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

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4
votes
1answer
252 views

Asymptotic approximation for confluent hypergeometric function

I have the following nasty expression that I would like to expand in powers of $\frac{1}{N}$: \begin{align} \frac{2^{\frac{3}{2}} 3^{\frac{1}{2}} \Biggl[ \sqrt{u} \cdot ...
4
votes
2answers
197 views

How to verify integral with hypergeometric function

Trying to evaluate the following integral, Mathematica returns this result: $$ \int \frac{e^{-\tau \omega}}{1+e^{-\beta \omega}} d \omega = \frac{e^{(\beta - \tau) \omega} \cdot {}_2F_1(1, ...
19
votes
4answers
1k views

Interesting integral related to the Omega Constant/Lambert W Function

I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry. ...
12
votes
1answer
440 views

elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been ...
1
vote
1answer
79 views

Nonhomogeneous equation involving logarithm

This is probably a lame question, but what is the general approach to solving $\log z +z \sigma +1=0$ for $z$? Wolfram Alpha obtains a Lambert W-function, but I don't quite see how.
0
votes
1answer
201 views

Coordinate scaling in incomplete gamma function integral

I'm faced with the integral $$\mathcal{I} = \int_0^\infty \mathrm d x \; e^{-\beta \, e^x - \mu x} \;,\quad \Re(\beta) > 0 \;.$$ The solution can be looked up. It reads $$\mathcal{I} = \beta^\mu ...
8
votes
3answers
281 views

Evaluating $\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)}$

On a lark (as a followup to this question), I was playing around with Wolfram alpha, and it seems that $$\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)} = ...
1
vote
3answers
1k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
13
votes
3answers
5k views

Analytic continuation- Easy explanation?

Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said, The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
4
votes
2answers
885 views

Tables of Hypergeometric Functions

I'm looking for a book, set of tables, or other reference which contains a comprehensive list of hypergeometric identities; that is, something which allows a hypergeometric fucntion to be expressed in ...
3
votes
1answer
105 views

How to show $ \frac{Q_{k}(1-z^2)-zP_{k}(1-z^2)}{Q_{k}(1-z^2)+zP_{k}(1-z^2)}=\left(\frac{1-z}{1+z}\right)^{2k+1} $ analytically?

Let \begin{eqnarray*} P_{k}(z)={_2F_1}(-k,\frac{1}{2}-k; -2k; z), \ \label{e0} Q_{k}(z)={_2F_1}(-k,-\frac{1}{2}-k; -2k; z) , \end{eqnarray*} where $k\ge 1$ is an integer. How to show ...
9
votes
0answers
328 views

Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$

A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is: $$\int_{0}^{\infty} e^{- \lambda ...
1
vote
3answers
196 views

better understanding of incomplete gamma function $\Gamma(0,x)$

By definition incomplete Gamma function is:$$\Gamma(0,x)=\int_{x}^{\infty}t^{-1}e^{-t}dt $$ I have an expression which includes $$\Gamma(0,r(A)e^{i\phi(A)}),$$ where $A>0$ is a parameter, and ...
15
votes
3answers
498 views

The graph of $x^{n}+y^{n}=r^{n}$ for sufficiently large $n$

The graph of the function $x^{n}+y^{n}=r^{n}$ for certain large values of $n$ looks suspiciously like a square. See this page from wolframalpha. Have any results been proven regarding this ...
2
votes
2answers
444 views

Representing affine transform of Legendre polynomials

I have a function defined as a set of weighted Legendre polynomials: $f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$. I have another function similarly defined with Legendre basis ...
2
votes
3answers
362 views
15
votes
2answers
2k views

Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)

I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
10
votes
2answers
989 views

Integral with spherical symmetry over cube

Is it possible to calculate the integral $$I = \int_{-1}^1 \mathrm dx \int_{-1}^1 \mathrm dy \int_{-1}^1 \mathrm dz \frac{1}{x^2 + y^2 + z^2}$$ analytically? I tried using spherical coordinates $$I ...
1
vote
2answers
2k views

How can I solve this equation (contains error function)?

Edited out incorrect formula Can someone please solve this equation for x? I have no idea what to do with the $\mathrm{erf}$ (error function). Edit: Hm, it did not work correctly... here is the ...
32
votes
2answers
5k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...
10
votes
2answers
555 views

Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$ for several values of k. I've noticed that that, for k a ...
3
votes
0answers
145 views

Relationship between Dixonian elliptic functions and Borwein cubic theta functions

In this paper, it says that the three Borwein cubic theta functions obey the identity $a(q)^{3}=b(q)^{3}+c(q)^{3}$, which is strongly reminiscent of the identity that Dixonian elliptic functions obey ...
27
votes
1answer
787 views

elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for ...
8
votes
1answer
193 views

Asymptotic behavior of $\Gamma^{-1}(x)$

For real $x,$ it's well-known that $$\Gamma^{-1}(x)\sim\frac{\log x}{\log\log x}$$ So a natural question is to bound $$G(x)=\Gamma^{-1}(x)\frac{\log\log x}{\log x}$$ which of course is 1 + o(1). ...
1
vote
2answers
263 views

expectation of incomplete gamma

Is the expectation of the (upper/lower) incomplete gamma function known? $$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$
4
votes
2answers
2k views

Integrating Legendre Polynomials over half range

Solving for the potential of a conducting sphere with hemispheres at opposite potentials, (not using Green's function) I am stuck at this point: $$I_l = V_1 \int_0^1 P_l(x)dx+V_2 \int_{-1}^0 ...
19
votes
2answers
929 views

doubly periodic functions as tessellations (other than parallelograms)

I think of a snapshot of a single period of a doubly periodic function as one parallelogram-shaped tile in a tessellation, could a function have a period that repeats like honeycomb or some other not ...
6
votes
1answer
174 views

How to prove the q-series identity?

How could I prove that $$(-q;q^2)_\infty (q;q)_\infty = 1 + 2 \sum_{i=1}^\infty (-1)^i q^{2 i^2}?$$ If that is too difficult is there a way to show $$(-q;q^2)_\infty (q;q)_\infty \equiv 1 \pmod 2?$$ ...
5
votes
0answers
166 views

Bounding function involving Beta functions

Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...
3
votes
1answer
401 views

How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk$, where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!
1
vote
1answer
71 views

Can one prove $\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min(1,\frac{c}{t})$?

Let $c>1/2$ be an arbitrary big fixed constant. Can one prove that for all $t\geq 1$: $$\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min\left(1,\frac{c}{t}\right)$$ for some small constant ...
9
votes
2answers
1k views

Proving and deriving a Gamma function

I'm having a hard time trying to prove this Gamma function and trying to derive the duplication formula: a.) Prove that $$\frac{\Gamma (p)\Gamma (p)}{\Gamma (2p)} = ...
14
votes
3answers
465 views

Can this integral $\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$ be written in the form of a elliptic integral

I am trying to find the magnetic field due to an elliptic loop of wire. How to do integrals of the type $$\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$$ Where a and b are ...
21
votes
1answer
586 views

Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — ...
1
vote
0answers
154 views

Orthogonal polynomial interpolation of a function

I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
3
votes
0answers
211 views

How did Bessel functions come to be denoted by $J_n$?

The $n$th Bessel function of the first kind is usually denoted $J_n(x)$. Where did the use of the letter $J$ to indicate the Bessel function come from?
1
vote
2answers
456 views

Integral of the fractional part of $\frac1x$ multiplied by $x$ on interval $(a,b), a\ge 0$.

I'm interested in finding the value of the integral of $\left\{\frac{1}{x}\right\}\cdot x$ (the fractional part of $\dfrac{1}{x}$ multiplied by $x$) on the interval $(a,b), a\ge 0$ the integral of ...
11
votes
1answer
602 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
5
votes
3answers
991 views

Continuous function with local maxima everywhere but no global maxima

Can there be such a function: $f \colon \mathbb R \to \mathbb R$ is continuous and non-constant. It has a local maxima everywhere, i.e., for all $x \in \mathbb R$ there is some $\delta_x>0$ such ...
6
votes
1answer
317 views

What's the sum of this power series?

What's the sum of this power series? $$f_k(x)=1-\frac{x^2}{k}+\frac{x^4}{k(k+1)\cdot2!}-\frac{x^6}{k(k+1)(k+2)\cdot3!}+\ldots$$ I'm just helping someone, I'm not good at math! :\
4
votes
2answers
325 views

Determination of inverse laplace transform using primitive functions

In How can you prove that a function has no closed form integral?, the accepted answer points to http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf where one can find a corollary by Liouville ...
15
votes
5answers
977 views

Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$

Can you help prove the functional equation: $$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$ Specifically, I am looking for a solution using complex ...
7
votes
2answers
2k views

Integral involving Modified Bessel Function of the First Kind

Why is this true? $$ \int_0^\infty e^{-\frac{1}{2}(b^2+x^2)} I_0(bx) x \,dx = 1 $$ Note that $I_0(x)$ is a modified bessel function of the first kind. The difficulty for me lies in a) translating ...
7
votes
1answer
310 views

Is there a gamma-like function for the q-factorial?

I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in ...
0
votes
2answers
435 views

Proving a Laguerre polynomial integral

After a fair bit of effort, I managed to prove that $$\int_0^\infty t^\alpha \exp(-t) L_n^{\alpha+1}(t)\mathrm dt=\Gamma(\alpha+1)$$ where $L_n^\alpha (t)$ is a generalized Laguerre polynomial, with ...
6
votes
1answer
929 views

How to decompose displaced Hermite-Gauss function into higher order HGs?

The Hermite-Gauss functions appear commonly in physics. These functions are formed from the product of a Hermite polynomial and a Gaussian: $$ u_n(x) = \left(\frac{2}{\pi w_0^2}\right)^{1/4} ...
3
votes
2answers
413 views

expression for the sum involving digamma function

I got this answer from WolframAlpha. Does anyone know how even to approach it to obtain the solution using digamma function. Please don't solve it, just show me in the right direction! $$ ...
4
votes
1answer
202 views

Closed form for some integrals related to the complementary error function

While studying the use of the trapezoidal rule for numerically evaluating the complementary error function $\mathrm{erfc}(z)$, the following integrals showed up when I was trying to derive expressions ...
3
votes
3answers
1k views

How to solve $n$ for $n^n = 2^c$?

How to solve $n$ for $n^n = 2^c$? What's the numerical method? I don't get this. For $c=1000$, $n$ should be approximately $140$, right?
10
votes
3answers
679 views

How does Lambert's W behave near ∞?

How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of $$\frac{W(x)}{\ln(x)}$$ near $\infty$ (but along the positive real line, if that ...