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

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6
votes
0answers
143 views

relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
2
votes
1answer
311 views

Special case of Meijer G function

I have an instance of the Meijer G function (using the definition from http://en.wikipedia.org/wiki/Meijer_G-Function, first equation there) that seems like, given its simplicity, it should be ...
0
votes
2answers
427 views

complete elliptic integral of the first kind

I'm looking for any "closed form" for the coefficient of the $\ell$-th power of $K(x)$, the complete elliptic integral of the first kind. Thanks.
7
votes
1answer
227 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $$f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$$ satisfies $$x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R.$$ How ...
3
votes
1answer
186 views

Deriving the form of the Exponential Integral from a given integral

The Wikipedia entry on Asymptotic Expansion outlines a detailed example, where it refers to the fact that the integral \begin{equation} \int_0^\infty \frac{e^{-w/t}}{1-w} \, dw \end{equation} ...
5
votes
0answers
158 views

Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
9
votes
1answer
578 views

Evaluate or simplify $\int\frac{1}{\ln x}\,dx$

I did a bit of work on this, but I'm not so sure about the parts towards the end. Starting with$$\int\frac{1}{\ln x}\,dx$$$$u=\ln x,1=\frac{dx}{du}\frac{1}{x},dx=x\,du,dx=e^{\ln ...
4
votes
0answers
152 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
2
votes
1answer
247 views

What does $ \langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu} $ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part. The $Y^l_m$ function is ...
1
vote
0answers
151 views

Proving or disproving that if $\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$ both $a$ and $b$ are integers.

I found some formula about special function very complicated, so I am curious how you people solve this by hand. $$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$$ but $a$ and $b$ are very ...
1
vote
2answers
5k views

Definition of Sinc function

I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: ...
8
votes
1answer
242 views

Can it be shown that $Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0$?

Background I am currently looking into the task of describing a transient temperature field $\theta(r,t)$ across the thickness $a \leq r \leq b$ of an infinitely long and hollow cylinder exposed to a ...
2
votes
1answer
403 views

A Curious Binomial Coefficient Sum

Let $k, l \leq n$ be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -l + 1}{n} = \binom{n - l + 1}{n} ...
1
vote
1answer
282 views

What is the fractional derivative of the function $\pi \cot (\pi x)$?

What is the fractional derivative of the function $\pi \cot (\pi x)$? I derived the following expression: $(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma ...
2
votes
1answer
264 views

how to evaluate $\int_0^1\sin(\frac{1}{x})dx$?

How can I evaluate the integral of $$\int_0^1\sin\left(\frac{1}{x}\right)dx.$$ Maybe it needs the cosine integral to evaluate it, but I cannot understand it very well. Thanks a lot.
4
votes
2answers
199 views

Could you give a application of a special function on number theory or analysis?

With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in ...
32
votes
2answers
2k views

Proof that $\sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$ regarding $\zeta(3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
6
votes
1answer
419 views

Was the definition of $\mathrm{erf}$ changed at some point?

I have seen two competing definitions of the error function. When I was an undergrad, Spiegel's Mathematical Handbook of formulas and tables (mine is the 1968 edition) was the definitive authority, ...
2
votes
1answer
169 views

Closed form of integral of $\operatorname{erfc} \log t$

Is there any closed form expression for the following integral? $$ \int\limits_t^\infty \left(1- \operatorname{erf}(\log x) \right )dx $$ or equivalently: $$ \int\limits_t^\infty ...
9
votes
2answers
1k views

Are there well known lower bounds for the upper incomplete gamma function?

Let $a >0, b >0$, and $r \in \mathbb{R}$. I am trying to find a lower bound for the integral $$\int_a^\infty y^{-r} \exp\left( - b(y-a)^2\right) \,\mathrm dy.$$ After consulting the Wikipedia ...
6
votes
1answer
3k views

Connection between Legendre polynomial and Bessel function

In Abramovitz and Stegun (Eq. 9.1.71) I found this curious relation $$\lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$$ valid for $x>0$. ...
2
votes
1answer
339 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, ...
6
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
474 views

Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
1
vote
2answers
251 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
389 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
278 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
149 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
331 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
414 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
253 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
446 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
258 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
474 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
491 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
815 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
275 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
173 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
458 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
193 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
217 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
323 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 ...
9
votes
3answers
4k 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
968 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
137 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 ...