0
votes
2answers
101 views

Extremely tough indefinite integral

This integral does indeed use special functions, so do include them here. Evaluate: $\int \frac{1}{\sqrt{x}\ln(x)} dx$ $x = {\sqrt{x}}^{2} \space \text{let} \space u = \sqrt{x}$ $= 2\int ...
0
votes
2answers
115 views

Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me ...
0
votes
0answers
41 views

Definite integral with doable improper case

Is there a way to evaluate one or both of the following integrals: $$ \int_{a_1}^{a_2} e^{ib_1(x+b_2 \sqrt{1+x^2})}dx \quad \text{and}\quad \int_{a_1}^{a_2}\frac{x}{\sqrt{1+x^2}} e^{ib_1(x+b_2 ...
2
votes
1answer
16 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
0
votes
1answer
43 views

Orthogonality of associated Legendre polynomials

Let $P_n(x)$ be the $n$-th degree Legendre polynomial. Let $k$ be a nonnegative integer less than or equal to both $n,m$. How to prove that $$ \int_{-1}^1 (1-x^2)^k D^kP_n(x) D^kP_m(x)\,dx = ...
5
votes
1answer
339 views

Generalized Legendre differential equation

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ which is ...
5
votes
3answers
124 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
2
votes
1answer
153 views

$\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ Legendre polynomial.

I was wondering whether there is a way to say what $$\int_{-1}^{1} x^{k} P_n(x)dx$$ is, where $k,n$ are positive integers or zero and $P_n$ is the n-th Legendre polynomial? I am looking for an ...
3
votes
1answer
41 views

Properties of a Mehler's type integral

When computing the resolvent of the Laplace beltrami opetator on $S^n$ for even dimension, $n=2k$, I came across the following integral $$ ...
2
votes
2answers
102 views

Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
3
votes
1answer
26 views

bessel function maximizer

I try to find global maximum for $ \frac{J_2(x)}{x^2} $ I suspect it happens at x=0 ( plotting the graph) where the value of the function is $ \frac{1}{8} $ I know local maximizers are at zeros of ...
0
votes
2answers
66 views

An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$

Let $T(x,y):=(t_1(x,y),t_2(x,y))$ be a continuous bijection, namely a homeomorphism on $[0,1]^2$. I am trying to find a $T$ such that $\det(J_T)=1$. (*) The trivial cases are $T(x,y)=(x,y)$, ...
4
votes
0answers
170 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
1
vote
0answers
30 views

Kummer U function: U(a,a+1/2,z)

Is there any way to simplify $U(a,a+1/2,z)$ or relate it to any other common special functions such as the incomplete beta function or the incomplete gamma function? Here, $U(a,b,z)$ is Kummer's U ...
2
votes
2answers
66 views

Characterizations of cosine and sine functions

Does the following set of rules characterize cosine and sine functions? $C(x)$ and $S(x)$ are $2\pi$-periodic, with $2\pi$ the smallest period. $C(x)$ is even and $S(x)$ is odd. $C(0)=1, S(0)=0$. ...
0
votes
2answers
81 views

Prove this integral, the Dirichlet's formula

Show that $$\int\int_R x^{p-1}y^{q-1}dxdy = \frac{\Gamma(\frac{p}{2})\Gamma(\frac{q}{2})}{\Gamma(\frac{p}{2}+\frac{q}{2}+1)},$$ where R is the region bounded by the first quadrant of the circle ...
13
votes
2answers
190 views

Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?

Ramanujan gave the following identities for the Dilogarithm function: $$ \begin{align*} \operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) ...
9
votes
1answer
146 views

Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...
3
votes
1answer
665 views

An example of almost periodic function

"I need a continuous almost periodic function $f(x)$ such that $f(x)$ exists as $x$ tends to infinity. But this function should not be constant, which is a trivial example." Definition of almost ...
3
votes
1answer
198 views

Bernoulli Map properties

I am referring to the function stated here http://en.wikipedia.org/wiki/Dyadic_transformation This map is defined on $[0,1]$ by $f_n(x)=nx [mod 1]$ There are three things I do not quite understand, ...
2
votes
1answer
128 views

Proving a function (involving Erf) has a unique real root

I am trying to prove that the following function: $$g(x) = x - \sqrt{\frac{2}{\pi}} e^{-x^2/2} ...
9
votes
1answer
194 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
0
votes
1answer
80 views

Rewriting $\delta(x,y)$ in terms of $\delta(r)$.

On my textbook is written: The function $\tau^{-1} u(x/\tau)$ is a rectangle function of height $\tau^{-1}$ and base $\tau$ and has unit area; as $\tau$ tends to zero a sequence of unit-area pulses ...
3
votes
1answer
143 views

Integral Representation of Bessel Function (K)

There is an integral representation for the modified Bessel function of the second (or third depending on who you talk to) kind (denoted $K_\nu$) that says: $$K_\nu(z) = ...
12
votes
1answer
392 views

Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$ where $G$ is the Catalan's Constant. Numerically, it's ...
28
votes
1answer
492 views

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
1
vote
0answers
61 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
3
votes
1answer
441 views

Fourier transform of the indicator of the unit ball

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.
1
vote
1answer
87 views

Looking for an analytical expression of this horror-integral

I have given a function $$a(m,n,\mu,\nu,p):=\frac{2p+1}{2}\frac{(p-m-\mu)!}{(p+m+\mu)!}\int_{-1}^1 P^m_n(x)P^\mu_{\nu}(x)P^{m+\mu}_p(x)dx.$$ (of course all parameters are appropriate integers, so that ...
1
vote
0answers
38 views

This limit of the hypergeometric function makes me stunning…

I am currently reading this paper Physics paper please have a look at the definition of (20) and then (36). In (36) they investigate the limit of the hypergeometric function ...
0
votes
2answers
83 views

Smooth step between $-1$ and $1$

I am currently interpolating the step between two functions from $-1$ and $1$ smoothly therefore I used $\tanh$. Since I am quite confident with the result, but interested in further ways to do this, ...
3
votes
1answer
100 views

On the absolute integrability of radially symmetric functions

Let $\phi:\mathbb R\to\mathbb R$ be an smooth, even function and $\int_\mathbb R|\phi(t)|^p\,\mathrm dt<\infty$, that is, $\phi$ is pth-power integrable in $\mathbb R$ iff $p\geq p_0$ for ...
1
vote
1answer
207 views

Express complex Bessel function in terms of functions taking real arguements

I want to use the Bessel function in C++. Since this one is not implemented there for complex arguments, I am looking for a way to express the bessel function(first and second kind) as: ...
3
votes
2answers
61 views

Prove approximation given by the physicist Max Born

In an old book about optics, I have found a nice approximation, that for large l one has: $$P_l(\cos(\theta)) \sim \sqrt{\frac{2}{l \pi \sin(\theta)}} \sin \left((l+\frac{1}{2}) \theta + ...
1
vote
0answers
287 views

Relation between the exponential function and the modified bessel function of second kind

I found the following sentence at the wikipedia page : Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα(this is the mod. bessel function of the ...
3
votes
1answer
71 views

Properties of the Fourier transform of a certain function

In my research I met the Fourier transform of the function $f(x)=(1+x^2)^{-1/2}$. I was not able to find its explicit formula. Is this a function known as a 'special function'? I would like to know if ...
13
votes
3answers
699 views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$ We can prove using the Beta-Function identity that $$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...
1
vote
1answer
264 views

Difference between Rician distribution and Gaussian distribution

could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated Thank ...
1
vote
1answer
138 views

Find local maxima of this quadratic function

How can I find local maxima of this quadratic function? $$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$ which ...
5
votes
1answer
193 views

A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$

Is there a special trick to calculate this integral? $$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$ for $\lambda>0$.
2
votes
1answer
70 views

When are the binomial coefficients equal to a generalization involving the Gamma function?

Let $\Gamma$ be the Gamma function and abbreviate $x!:=\Gamma(x+1)$, $x>-1$. For $\alpha>0$ lets generalize the binomial coefficients in the following way: $\binom{n+m}{n}_\alpha:=\frac{(\alpha ...
3
votes
3answers
95 views

The number $ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}$

For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number $a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can ...
0
votes
1answer
71 views

Ordinary generating function for Bernoulli polynomial

I know the exponential generating function for the Bernoulli polynomial $B_n(x)$:$$\frac{te^{tx}}{e^t-1}=\sum_{n=0}^\infty B_n(x)\frac{t^n}{n!}.$$ But is there an ordinary generating function? i.e a ...
1
vote
0answers
36 views

Sup and lim sup of a function defined by double series

It is unlikely that the following function has a closed form expression: $$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$ ...
2
votes
0answers
70 views

Closed form formula for a double series related to wave equation

Does anyone have a closed form formula for the double series $$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$ This is related ...
10
votes
1answer
209 views

The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)$

Does anyone have a proof that $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)=\frac{\pi}{8}$$
1
vote
1answer
108 views

Why does the Gamma function interpolate $(n-1)!$?

Why does the Gamma function interpolate $(n-1)!$ and not $n!$ instead? What is the historical reason?
2
votes
1answer
79 views

Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?

Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$ where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
6
votes
2answers
201 views

Property of sum $\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}$

Is it true that for all $n\in\mathbb{N}$, \begin{align}f(n)=\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}\end{align} is always rational. I have calculated via Mathematica, which says ...
28
votes
9answers
8k views

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ ...