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

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6
votes
1answer
305 views

Did Euler have an alpha function

I've heard of Euler Gamma function: $\Gamma(x)$, and Euler's beta function: $\text{B}(x,y)$. Did Euler have an alpha function?
2
votes
1answer
279 views

Question on legendre equation - part 2

I would like to know if is possible to have regular solutions of Legendre equation when the constant $l$ in the Legendre equation $(1-x^2)u''-2xu''+l(l+1)u=0$ is a non integer number? I am interested ...
2
votes
1answer
77 views

Question on Legendre equation

I have a doubt. If Legendre equation has a polynomial solution, is the constant $l$ in $l(l+1)$ necessarily a integer number? Asked in another way, is possible $l(l+1)$ be a integer if $l$ is not an ...
0
votes
1answer
311 views

Solve in terms of the Gamma function

Show: \begin{align*} \int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\,\mathrm d x &=\frac{\sqrt \pi}{4}\left(\frac{\Gamma ...
4
votes
0answers
394 views

Solving inhomogenous bessel equation

I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
6
votes
1answer
250 views

Definite integral involving Fresnel integrals

I am seeking to evaluate $\int_0^{\infty} f(x)/x^2 \, dx$ with $f(x)=1-\sqrt{\pi/6} \left(\cos (x) C\left(\sqrt{\frac{6 x}{\pi }} \right)+S\left(\sqrt{\frac{6 x}{\pi }} \right) \sin ...
3
votes
1answer
1k views

Relationship between Legendre polynomials and Legendre functions of the second kind

I'm taking an ODE course at the moment, and my instructor gave us the following problem: Derive the following formula for Legendre functions $Q_n(x)$ of the second kind: $$Q_n(x) = P_n(x) ...
9
votes
0answers
468 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
1
vote
0answers
128 views

Question on arguments of the Gamma Function

I came accross with this site about Gamma function. I just want to verify, clarify, whatever you may want to call it. It says you can compute for the gamma value for a negative argument using ...
0
votes
1answer
202 views

Expressing solution to an inequality with Lambert W function

I'm new to Lambert functions, any ideas on how to solve this are welcome: $$ \theta \rho^{\theta}+r \theta>v $$ where $\theta \in \mathbb{R}^{+}, -1<r,v<1, \ 0<\rho<1$. I've tried ...
3
votes
0answers
140 views

Modified Bessel function

I use the standard notations. When $x$ is real then by definition $$ I_{\nu}(x)=e^{-\nu\pi i/2}J_{\nu}(ix). $$ I want to define $I_{\nu}$ for complex $z$. Watson (Treatise of the Theory of Bessel ...
2
votes
1answer
90 views

To solve $U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0 $

$$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$$ $$\frac{\partial}{\partial t }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$$ ...
6
votes
2answers
235 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 ...
43
votes
9answers
17k 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 $$ ...
1
vote
2answers
948 views

How do I show the Wronskian of $(J_{a}(x),Y_{a}(x)) = \dfrac {2} {\pi x}$

Based of using my undergrad class notes. I know that the wronskian of $(J_{a}(x),Y_{a}(x))$ is $ W(J_{a}(x),Y_{a}(x)) = \left| \begin{matrix} J_{a}(x) & Y_{a}(x) \\ J_{a}'(x) & ...
6
votes
2answers
401 views

Bounding the Gamma Function

I'm trying to verify a bound for the gamma function $$ \Gamma(z) = \int_0^\infty e^{-t}t^{z - 1}\;dt. $$ In particular, for real $m \geq 1$, I'd like to show that $$ \Gamma(m + 1) \leq ...
4
votes
3answers
430 views

Limiting behavior of gamma function

I am trying to determine whether $\Gamma(x+iy)\rightarrow 0$ as $y\rightarrow\infty$. How should I go about doing it? I was trying to see if I could get anything from ...
1
vote
0answers
78 views

for what $\nu$ does Riemann-Liouville differintegral of digamma function $\psi(z)$ exist?

For what values of $\nu$ does the Riemann-Liouville differintegral $_{-\infty}D_{z}^\nu$ of the digamma function $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ exist, with $c=-\infty$? All I've got so far is ...
3
votes
0answers
306 views

How to prove Gegenbauer's addition theorem?

How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\over 2 r_1 r_2}\int_{|r_1 - r_2|}^{r_1+r_2} e^{-{r\over D}} P_k\left(r_1^2 - r^2 + r_2^2 \over 2 r_1 ...
0
votes
1answer
137 views

How can I get $B(x,y+1)= \frac{y}{x+y} B(x,y)$ using integration by parts?

So far I have $B(x,y+1) = \int_{0}^{1} t^{x-1}(1-t)^{(y+1)-1} dt =\int_{0}^{1} t^{x-1}(1-t)^{y} dt $ for $x,y>0$. I tried doing integration by parts by letting $dv = (1-t)^y$ and $u=t^{x-1}$ but it ...
0
votes
1answer
56 views

How to find $\lim_{x \rightarrow -N} J_{a} = (-1)^N J_{N}$?

So far I have $$\lim_{a \rightarrow -N} J_{a} = \lim_{a \rightarrow -N} \mid \frac{x}{2} \mid^a \sum_{k=0}^{\infty} \frac{(-x^2/4)^k}{k! \; \Gamma(a+k+1)} = \mid \frac{x}{2} \mid^{-N} ...
5
votes
0answers
148 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
1
vote
1answer
309 views

Limit of a summation wth Gamma function

Can anyone prove this (I'm very confident that it is correct) or have any idea how this can be handled: $$ \lim_{n \rightarrow \infty} \frac{1}{n-1}\sum_{i=1}^{n-1} \frac{1}{(\alpha-1)(n-i) -1} ...
1
vote
1answer
212 views

To find closed form of $f(x)=\int_0^{\frac{\pi}{2}} e^{\sqrt{1-x^2 \sin^2 t}}\, dt$ as known functions

$$f(x)=\int_0^{\frac{\pi}{2}} e^{\sqrt{1-x^2 \sin^2 t}}\, dt$$ $u=\sin t$ $$f(x)=\int_0^{1} \cfrac{e^{\sqrt{1-x^2 u^2}}}{\sqrt{1-u^2}}\, du$$ $$f'(x)=\int_0^{1} \frac{-xu^2}{\sqrt{1-x^2 u^2 ...
1
vote
1answer
66 views

Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$

Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$ Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W ...
1
vote
2answers
160 views

An injective map where each value is mapped to many others?

I want "something" ("something" because maybe it is not really a mathematical function, called F in the above image) that can describe what is shown on the image. A given value from a domain Xi can ...
0
votes
1answer
88 views

Two $\psi$ functions

This is either a notation/history question or a point of confusion. In (for example) Ramanujan's proof of Bertrand's postulate, he uses the following notation: $\log [x]!$ means $\log ([x]!),$ in ...
1
vote
0answers
143 views

Solving for $x$ in $y=x^x(\ln x + 1)$ (Lambert W?)

I made a bunch of problems exercising the Lambert W-function in the solution, because I like to exercise to new concepts that I learn about. One that I came up with was rearranging $y = x^x(\ln x + ...
1
vote
2answers
2k views

Problems regarding integrals involving Legendre polynomials

I am finding difficulty doing this integral involving Legendre polynomials. $$\int_{-1}^1 x^2 P_{n-1}(x)P_{n+1}(x)dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}$$ I have two strategies in my mind both of ...
1
vote
2answers
1k views

Proving a property of Legendre polynomials containing its derivatives

I am trying to prove the following property of Legendre polynomials. $$nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$$ My guess is that I somehow have to use the Bonnets recursion formula ...
1
vote
1answer
233 views

How does $\zeta(1 - s)$ become $(-1/s + \cdots)$?

Why is $$\zeta(1 - s) = -\frac{1}{s} + \cdots$$ for small negative values of $s$? A detailed explanation would be appreciated.
4
votes
1answer
137 views

Nagura's paper--can we substitute for the original upper bound?

This question concerns two results about primes. The first is J. Nagura's 1952 result, that there is a prime on the interval $[x, (1+1/5)x] $ for $x> 2103,$ which depends on the result derived ...
6
votes
1answer
2k views

Orthogonality of Bessel functions

The orthogonality for Bessel functions is given by $\int_0 ^1 rJ_n(k_1r)J_n(k_2r) dr=0,\ (k_1 \neq k_2)\\ \neq 0, (k_1=k_2,\ J_n(k_1)=J_n(k_2)=0\ \mbox{or}\ J'_n(k_1)=J'_n(k_2)=0)$ This suggests a ...
2
votes
0answers
263 views

Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability ...
1
vote
2answers
811 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?
2
votes
1answer
144 views

Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.
7
votes
1answer
1k views

Median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
5
votes
3answers
438 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
1
vote
1answer
129 views

How to compute $\int\frac{x^7}{\sin(x)} dx$ efficiently?

How to compute $\int\frac{x^7}{\sin(x)} dx$ efficiently ? We need $Polylog$ for this.
7
votes
2answers
717 views

How to prove asymptotic limit of an incomplete Gamma function

How can I prove: $$\lim_{z\to\infty}{\Gamma(z, x)\over\Gamma(z)} = 1$$ ? Here $\Gamma(z, x)$ is the upper incomplete gamma function and $\Gamma(z)$ is the gamma function. This must be something ...
2
votes
2answers
421 views

What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
1
vote
0answers
46 views

Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $ It's my ...
2
votes
0answers
72 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
0
votes
1answer
88 views

Limit of an integral containing a product

I'm stuck on the following problem: given the integral: $$I(N)=\int \prod_{k=1}^N \left(k-\frac{k}{x}\right) \, dx$$ calculate the following limit: $$I_{\infty}(N)=\lim_{x\to\infty}I(N)$$ I know that ...
1
vote
1answer
205 views

simpler expression for terminating 3F2 series with negative unit argument

I am hoping to evaluate a simpler expression for the following: $${}_3F_2(-n,1,1; a, (a+3)/2; -1)$$ Here $n,a \in \mathbb{N}$ and $a$ is odd. I am also interested in the asymptotics in $n \in ...
3
votes
1answer
749 views

Calculating digamma and trigamma functions

What is the best method to calculate the value of digamma and trigamma functions? Wikipedia suggests using recurrence relations $\psi_0(x+1) = \psi_0(x) + 1/x$, $\quad\psi_1(x+1) = \psi_1(x) - 1/x^2$ ...
1
vote
0answers
228 views

Integral of Hankel functions

The integral: $$S(x)=\int_0^x H^{(1)}_{k+1}(\eta)H^{(2)}_{k-1}(\eta)d\eta$$ can be expressed as a combination of Hypergeometric functions and trigonometric functions. I have some difficulty to ...
6
votes
1answer
324 views

Integral related to the modified Bessel function

I would like to solve the integral $$F_n(\kappa,\theta,\phi)=\int_{-\pi}^{\pi}{\rm e}^{\kappa\cos(x-\theta)}\cos(n\, x-\phi)\,{\rm d}x$$ that appears related to the identity ...
0
votes
1answer
85 views

Recurrence equation and special functions

Can someone give me a proof or a hint on why the recurrence equation: $$g(k+2)=k*g(k+1)-g(k)$$ has the solution: $$g(k)=c_1 {_0\tilde F_1}(;k;-1)+c_2 Y_{k-1}(2)$$ where ${_0\tilde F_1}(;a;x)$ is the ...
2
votes
1answer
139 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...