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

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17
votes
1answer
384 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
6
votes
3answers
1k views

if $w(x)=\sum \limits_{n=1}^\infty e^{-n^2\pi x}$ then how to show that $\frac {1+2w(x)} {1+2w(\frac{1}{x})}=\frac{1}{\sqrt{x}}$

$w(x)=\sum \limits_{n=1}^\infty e^{-n^2\pi x}$ $\frac {1+2w(x)} {1+2w(\frac{1}{x})}=\frac{1}{\sqrt{x}}$ I wonder how can be proved such a beautiful relation as shown in wolfram page I need to ...
7
votes
1answer
299 views

$f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$?

$$f(x)=\int_0^{+\infty} e^{-(t+\frac{1}{t})x}\;dt$$ if while $ x>0 $ , $ f(x) $ has values I noticed some interesting relations for $f(x)$ as shown below: $$ \begin{align} t & ...
4
votes
2answers
895 views

Resources for learning Elliptic Integrals

During a quiz my Calc 3 professor made a typo. He corrected it in class, but he offered a challenge to anyone who could solve the integral. The (original) question was: Find the length of the curve ...
2
votes
1answer
259 views

Question on Inverse Pochhammer Symbol

I struggled quite a while without success, so I highly appreciate if anybody can help me, proving that: $$ \frac{1}{\left(1+1/n \right)^{(M)}}=\sum_{k=0}^M \frac{(-1)^k}{k!(M-k)!(nk+1)}, $$ where ...
4
votes
1answer
194 views

Infinite series representation. Limited or not?

Recently I've found the following. Let $n < m$. Then the integral $$\int\limits_0^\infty {\frac{{{x^{n-1}}}}{{1 + {x^m}}}} dx$$ converges and its value is (using the $B$ and $\Gamma$ function ...
1
vote
0answers
209 views

analytic continuation of an integral involving the mittag-leffler function

i have posted this question on MO, and i didn't get an answer . we have the following integral : $$I(s)=\int_{0}^{\infty} \frac{s}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)dx -\lim_{z \to 1 ...
4
votes
1answer
373 views

Using the Lambert W to express a solution of a differential equation.

I solved a differential equation some time ago and I need to solve for $y$. How can we solve for $y$ using the Lambert W function? $$C_1+x = e^y+Cy$$
28
votes
2answers
2k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
3
votes
2answers
420 views

What is this generalized multivariable hypergeometric function?

I'm looking for any kind of reference on a multivariable generalization of a (confluent) hypergeometric function. To be specific, Horn's List is a list of 34 two-variable hypergeometric functions, 20 ...
1
vote
1answer
151 views

Is Bessel function $J_0(n)$ absolutely summable?

Is the Bessel function $J_0(n)$ absolutely summable i.e $\sum_{n=0}^{\infty}|J_0(n)| < \rm C$? Since $\lim\limits_{n \to\infty} J_0(n) = 0$, I'd assume the absolute sum converges to a constant ...
3
votes
1answer
427 views

Concave functions on discrete domain

We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property: $f(x+y) \leq f(x) + f(y) $ for all natural numbers $x$ and ...
9
votes
2answers
621 views

Roots of the incomplete gamma function

Is there any way that one can describe all the roots of the incomplete gamma function $\Gamma(n,z)$, for $n\in \mathbb{N}$, analytically?
4
votes
1answer
2k views

Projection of Gaussian in Spherical Coordinates

Consider a point with spherical coordinates $\vec{r}_0=(r_0, \theta_0, 0)$. The spherical gaussian distribution centered at $\vec{r}_0$ is $f(\vec{r})=Ne^{|\vec{r}-\vec{r}_0|^2/A}$, where $N$ is the ...
1
vote
1answer
83 views

Show the convergence of a series

Given the series: $$S(\lambda,\Phi)=\sum_{n=0}^{\infty}{J_n}(\lambda)e^{in\Phi}$$ where the $J_n(\lambda)$ is the Bessel function of order $n$ I have some difficulty to give a proof of its ...
2
votes
3answers
674 views

What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"? $\displaystyle \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$
4
votes
1answer
957 views

Limit for gamma function

How can I prove that $$\displaystyle \Gamma(z)=\lim_{n \to \infty} \displaystyle \int_0^n \left( 1-\frac{t}{n}\right)^n t^{z-1}\ \text{d} t\;=\displaystyle \int_0^{\infty} e^{-t} t^{z-1}\ \text{d} ...
2
votes
1answer
312 views

integral related to Beta function and binomial

I have not frequented the board lately due to being very busy. But, I ran across what I think is an interesting problem. $\displaystyle \int_{0}^{k}(tx+1-x)^{n}dx$, where $n\in \mathbb{Z}^{+}$ and ...
3
votes
1answer
3k views

Steps in evaluating the integral of complementary error function?

Could you please check the below and show me any errors? $$ \int_ x^ \infty {\rm erfc} ~(t) ~dt ~=\int_ x^ \infty \left[\frac{2}{\sqrt\pi} \int_ t^ \infty e^{-u^2} du \right]\ dt $$ If I let dv=dt and ...
5
votes
1answer
325 views

Solving Shallow water Equations with Hermite polynomials

I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line. The main equation is $$\frac{d^2\eta}{dt^2} + ...
5
votes
3answers
442 views

Theory of the Mathieu Operator

How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ...
4
votes
1answer
244 views

Zeta function identity

How does one prove the zeta function identity $$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
4
votes
1answer
230 views

Heuristic\iterated construction of the Weierstrass nowhere differentiable function.

I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example ...
4
votes
1answer
158 views

What is the reason to use hypergeometric functions?

I would be grateful if anyone could explain the purpose of using hypergeometric functions. If a function exists in closed form, e.g. $\sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert ...
0
votes
1answer
303 views

Maclaurin series involving an elliptic integral

I have been asked to find the Maclaurin series expansion of a term involving an elliptic integral, I would be grateful for any help as I am unsure as to how to even start this question. The term I ...
12
votes
1answer
845 views

Recursive solutions to linear ODE.

When finding the solutions to the simple ODE $$ y'- mxy= x^n \text{ ; } y(0) = 0$$ I found the following: Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define ...
2
votes
1answer
301 views

Integration and $\gamma(n,x)$

It isn't hard to prove that: $$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$ Or put in a different way: $$\int_0^x e^{-t} \frac{t^n}{n!} dt = ...
3
votes
1answer
342 views

Integrals $\int\limits_0^t {{e^u}\log u\operatorname d \!u} $ and $\int\limits_0^t {{e^{ - u}}\log u\operatorname d \!u} $

Ok, I want to find $$\int\limits_0^t {{e^u}\log udu} $$ and $$\int\limits_0^t {{e^{ - u}}\log udu} $$ I'm thinking as follows $$d\left( {{e^u}\log u} \right) = {e^u}\log udu + ...
2
votes
2answers
931 views

Spherical Bessel Zeros

I was wondering if there is a known closed form solution for the zeros of the spherical Bessel functions. While doing a quantum assignment, I came across them as a solution for the spherical infinite ...
8
votes
3answers
783 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
12
votes
2answers
1k views

The roots of Hermite polynomials are all real?

The Hermite polynomials are defined as $$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$ How does one prove that all the roots of the Hermite polynomial are real?
3
votes
1answer
700 views

Integral representation of modified bessel function

How can I obtain $$I_n(x)= \frac{1}{2\pi}\int_{0}^{2\pi}e^{in\theta}e^{x \cos\theta}\,d\theta ,$$ from the integral $$J_n (x) =\frac{1}{2\pi} \int_{0}^{2\pi}e^{-in\theta}e^{ix \sin\theta}\,d\theta .$$ ...
1
vote
1answer
387 views

Verifying identities for Riemann zeta function

I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
8
votes
1answer
791 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
1
vote
1answer
2k views

Associated Legendre polynomials of fractional order

My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers $m$ and $l$; i.e., $P_l^m(x)$, for $x \in [-1,1]$. Usually one assumes that $m$ and $l$ are ...
1
vote
1answer
153 views

distribution function

Let$(X,\mu)$ be a measure space. $f:X \to \mathbb{R}$ be a measure function. For every $t\in \mathbb{R}$ the distribution function $F$ of $f$ is defined as $ F(t)=\mu\{x \in X:f(x)<t\}.$ I have ...
0
votes
1answer
235 views

Create 'smooth breakpoint function' by using integral?

Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a ...
1
vote
3answers
105 views

Oddities in the Definition of IntegralCosinus ${\rm Ci}(x)$

Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ ...
1
vote
2answers
555 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
4
votes
1answer
158 views

Solving an integral using generating functions, coefficients of equivalent series don't match!?

Please don't be frightened by the length of this, I just wanted to provide ample detail. If you want, you can skip the derivation and go straight to the result at the bottom. I have $$ ...
10
votes
3answers
911 views

Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
6
votes
1answer
911 views

On the completeness of the generalized Laguerre polynomials

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \begin{equation} \int_0^{\infty} ...
1
vote
1answer
429 views

Advanced application of the Binomial Theorem

I'm trying to solve the following integral: $$ \int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx $$ Where $C_{n}^{\lambda}(x)$ is a ...
2
votes
1answer
200 views

real-value solution

I have this integral $$\int\frac{dz}{\sqrt{(z^{2}-\rho^{2})(\lambda^{2} - z^{2})}}$$ and parameters obey the following conditions $$z= \exp[k\varphi],$$ $$\lambda^{2} = \frac{b + \sqrt{b^{2} - ...
1
vote
1answer
116 views

Problem with the Limits of the Hypergeometric Series

I'm new to the hypergeometric series, and I'm trying to decipher a proof in which the author identifies a particular finite sum as a a hypergeometric series. The particular summation is: $$ ...
5
votes
1answer
160 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma ...
2
votes
1answer
259 views

Closed form formula for series involving derivatives of reciprocal gamma function

How to get closed form for the sum $\displaystyle{\sum\limits_{k = 1}^\infty {\frac{{{p^k}}} {{\left( {2k} \right)!!}}\frac{{{d^k}}} {{d{s^k}}}{{\left. {\frac{1} {{\Gamma \left( s \right)}}} ...
23
votes
1answer
1k views

How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ...
5
votes
1answer
313 views

Orthogonality of the Gegenbauer Polynomials

Typically the orthogonality relation for the Gegenbauer polynomials is given as: $$ ...
7
votes
3answers
446 views

Reference requests: Jitsuro Nagura

I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev $\psi$ function says that ...