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

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13
votes
1answer
2k views

Quotient of gamma functions?

I'm sorry if this is a simple question, but this page on Wolfram Research states that it follows from Stirling's formula that: $$ \frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta $$ for large $x$, ...
11
votes
4answers
383 views

Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $

What is $ \displaystyle\int_0^1 \frac{\ln(1+bx)}{1+x} dx $? I call it $f(b)$ and differentiate with respect to be $b,$ a bit of partial fractions and the $x$ integral can be done. Then I ...
4
votes
1answer
240 views

What special function is this?

Assume that $\zeta$ is a positive real number and $a = \frac{2 \pi}{\alpha_{\text{max}}}$ for $0 < \alpha_{\text{max}} < \frac{\pi}{2}$. In other words $a > 4$. Is there a special function ...
1
vote
1answer
72 views

Nonlinear equation containing parametric integral

I have the following equation: $$I(k,x)=\int_0^xJ_k(\tau^k)d\tau=\alpha$$ where $\alpha$ is a given constant $\alpha\in \mathbb{R}$ and $k$ integer with $k\gt 0$. $J_k(x)$ is the Bessel function of ...
3
votes
1answer
79 views

Asymptotics of a solution

Let $x(n)$ be the solution to the following equation $$ x=-\frac{\log(x)}{n} \quad \quad \quad \quad (1) $$ as a function of $n,$ where $n \in \mathbb N.$ How would you find the asymptotic behaviour ...
0
votes
1answer
129 views

Are there names for the indices of the spherical harmonics?

I know that physicists call $\ell$ and $m$ the "azimuthal" and "magnetic" quantum numbers, respectively. But those sound very physics-y. (I am actually a physicist, but still.) Are there names for ...
0
votes
1answer
142 views

Finding $\int_{-\infty}^{\infty} e^{-x^2/2}x^{2n}\,\mathrm dx$ by symmetry

I can easily show that (substituting $\frac{x^2}{2} = t $ using the identity for Gamma function of $n+\frac{1}{2}$, then further expanding $\Gamma(n+\frac{1}{2})=\dfrac{(2n-1)!! \sqrt{\pi}}{2^n}$ and ...
4
votes
2answers
254 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
2
votes
2answers
385 views

On the modulus of $\Gamma(z)$

In about two weeks, I'm going to be giving a presentation on the complex-valued Gamma function $\Gamma(z)$. By definition, I know that $$\Gamma(z)= \int_0^\infty e^{-t}t^{z-1}dt.$$ Now if I let ...
2
votes
0answers
240 views

Asymptotic Series of Confluent Hypergeometric Function

I am using a generating function method to try and solve a recurrence. I have solved the resulting differential equation to find the generating function takes the form: $$A(z) = \frac{\, ...
3
votes
1answer
340 views

An example of divergent series with the Lerch function

I am often working with divergent series all around being this the bread and butter for a theoretical physicist. Thanks to the excellent work of Hardy these have lost their mystical Aurea and so, they ...
4
votes
1answer
823 views

The area of the superellipse

I'm watching this video, where D. Knuth explains the connection of $\pi$ and factorials, and other matters (it is very interesting). Almost at the end of the talk he says the area of the superellipse ...
16
votes
1answer
375 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
293 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
845 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
191 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
197 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
356 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
396 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
415 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
606 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
82 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
659 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
908 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
303 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 ...
4
votes
1answer
314 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
427 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
238 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
228 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
155 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
298 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
815 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
294 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
336 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
829 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
724 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 ...
11
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
693 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
379 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
760 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
1k 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
147 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
223 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
103 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: $$ ...