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
321 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
639 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
135 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
188 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
314 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
454 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
593 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
235 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
89 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
229 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? ...
3
votes
1answer
189 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 ...
7
votes
2answers
934 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
481 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 ...
1
vote
4answers
476 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
8k 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: $$\displaystyle \sum_x{\Gamma(x)}=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}{e}+C$$ ...
8
votes
0answers
352 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
348 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
339 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
456 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 ...
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 ...
3
votes
1answer
91 views

Integrating a differential equation?

How does $(xJ_0'(x))'+xJ_0(x)=0\implies\int\nolimits_0^1 x J_0(ax)J_0(bx) dx={bJ_0(a)J_0'(b)-aJ_0(b)J_0'(a)\over{a^2-b^2}}?$ Thanks. Perhaps int by parts? But how do I get the RHS form?
1
vote
0answers
297 views

Nested Integral of exponential function with trigonometric identities

Is there any possibility to simplify the following integral or any function that is equivalent to the following integral? $$ ...
6
votes
1answer
837 views

Inverse of the polylogarithm

The polylogarithm can be defined using the power series $$ \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. $$ Contiguous polylogs have the ladder operators $$ \operatorname{Li}_{s+1}(z) ...
13
votes
4answers
553 views

Intriguing polynomials coming from a combinatorial physics problem

For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define $$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$ ...
1
vote
0answers
88 views

Computing the PDF of a product of the sum of 2 Nakagmi-m R.V.s with a Normal R.V

I really have two questions: One is about computing a PDF and the second is about how to sum a series involving $K_v(x)$ that the PDF in question seems to contain. I have come across the following ...
2
votes
1answer
149 views

Solving for $y$ in $y = x \ln(y)$

I want to solve $y = x \ln(y)$ for $y$ in terms of $x$. Wolfram Alpha kindly produces this plot with the solution, $y = -x W(-\frac{1}{x})$, where $W$ is the Lambert function. However, that only ...
8
votes
2answers
557 views

Is there a combinatorial way to see the link between the beta and gamma functions?

The Wikipedia page on the beta function gives a simple formula for it in terms of the gamma function. Using that and the fact that $\Gamma(n+1)=n!$, I can prove the following formula: $$ ...
7
votes
1answer
778 views

the limit of the ratio of two $\Gamma(x)$ functions

I am interested in the quantity $$ a_{n} = \sqrt{n/2} \frac{\Gamma((n-1)/2)}{\Gamma(n/2)}$$ (this is the geometric bias of the non-central t-distribution with $n$ d.f.) After some plotting, my hunch ...
4
votes
2answers
540 views

Digamma function integral

Does anyone how to get a finite value to this integral ? $ \int\nolimits_{0}^{\infty} dx \frac{ \Psi (1/4+ix/2) +\Psi (1/4-ix/2)}{x^{2}+1/4} $ i have tried residue theorem but i got nonsenses :( can ...
15
votes
2answers
1k views

Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
4
votes
0answers
217 views

please help with the a gamma function since i don't even have the idea?

How to prove: $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{\Gamma(\alpha_1+x)}{\beta_1^{\alpha_1+x}}\, \frac{\Gamma(\alpha_2-x)}{\beta_2^{\alpha_2-x}}\, ...
3
votes
1answer
293 views

Conceptually, what is the difference between the Beta function and the Beta distribution?

I have read the Wikipedia pages on the Beta function and the Beta distribution, but I'm still not sure I have a good intuition for what's going on. I'm am hoping someone will be kind enough to ...
5
votes
1answer
145 views

orthonormal polynomials

Here is the question: Suppose $P_0, P_1, P_2, \dots$ are polynomials orthonormal with respect to the inner product $$(f,g)=\int_a^b f(x)g(x)W(x)dx,$$ where $W(x) > 0$ is a weight function and ...
8
votes
2answers
163 views

Is there a neat way to show $\int_{-1}^1 \frac{ U_n(z) U_n(z)}{\sqrt{1-z^2}} \mathrm{d} z = \pi (n+1)$

In answering a question on math.SE, I attempted to find integral of Fejér kernel by using $$ K_n(t) = \frac{1}{n} U_{n-1}^2\left( \cos \frac{t}{2} \right) $$ where $U_n(z)$ stands for the ...
3
votes
1answer
187 views

Solving for a variable under $\Gamma(z)$

My friend came to see me regarding some calculation in probability where he would like to know if it is possible to solve for a variable analytically under the gamma function. By this I mean say we ...
3
votes
1answer
986 views

Differentiation of generating function of Hermite's polynomials

The generating function of Hermite's polynomials is given by $G(x,t)=e^{2xt-t^2}$ for $x, t \in \mathbf{R}$. It is known that $\displaystyle G(x,t)=\sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}$ for $x, t ...
2
votes
0answers
106 views

On a certain lemma concerning Incomplete Hypergeometric Functions of Fox type

I am reposting this from MO, since, I guess, my question might have been too elementary for MO and did not receive any reaction at all for the past 30 hours. Here is the link. Here is my post: I am ...
13
votes
5answers
1k views

Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF

As title. Can anyone supply a simple proof that $$x \Phi(x) + \Phi'(x) \geq 0 \quad \forall x\in\mathbb{R}$$ where $\Phi$ is the standard normal CDF, i.e. $$\Phi(x) = \int_{-\infty}^x ...
7
votes
2answers
463 views

An integral about Bessel function

Is there somebody who knows the solution for the integral $$\int_0^\infty\frac{J^3_1(ax)J_0(bx)}{x^2} dx$$ where $a>0,b>0$ and $J(\cdot)$ the bessel function of the first kind with integer ...
3
votes
1answer
753 views

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ...
3
votes
2answers
870 views

Hankel function in terms of planewaves

It is well know that planewaves are a complete basis for solutions to the wave equation. Let us assume a 2D space, and at fixed temporal frequency, the equation reduces to the Helmholtz equation. In ...
8
votes
2answers
578 views

Why is this sum equal to the Logarithmic Integral?

I am using this sum: $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\left((-1)^{k-1} (n-1) + \sum_{j=1}^{k-1}\frac{(-1)^{j+k-1}n (\log n)^j}{j!}\right)$$ Empirically, this is precisely equal to ...
1
vote
1answer
154 views

What are the $\ker$, $\mathrm{kei}$ functions?

In a book titled 'Ordinary Differential Equations and Useful Polynomials', under the chapter 'Bessel's function', the author has introduced four new functions $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, ...
7
votes
1answer
978 views

Variations on the Stirling's formula for $\Gamma(z)$

I am currently reading some material that makes heavy usage of Hypergeometric functions, and there is one particular point about applying Stirling's approximation to various terms consisting of ...
10
votes
2answers
2k views

Integral of product of two error functions (erf)

In the course of my research I came across the following integral: $$\int\nolimits_{-\infty}^{\infty}\operatorname{erf}(a+x)\operatorname{erf}(a-x)dx$$ where ...