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
2k views

Multivariate integrals involving Dirac delta functions

I'm interested in the behavior of Dirac deltafunctions within multivariate integrals. Here is a simple example to which I do not know the answer: $$\iint\limits_{[0,1]\times [0,1]} \delta\left(x - ...
12
votes
3answers
595 views

Calculating $ \int _{0} ^{\infty} \frac{x^{3}}{e^{x}-1}\;dx$

how to calculate $$\int_0^\infty \frac{x^{3}}{e^{x}-1} \; dx$$ Be $q:= e^{z}-1 , p:= z^{3}$ , then $e^{z} = 1 $ if $z= 2\pi n i $, so the residue at 0 is : $$\frac{p(z_{0})}{q'(z_{0})} = 2\pi ...
4
votes
1answer
782 views

Finding a generating function for the Laguerre polynomials

I've started learning some quantum physics and one often encounters special functions (like Legendre polynomials, Laguerre polynomials, Bessel functions, ...). Many calculations with these functions ...
1
vote
1answer
104 views

Find function if known some limit

I have trouble with the following problem. Let $f=f(p)$, $p>1$, $0<f<1$, and $\lim_{p\to\infty}f(p)=1$. Find $f(p)$, such that $$\lim_{p\to\infty}p\left(1-\sqrt{p ...
4
votes
1answer
346 views

How can I solve this integral equation in terms of Hermite polynomials?

It must be proven that the solution of the integral equation $$f(x)=\int_{-\infty}^{+\infty} e^{-(x-t)^2} g(t)dt$$ is $$g(x)=\frac{1}{\sqrt{}\pi}\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{2^nn!} H_n(x)$$ ...
2
votes
1answer
144 views

Reason behind the reciprocity of series

This question may appear to be a silly one for experts. From long back I have been observing all kinds of series but every-series contain a reciprocal part, I mean the " one over something " , is ...
2
votes
1answer
373 views

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, ...
1
vote
1answer
295 views

sinc function in terms of Hermite function

Is there any formula which represent the sinc function $\operatorname{\rm sinc}(x)=\dfrac{\sin(\pi x)}{\pi x}$ (its expansion) in terms of the Chebychev-Hermite function?
3
votes
1answer
268 views

What's the reasoning for this recurrence on $q$-multinomial coefficients?

I'm familiar with the recurrence for binomial coefficients based on Pascal's triangle. However, in general, there is the recurrence for $q$-multinomial coefficients given by $$ ...
2
votes
3answers
482 views

What is the analytic form of MeijerG in Mathematica?

I know that the official standard formula can be found here, but I am having a very hard time simplifying this special case: ...
4
votes
3answers
1k views

Taylor Series of Ratio of Bessel Functions

In attempting to solve a recursion relation I have used a generating function method. This resulted in a differential equation to which I have the solution, and now I need to calculate the Taylor ...
1
vote
2answers
88 views

Proving the problem of integration

From a journal, they proved this equality: $$ \frac{z}{\alpha -1}\left(\int_0^1 \frac{t^{\frac{1}{\alpha}}}{1-tz} dt -\alpha \int_0^1 \frac{v}{1-vz} dv\right) = \int_0^1 t^{\frac{1}{\alpha}} ...
6
votes
1answer
275 views

Laguerre polynomials and inclusion-exclusion

Does anyone know a reference for the solution of the generalized derangement problem via Laguerre polynomials? The Wikipedia article here says that this is an application of inclusion-exclusion, but ...
0
votes
1answer
81 views

questions about a sum of logarithmic integrals

Consider the following sum, where $\operatorname{li}((x)$ is the logarithmic integral function: $$\operatorname{lisum}(x) = \sum_{k=1} ^{\lfloor\sqrt x\rfloor} \operatorname{li}((x/k)$$ For small ...
2
votes
0answers
196 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
1
vote
0answers
249 views

Bound on Bessel function of the first order

Let $I_1(z)$ be the Bessel function of the first order with purely imaginary argument. Can we explicitly bound $I_1$ on $[0,x]$, where $x>0$ is a real number in terms of $x$?
6
votes
1answer
242 views

Proof involving the logarithmic integral

Another exercise from Apostol's book, this time we're supposed to prove $$\mathrm{Li}(x)=\frac{x}{\log x}+\int_2^x \frac{dt}{\log^2t}-\frac{2}{\log 2}.$$ which is easy to do via integration by ...
3
votes
3answers
282 views

Approximate Riemann zeta function

Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$. In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function. My question is: Is there a Functional equation for ...
6
votes
1answer
303 views

How do I express $\int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz$ in terms of named functions?

Recently I derived an expression for a particular probability density function. The expression contains the integral $$ f(t,v,a) = \int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz = 2a ...
27
votes
7answers
2k views

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...
1
vote
1answer
316 views

Contour Integral with confluent hypergeometric function

Can we get a closed form for the following contour integral?. Let us assume that n is a non-negative integer, $\frac{1}{2\pi ...
4
votes
2answers
846 views

The relationship between Legendre Polynomials and monomial basis polynomials

I am currently doing filter designs and stumbled across this mathematical problem which I cannot understand. I was hoping for some insight from experts around this field to help me with this. ...
1
vote
1answer
513 views

Help interpreting a gamma distribution

The following is from an article I'm reading and is the conditional density of a random variable that is distributed according to a gamma distribution, conditional on the value of a parameter $t$. ...
1
vote
0answers
69 views

Evaluation of this integral

Given the integral $$\int_{-\infty}^{\infty} dx \;\psi(1/4+ix/2)\exp(-ax^2)$$ How can I evaluate that in the limit $ a\to 0$ and $ a\to \infty$? Here $\psi(x)$ is the digamma function. Thanks.
2
votes
1answer
327 views

About integration related to the gamma function

I would like to compute the integral $$ \int_{0}^{\infty}\frac{1}{\sqrt{2t}}e^{-\frac{1}{2t}}dt $$ which wolfram alpha says that it does not converge. However by letting $x=1/2t$ I get ...
2
votes
2answers
716 views

Euler Gamma function $\Gamma(z)$ on $\mathbb{C}$

I'm working on an exercise about the Gamma Function from Euler. First, $\Gamma (z)= \int_0^\infty e^{-t}t^{z-1}dt$. Now, if we consider the "similar" function $\int_{\frac{1}{n}}^\infty ...
1
vote
1answer
137 views

does this integration converge?

Consider the first kind Bessel function $J_0$, one way to define it is $J_0(x)=1/\pi \int_0^\pi \cos(x \sin t)\;dt$. My question is, $\int_0^n J_0(x)\;dx$ converge when $n$ tends to infinity? For ...
2
votes
0answers
192 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...
14
votes
2answers
320 views

tough integral involving the Cosine integral

I ran across an integral on a German math site that has a friend of mine and I quite stuck. They give, without derivation, $$\int_0^\infty \mathrm{Ci}(\alpha x)\mathrm{Ci}(\beta x)dx=\frac{\pi}{2 ...
1
vote
1answer
110 views

Integration error spotting

What have I done wrong?  I have to evaluate the following integral: $$\int\limits_0^\infty\int\limits_0^{2\pi} \phi(r^2)\delta'(r^2-a)\delta\left(\theta- \left(n+{1\over 2}\right){\pi\over ...
5
votes
1answer
466 views

Dirac delta function relation

Is there a way of showing that for a monotonic (not necessarily strictly monotonic) function $f(x)$ defined on $[a,b]$ which has a simple zero at $c\in (a,b)$ that $\int\limits_a^b ...
0
votes
1answer
49 views

Bessels and initial conditions

I'd like to know if I have got the following ideas right: 1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial ...
10
votes
2answers
608 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
1
vote
1answer
242 views

Bessel's function

I read that $\int\limits_0^1 xJ_n(j_{na}x) J_n(j_{nb}x) dx={1\over 2}\delta_{ab}[J_n'(j_{na})]^2$, where $j_{na},j_{nb}$ are zeros of $J_n$, the Bessel function of the $n$th degree. Is there a ...
1
vote
0answers
90 views

Fourier analysis confusion

I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory: Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty ...
5
votes
1answer
239 views

Lower bounding a ratio of gamma functions

I am trying to show that the following function has a lower bound of $\ \frac{1}{2}$ for all $c\geq 2$. Or, alternatively, that that function increases with $\ c$: ...
5
votes
1answer
1k views

Spherical harmonics give all the irreducible representations of $SO(3)$?

It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? ...
4
votes
1answer
199 views

Special functions as representations of Lie Groups

-The spherical harmonics $Y_{lm}$ are complete on $L^2(S^2)$. They are also a representation of the (compact) Lie group $SO_3 (\mathbf{R})$. -The functions $e^{i n x}$ are complete on ...
8
votes
2answers
1k views

Integrating a product of exponentials and error functions

I have the following integral $$ \int\limits_0^\infty x^2\exp(-\delta x^2)\operatorname{erf}(\gamma x)\,dx. $$ Ideally, I would like a closed-form in terms of common functions, but a series answer ...
4
votes
2answers
522 views

Writing complete elliptic integral of first kind as a hypergeometric function

I am trying to show $K=\int_{0}^{\frac{\pi}{2}}\frac{dz}{\sqrt{1-k^{2}\sin^{2}(z)}}$ can be written as $\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})$. First I used ...
2
votes
4answers
485 views

(Should be easy) Legendre polynomial integration debugging

I am trying to evaluate the following integral: $$I_n=\int\limits_{-1}^1 f(x)P_n(x)dx$$ where $f(x)=1$ for $x\in[-1,0)$ and $f(x)=-1$ for $x\in(0,1]$ and $P_n(x)$ is the Legendre polynomial of degree ...
5
votes
2answers
9k views

Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
2
votes
0answers
181 views

Can we confirm $\sum_x{x!}$?

According to Wikipedia's article on indefinite sums, they list the following formula near the bottom of the page: $$\displaystyle \sum_x{\Gamma(x)}=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}{e}+C$$ ...
9
votes
0answers
358 views

New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$ This is most closely related ...
8
votes
1answer
353 views

Prove $f(x)=\int\frac{e^x}{x}\mathrm dx$ is not an elementary function

How do I prove that the exponential integral $$f(x)=\int \frac{e^x}{x}\mathrm dx$$ is not an elementary function? Also, what are the general methods and tricks to prove that an integral or solution ...
12
votes
3answers
340 views

How this integral $ \int_0^z\frac{1-e^x}{x} dx$ is connected to the Gamma function and Euler constant?

This is my first question in this forum; I hope it is an appropriate question. The Wolframalpha website tells me that $$ \int\nolimits_0^z\frac{1-e^x}{x} dx = \log (-z)+\Gamma(0, -z)+\gamma\quad ...
6
votes
2answers
490 views

Beta Function — finding a lower bound based on parameters

I would like to show that $$ 1-\frac{1}{c}Beta\left(c+1,\frac{1}{c}\right) \geq \frac{1}{c+1}.$$ for all $c \geq 2$. I have plotted it out for $c$ up through 200, and it seems to hold. Does anyone ...
1
vote
1answer
87 views

What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 ...
8
votes
1answer
1k views

On the growth of the Jacobi theta function

So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5: Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function $$\theta(z | r) = ...
1
vote
0answers
51 views

Further simplifiable?

I have an equation $(xy')'+kxy=xf(x)$ where $k$ is not an eigenvalue; $y(x)$ and $f(x)$ are subjected to boundary conditions (i) bounded as $x\to 0$; (ii) $y(1)=0=f(1)$ I want to get the ...