the definition of the exponential function $$ e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} \\ \Rightarrow 1 = ...
0
votes
0answers
42 views

A discrete fourier-bessel series?

A function $f$ on an interval $[0,b]$ can be expanded as a sum of Bessel functions, using the inner product $$\int_0^b f(x) g(x) x\mathrm dx$$ under which these functions are orthogonal, for example ...
3
votes
1answer
50 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...
1
vote
4answers
50 views

construction of a curve connecting two points

Let $a,b,c$ be positive reals numbers. Assume $a<b$. I'm trying to construct a $C^1$ function (meaning a function with continuous derivative) $f$ with the following properties: $f$ is increasing ...
0
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0answers
13 views

In what ways can I extend the error function to accept complex arguments?

What are the different approaches to extending the error function to accept complex arguments? When should I favor using one approach over another?
0
votes
1answer
16 views

How to define formula for decimal places

I need to define a formula for a half unit of the smallest decimal place in unit price ($UP$), or understand if this can be defined with a formula? What I have is this $$ T_{min,max} = Amt +/- (Qty ...
0
votes
0answers
109 views

Integral of the product of two Normal distribution CDF (erf)

How do I solve the following? $$ \lim_{x \rightarrow \infty} \int_0^{x} \left[ 1 + \text{erf} \left( \frac{\epsilon - a}{b} \right) \right] \left[ 1 + \text{erf} \left( \frac{\epsilon - c}{d} \right) ...
0
votes
0answers
15 views

Integral with the Floor Function in the Limits

I have $$\int d\mathbb{r}_{i}\int\mathbb{d}r_{j} f\left(x_{i},y_{i},x_{j},y_{j}\right)$$ where $d\mathbb{r}_{i}=dx_{i}dy_{i}$, and similarly for $d\mathbb{r}_{j}$. If I want to integrate $f$ over ...
1
vote
1answer
73 views

How to bound the uniform convergence on $[0,1]$ of the Bernstein polynomials of $ e^x $ to $e^x$

I have a question: How can we prove that the Bernstein polynomial $$p_{n}(x)=\sum_{l=0}^{n} e^{l\over n}\begin{pmatrix} n\\ l \end{pmatrix}x^l(1-x)^{n-l}$$ uniformly converges $e^x$ in the interval ...
0
votes
2answers
38 views

Why is the imaginary part of the logarithm of the gamma function a square wave?

I just stumpled upon it and it made me curious. Why is the imaginary part of $\ln(\Gamma(x))$ a square wave for $x < 0$ ? The square wave has a period of 2 and a amplitude of $\pi/2$. How can one ...
4
votes
2answers
32 views

Simplification of a combination of 6 values of the gamma function

I'm trying to simplify this combination of gamma functions: ...
1
vote
1answer
33 views

$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients

Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest. Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where ...
6
votes
0answers
54 views

Function preserving exponentiation

I'm wondering what kind of function preserves exponentiation, i.e., what is an $f$ such that $f(a^b)=f(a)^{f(b)}$?
0
votes
0answers
19 views

Function property of $o$

I have a doubt regarding $o$-function. Could we write $o(\|\theta h)\|)=\theta \ o(\|h\|)$ ?
1
vote
2answers
38 views

Solution to simple functional equations

What is $\psi$ in functional equation: $$\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{n+x}}=1/2\,\Psi \left( 1/2+x/2 \right) -1/2\,\Psi \left( x/2 \right)?$$
1
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0answers
14 views

Important Functions That Are Multivariable Integrals

There are lots of "important" functions of one variable that are defined in terms of integrals for which no closed form exists, like the Gamma Function and the normal distribution. Are there any such ...
0
votes
1answer
58 views

Definite integral involving modified bessel function of the first kind and its logarithm

I'm trying to solve the following integral $$ T=\int_0^\infty \exp(-a x^2) I_1(b x) \log(I_1(b x))\, dx $$ where $I_1(x)$ is the modified Bessel function of the first kind and order one, and $a$, $b$ ...
0
votes
0answers
29 views

on convergence of Integral involving Bessel function

For $r>0$ suppose $$\chi(r)=\sqrt{r}J_{\nu}(re^{i\frac{\pi}{4}})$$ where $J_{\nu}$ is a Bessel function of order $\nu$ (which is complex number). I am trying to find for what values of $\nu$ is ...
3
votes
3answers
114 views

Which methods can be used to evaluate the following integral?

How can I evaluate the following integral $$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$ I know the answer is $\sqrt{2\pi}$.
4
votes
2answers
75 views

Simplify $\frac{\Gamma(n)}{\Gamma(n+a)}$ with $a\in\mathbb C$.

How can simplify the following expression? $$\frac{\Gamma(n)}{\Gamma(n+a)}\sim \cdots\text{ ?}$$ Where $a\in\mathbb C$, $n\in \mathbb N$. Any suggestions please? I propose the following. We have ...
1
vote
1answer
56 views

Does a function relationship for a specific $y$ hold for any?

Let $f:\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ , continous $\forall$ $y$ in $\mathbb{R}$, $\exists$ $a,b$ such that $f(x,y) = ax + b$ $\forall$ $x$ in $\mathbb{R}$, $\exists$ $a',b'$ such ...
2
votes
0answers
39 views

Fourier transform all steps walkthrough for wave vector $k$ and $x$

Below is my walkthrough of a fourier transform. My problem is that I want to do all the similar steps for a fourier transform between position x and the wave vector k. That is working on a solution of ...
1
vote
1answer
30 views

is split function derivable

$ f(x) = \begin{cases} \frac{sin(x)}{x}, & x \ne0 \\ x+1, & x=0 \end{cases}$ I know that the function is a continuous function in R. But is this function derivable at x=0? I am not sure.. ...
1
vote
0answers
39 views

Rodrigues formula Associated Laguerre polynomial

Could you find the rodriguez formula of $$L_n^{\beta }\left(x^2\right)$$ knowing that $$\frac{\left(e^x x^{-\beta }\right) \frac{\partial ^n\left(e^{-x} x^{\beta }\right)}{\partial ...
7
votes
2answers
192 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: I tried to evaluate the integral $$\begin{align} ...
2
votes
0answers
65 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
0answers
29 views

Simplify $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$

EDIT: I would love to find an analytical solution for this definite integral: $$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$$ with $\delta>0,\, R>0,\,b>0$. Does ...
1
vote
0answers
29 views

Infinite product representation for the Sine Integral $\mathrm{Si}(z)$

The infinite series representation of the sine integral (http://en.wikipedia.org/wiki/Trigonometric_integral, previous m.se question: Is there any infinite series representation of the sine ...
0
votes
1answer
35 views

Legendre's Chi- Function

I want to get the numerical value(twenty at thirty decimals) of $$\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ Thanks you very much.
0
votes
0answers
14 views

What is the region ( area) of integration in Double mellin Barnes integral?

In double Mellin-Barnes contour integrals how to decide the region of integration?
4
votes
1answer
87 views

Infinite sum of products of four Bessel functions

The discrete Schrödinger equation for two interacting electrons in 1D under an electric field reads $$ E\psi_{mn}=[(m+n)F+U\delta_{mn}]\psi_{mn}-\psi_{m+1,n}-\psi_{m-1,n} -\psi_{m,n+1}-\psi_{m,n-1}\ . ...
0
votes
1answer
19 views

Generalizing 1D function to higher dimensions

I have a function in 1D given by $f(x) = \tanh(x-x_1) + \tanh(x-x_2)$. I want to generalize this to two dimensions, such that it describes a circle. The function $f(x,y)$ has to have a form such that ...
1
vote
1answer
48 views

get a integral from another

if $\int\limits_{0}^{+\infty}x^3e^{-\alpha x^2} dx=\frac{1}{2A}$ then $\int\limits_{0}^{+\infty}x^4e^{-\alpha x^2} dx=$ i tried to use integration by parts $$\begin{align} ...
2
votes
0answers
42 views

Comprehensive summary of where the function $\pi^{-\frac x\pi}$ can be encountered

I am studying the special functions, including the Riemann Xi and Zeta, and everywhere a function $\pi^{-\frac x\pi}$ pops up, usually as multiplier to the Gamma function. But yet I am not sure this ...
2
votes
1answer
158 views

Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

How would one deal with such an integral? $$I(k)\equiv \int_0^\infty r^n e^{-r(1+\mu)} e^{-{\mathrm i} kr}\:{}_1F_1({\mathrm i}/k+1;2;2{\mathrm i} kr) \, \mathrm{d} r$$ Here $n\in\{0,1\}$, $\mu\in ...
0
votes
0answers
27 views

evaluation of the limit $\lim_{u\to 0}\phantom{ }_0F_1(1,-u)u^{1-\alpha/2}$

What are the values of the limits \begin{equation} \lim_{u\to 0}\phantom{}_0F_1(1,-u)u^{1-\alpha/2}=?\\ \lim_{u\to \infty}\phantom{}_0F_1(1,-u)u^{1-\alpha/2}=? \end{equation} where ...
1
vote
1answer
31 views

Where can I find simple integration problems (and other computational exercises) involving special functions?

Working lots of computational exercises in my pre-calculus and calculus classes has given me a great deal of intuition in dealing with elementary functions. Thanks to these years of practice, I can ...
2
votes
1answer
48 views

$\frac{\sin(nx)}{\sin(x)}=(-4)^{(n-1)/2} \prod_{1\leq j \leq (n-1)/2}(\, \sin^2(x)-\sin^2(\frac{2\pi j}{n})\,)$

In Serre's A Course in Arithmetic, it states For $n$ odd and positive integer, proof that $\frac{\sin(nx)}{\sin(x)}=(-4)^{(n-1)/2} \prod_{1\leq j \leq (n-1)/2}(\,\sin^2(x)-\sin^2(\frac{2\pi ...
2
votes
1answer
48 views

Limit of Ratio of Chebyshev Polynomials

I have been trying to compute the limit $$\lim_{n\to\infty}{{U_n(x)^2}\over{U_{n-1}(x)^2+U_n(x)^2}}$$ where $U_n(x)$ is the $n$-th Chebyshev polynomial of the second kind and $x\ge 1$. Using software ...
2
votes
0answers
19 views

How to find a $\theta$ function verifying this property?

Let $r>4$ and $n>1$ be positive integers. Intuitively, the infinite sum $$S=\sum_{m=1}^{∞}\frac{2m}{r^{m^2}}$$ is related to a $\theta$ function. However, I cannot find a way to calculate this ...
2
votes
1answer
87 views

finding a harmonic sum using residues/complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{H_n}{n^2}$$ Using complex analysis. I just needs hints, I have no attempts, but I believe is has to do with residues.
0
votes
0answers
12 views

Integrals involving Marqum Q and Gaussian functions

I want to evaluate the following double integral. Any closed form solution exists? $$I(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-y^2}e^{-z^2} Q_1 \left[\sqrt{y^2+z^2},\sqrt{2x}\right]^2 ...
3
votes
1answer
65 views

the integral of $\sin(z^2) \exp\left({-4z^2xy \over y^2-x^2}\right)$ can be written in Fresnel integrals?

$$\int_{0}^{(y^2-x^2)/ 4t}s^{-1/2} \sin(s) \exp\left({-4sxy \over y^2-x^2}\right)\mathrm{d}s=2\int_{0}^{\sqrt{(y^2-x^2)/4t}}\sin(z^2) \exp\left({-4z^2xy \over y^2-x^2}\right)\mathrm{d}z$$ I applied ...
3
votes
0answers
32 views

Software to compute spherical harmonics in higher than 3 dimensions (100 or maybe 500 dimensions)?

I have been trying to find an implementation of Spherical harmonics for higher dimensional data but I couldnt find anything in Sage, Mathematica, Matlab. Does anyone have any idea of a standard/fast ...
3
votes
1answer
30 views

How to Derive this Digamma identity?

I dont see the transition from $(-z)^k$ in the fist sum to the transition to $(z+2)^k$ in the second sum? How is that derived?
2
votes
0answers
22 views

Residue of $f(z)$ using Laurent Series at $z=-2$ [duplicate]

Calculate the residue of: $$f(z) = \frac{\psi(-z)}{(z+1)(z+2)^3} \space \text{at} \space z=-2$$ Where $\psi(z)$ is the digamma function, and $\zeta(z)$ is the Riemann-zeta function (below). The ...
0
votes
0answers
56 views

Lambert W function, W(x), representation for entire domain

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express ...
1
vote
2answers
44 views

Derivated function $f$ so that $f(x+y)=f(x)f(y)$ and $f'(x)f(y)=f(x)f'(y)$ for all $x,y \in \Bbb R$

Let $f:\Bbb R\to \Bbb R$ a derivated function in all $\Bbb R$ that satisfies the condition $$f(x+y)=f(x)f(y),\;\,\,\text{for all $x,y \in \Bbb R$}$$ I already tried that $f'(x)f(y)=f(x)f'(y)$ for ...
0
votes
1answer
20 views

Question Mark Function and continued fraction representations

One could defined Minkowki's question mark question by : $$?(x) = a_0 + 2 \sum_{n= 1}^\infty \dfrac{(-1)^{n+1}}{2^{a_0 +\dots +a_k}},$$ with $x = [a_0;a_1,a_2,\dots]$. Is Minkowski's question mark ...