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

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3
votes
1answer
353 views

Integrals $\int\limits_0^t {{e^u}\log u\operatorname d \!u} $ and $\int\limits_0^t {{e^{ - u}}\log u\operatorname d \!u} $

Ok, I want to find $$\int\limits_0^t {{e^u}\log udu} $$ and $$\int\limits_0^t {{e^{ - u}}\log udu} $$ I'm thinking as follows $$d\left( {{e^u}\log u} \right) = {e^u}\log udu + \frac{{{e^u}}}{u}du$$...
2
votes
2answers
1k views

Spherical Bessel Zeros

I was wondering if there is a known closed form solution for the zeros of the spherical Bessel functions. While doing a quantum assignment, I came across them as a solution for the spherical infinite ...
8
votes
3answers
834 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
12
votes
2answers
1k views

The roots of Hermite polynomials are all real?

The Hermite polynomials are defined as $$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$ How does one prove that all the roots of the Hermite polynomial are real?
3
votes
1answer
708 views

Integral representation of modified bessel function

How can I obtain $$I_n(x)= \frac{1}{2\pi}\int_{0}^{2\pi}e^{in\theta}e^{x \cos\theta}\,d\theta ,$$ from the integral $$J_n (x) =\frac{1}{2\pi} \int_{0}^{2\pi}e^{-in\theta}e^{ix \sin\theta}\,d\theta .$$ ...
1
vote
1answer
391 views

Verifying identities for Riemann zeta function

I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
8
votes
1answer
805 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
1
vote
1answer
2k views

Associated Legendre polynomials of fractional order

My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers $m$ and $l$; i.e., $P_l^m(x)$, for $x \in [-1,1]$. Usually one assumes that $m$ and $l$ are ...
1
vote
1answer
157 views

distribution function

Let$(X,\mu)$ be a measure space. $f:X \to \mathbb{R}$ be a measure function. For every $t\in \mathbb{R}$ the distribution function $F$ of $f$ is defined as $ F(t)=\mu\{x \in X:f(x)<t\}.$ I have ...
0
votes
1answer
248 views

Create 'smooth breakpoint function' by using integral?

Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a ...
1
vote
3answers
106 views

Oddities in the Definition of IntegralCosinus ${\rm Ci}(x)$

Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ \begin{...
1
vote
2answers
579 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
4
votes
1answer
159 views

Solving an integral using generating functions, coefficients of equivalent series don't match!?

Please don't be frightened by the length of this, I just wanted to provide ample detail. If you want, you can skip the derivation and go straight to the result at the bottom. I have $$ \begin{...
10
votes
3answers
952 views

Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
7
votes
1answer
969 views

On the completeness of the generalized Laguerre polynomials

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \begin{equation} \int_0^{\infty} e^{-x}x^kL_n^k(x)L_{m}^k(x)dx=\...
1
vote
1answer
433 views

Advanced application of the Binomial Theorem

I'm trying to solve the following integral: $$ \int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx $$ Where $C_{n}^{\lambda}(x)$ is a ...
2
votes
1answer
200 views

real-value solution

I have this integral $$\int\frac{dz}{\sqrt{(z^{2}-\rho^{2})(\lambda^{2} - z^{2})}}$$ and parameters obey the following conditions $$z= \exp[k\varphi],$$ $$\lambda^{2} = \frac{b + \sqrt{b^{2} - 4ac}}{...
1
vote
1answer
117 views

Problem with the Limits of the Hypergeometric Series

I'm new to the hypergeometric series, and I'm trying to decipher a proof in which the author identifies a particular finite sum as a a hypergeometric series. The particular summation is: $$ \begin{...
5
votes
1answer
163 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma \left(\...
2
votes
1answer
266 views

Closed form formula for series involving derivatives of reciprocal gamma function

How to get closed form for the sum $\displaystyle{\sum\limits_{k = 1}^\infty {\frac{{{p^k}}} {{\left( {2k} \right)!!}}\frac{{{d^k}}} {{d{s^k}}}{{\left. {\frac{1} {{\Gamma \left( s \right)}}} \...
25
votes
1answer
1k views

How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ...
5
votes
1answer
330 views

Orthogonality of the Gegenbauer Polynomials

Typically the orthogonality relation for the Gegenbauer polynomials is given as: $$ \int_{-1}^{1}C_{n}^{\alpha}(x)C_{m}^{\alpha}(x)\cdot(1-x^2)^{\alpha-1/2}dx=\frac{\pi2^{1-2\alpha}\Gamma(2\alpha+n)}...
7
votes
3answers
446 views

Reference requests: Jitsuro Nagura

I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev $\psi$ function says that ...
0
votes
1answer
82 views

Trigonometric term in digamma function $\psi_{0}(-n)$

Solutions to expressions s.a. $$ S(n)=\sum_{k=1}^{n}\frac{1}{k-r} = \psi_{0}(n-r+1)- \psi_{0}(1-r), $$ involves digamma function. For positive values it has the largest term $O(\log(n))$, but ...
0
votes
1answer
150 views

Integration about standard normal

Let $N(x)$ denote the cdf of standard normal and $n(x)$ denote the pdf of standard normal. How to evaluate the integral $\int\limits_{-\infty}^\infty N(a+x) n(x) \mathrm{d} x$ ? Thanks a lot!
21
votes
1answer
556 views

Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
0
votes
1answer
120 views

Looking for a magnifyer function

I am looking for a function $f$ having the following characteristics: $f$ defined on $[0,1]$ $f(0)=0$ $f(1)=1$ $ \forall x \in ]0,1[, x <f(x) < 1$ $f$ differentiable on $]0,1]$ $f'>0$ $...
3
votes
1answer
324 views

Proof of an inequality under Equivalence of Weierstrass and Euler's Definitions of Gamma Function

I am using my lecturer's notes on Special Functions. When dealing with Gamma Functions under the title Equivalence of Wierstrass and Euler's Definitions, he has used a lemma (with no proof). I tried ...
8
votes
1answer
831 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and $\operatorname{li}...
4
votes
1answer
642 views

Calculating the limit of the derivative of a sum of digamma functions

The (real) digamma function is new to me and I notice that under some circumstances a sum of $\log\Gamma$ functions will have a derivative consisting of a sum of digamma functions that converge to a ...
2
votes
0answers
52 views

examples of 'continuous bases of functions,' like the Fourier transform

For suitable choice of a one-parameter family of functions $\{ g_w:\mathbb{R} \rightarrow \mathbb{C} \}_{w\in \mathbb{R}}$, the following two statements are equivalent (modulo sets of measure $0$): $\...
1
vote
1answer
892 views

Square of the hypergeometric function

Given the hypergeometric function $\,_2F_1$, Pochhammer symbol $(m)_n$, and $0<a< 1$, anybody knows how to prove that, $\,_2F_1(a,1-a;1;z) = \sqrt{\sum_{n=0}^\infty \frac{(a)_n (1-a)_n (\tfrac{...
7
votes
5answers
970 views

How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$?

How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
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 - y\...
12
votes
3answers
597 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 i ...
4
votes
1answer
796 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 f(p)}\frac{\Gamma(p+1)}{\...
4
votes
1answer
350 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
145 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
375 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
300 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
274 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 $$ \binom{n}{m_1,m_2,\...
2
votes
3answers
505 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}} \left(\...
6
votes
1answer
281 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 $x$...
2
votes
0answers
199 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 + (\sqrt{5}\,\eta(5\...
1
vote
0answers
251 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
252 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 ...