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
1answer
344 views

How to show integral of different order Hankel transformed functions are equal?

Say I have a function $p_v(r) \in L^2(\mathbb{R})$ given by $$p_v(r) = \int_0^\infty P(k) J_v(rk)\,k\,dk$$ From mucking around in MATLAB it seems the following is true: $$\int_{r=0}^\infty ...
1
vote
1answer
100 views

Why exponentiation is considered elementary funtion?

Why exponentiation and power function with non-integer power are considered elementary functions while some other functions like Bernoulli polynomials generalized to non-integer order, polylogarithm, ...
7
votes
1answer
1k views

Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma ...
2
votes
0answers
494 views

Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
1
vote
2answers
252 views

Bounds on integral $x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy$

Consider the function $$ I(a,x) = x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy $$ where $x \geq 1$, and $a \geq 0$. I am not really interested in the parameter $x$, so define $$ I(a) = \sup_{x \geq 1} ...
8
votes
2answers
2k views

An explanation of spherical harmonics?

Could somebody please explain spherical harmonics in a simpler manner than it is demonstrated on various websites (like the Wikipedia page which simply overflows my buffer with symbols). I've tried ...
5
votes
3answers
398 views

Proof that $Γ'(1) = -γ$?

I know that $Γ'(1) = -γ$, but how does one prove this? Starting from the basics, we have that: $$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$ How do we differentiate this? How do we then find that ...
6
votes
4answers
285 views

Gamma identity $\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p})dx=\Gamma(p+1)$

I ran across what appears to be another Gamma identity. Show that $$\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p}) \,\mathrm dx=\Gamma(p+1)=p!$$ I tried several different subs and ...
4
votes
2answers
150 views

'Error term' in zeta function [duplicate]

Possible Duplicate: What is the expression of $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$? Asymptotic formulas for the n-th harmonic number are well-known: $$ \sum_{k=1}^n\frac1n=\log ...
1
vote
2answers
336 views

Evaluating $\zeta(0)$ using the functional equation of Riemann-Zeta function.

$$\zeta(it)=2it\pi it−1\sin(i\pi t/2)\Gamma(1−it)\zeta(1−it).$$ Everything on the RHS is never zero, Does that means LHS has no zeros, since $\sin(s)$ has a simple zero at $s=0$ while $\zeta(1−s)$ ...
6
votes
2answers
435 views

Inequality involving the regularized gamma function

Prove that $$Q(x,\ln 2) := \frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} dt} \geqslant 1 - 2^{-x}$$ for all $x\geqslant 1$. ($Q$ is the regularized gamma function.) ...
4
votes
1answer
281 views

Extending partial sums of the Taylor series of $e^x$ to a smooth function on $\mathbb{R}^2$?

Is there a smooth function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,n)$, where $n\in\mathbb{N}$, is the truncated Taylor series of $e^x$, namely $1+ x + \frac{x^2}{2} + \dotsb + \frac{x^n}{n!}$, ...
11
votes
1answer
460 views

Integral of digamma function

I was attempting to evaluate a series $$\sum_{n=1}^\infty \frac{1}{n} \ln\left(1+\frac{1}{n}\right)$$ Since $$\frac{1}{n}\ln\left(1+\frac{1}{n}\right)=\int_0^1 \frac{1}{n(n+t)}dt,$$ I rewrote it as ...
1
vote
0answers
259 views

How is the Riemann-Siegel formula applied?

What is the application of the Riemann-Siegel formula: $$ \zeta(s) = \sum_{n=1}^N\frac{1}{n^s} + \gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}} + R(s) , $$ where $ \displaystyle\gamma(s) = ...
1
vote
1answer
479 views

An integral of a complementary error function

I really appreciate it if someone help me solving this integral: $$ \int \frac 1x \cdot \operatorname{Erfc}^n x\, dx,$$ where $\operatorname{Erfc}$ is the complementary error function, defined as ...
4
votes
1answer
4k views

Taylor Expansion of Error Function

I am working on a question that involves finding the Taylor expansion of the error function. The question is stated as follows The error function is defined by $\mathrm{erf}(x):=\frac ...
4
votes
2answers
501 views

Finding some orthogonality in a convolution-like integral over Legendre polynomials

I encountered the following integral in my (physics) research, and I've yet to find an analytic solution: $$I(n_1,n_2,n_3) = \int_{-1}^{1} d(\cos\theta_1) \int_{-1}^{1} d(\cos\theta_2) ...
0
votes
2answers
854 views

Looking for function of bell-like curve that peaks quickly.

I'm writing a little Sage/Python script that would graph the cumulative effects of taking a particular medication at different time intervals / doses. Right now, I'm using the following equation: ...
4
votes
2answers
1k views

Definition of the gamma function

I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ ...
5
votes
2answers
322 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
1
vote
1answer
175 views

How to evaluate $\sum J_0(\alpha n) z^{-n}$ in closed form?

I need to evaluate $\sum_{n = -\infty}^{\infty} J_0(\alpha n) z^{-n}$ in closed form, where $z$ is complex variable and $J_0()$ is the zeroth order Bessel function of the first kind. How do I evaluate ...
4
votes
1answer
473 views

An addition property of Weierstrass $\wp$

I want to show $$ \left( \begin{array}{ccccc} &1 &\wp(v) &\wp'(v) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(v+w) &-\wp'(v+w) \end{array} \right)=0 $$ ...
2
votes
1answer
197 views

integration question about dilogarithm

I want to show that $$\operatorname{Li}_2(z)=z\int_{0}^{\infty}\frac{x}{e^x-z}dx$$ It is the integral of the Bose–Einstein distribution in dilogarithm case. Thank you!
3
votes
1answer
235 views

Dilogarithm asymptotics for an exponential parameter.

So this question is about this dilogarithm function. Assume the argument $z$ is real then I want to show the formula $$\operatorname{Li}_2(e^{-z})=\frac{\pi^2}{6} + z\log z -z+O(z^2) $$ as $z$ ...
15
votes
1answer
2k views

Residue of $z^2 e^{1/\sin z}$ at $z=\pi$

A while back I was working through many problems in Mathews and Walker's Mathematical Methods of Physics. In the appendix is this problem: A-6. Find the residue of the function $z^2 e^{1/\sin z}$ ...
8
votes
1answer
354 views

Does the series of squares of Legendre polynomials converge?

I am a physicist working on an electrostatic problem and this series popped up: $\sum^{\infty}_{l=0} (P_l(x))^2$ where $P_l$ is the $l$-th Legendre polynomial. Computing this numerically I think the ...
10
votes
3answers
5k views

How to come up with the gamma function?

It always puzzles me, how the Gamma functions's inventor came up with it's definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
1
vote
1answer
981 views

How to show that functions of this type are strictly decreasing

Let $f:[0,\infty)\to \mathbf{R}$ be defined by $$ f(x) = \frac{1}{x+1} \int_x^\infty g(r,x) dr,$$ where $g(r,x)$ is a "nice" function and all of this makes sense. Suppose that I want to show that ...
3
votes
0answers
139 views

The polynomial where only the terms in the multinomial series where each variable's exponent is $>0$ are kept?

I'm wondering if there's a special polynomial with a name out there with $x_1,x_2,\ldots,x_k$ as variables that's defined like this: $$ \sum_{\substack{i_1>0,i_2>0, \ldots,i_k>0 \\ i_1 ...
2
votes
2answers
268 views

How big is the integral $\int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} dx$

I can't seem to get Maple to approximate the integral $$\int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} dx.$$ Could somebody tell me why? This integral "should be" well-defined. (My ...
4
votes
2answers
339 views

Is there a precise mathematical connection between hypergeometric functions and modular forms

I've been playing around with Gauss' hypergeometric series $F(a,b,c,z)$ these days and I was wondering if there is some relation with the theory of modular forms.
0
votes
1answer
86 views

Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$

Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)? I'm aware of Euler's formula: $$F(5/4,3/4; 2, z) = ...
10
votes
0answers
797 views

How to compute this integral of Bessel functions?

I have $\alpha_\max$ a real number between $0$ and $\frac\pi2$. Furthermore $\zeta$ and $\xi$ are positive real numbers. Now I would like compute the integral $$\int_0^{\alpha_\max} \mathrm{e}^{i ...
5
votes
2answers
1k views

Convergence of $\Gamma(p)$ for $0<p\leq 1$ and divergence for $p \leq0$.

Can someone show me a proof or any clear resource about convergence of gamma function for values of $p$ less than zero. If possible I need proofs using integration by parts. My problem evaluating ...
13
votes
1answer
3k 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
387 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
242 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
131 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
144 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
256 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
397 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
243 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
341 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
874 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 ...
17
votes
1answer
388 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
304 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
909 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 ...