Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [special-functions]

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

0
votes
1answer
19 views

Show that second linearly independent solution to legendre ODE is $Q_n(x)=P_n(x)\int^x \frac{dx}{(1-x^2)[P_n(x)]^2}$

$$Q_n(x)=P_n(x)\displaystyle\int^x \dfrac{dx}{(1-x^2)[P_n(x)]^2}$$ The form looks like Green's function, or general solution after the variation of parameters method. I couldnot figure it out. I ...
0
votes
0answers
8 views

Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?

The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,...
0
votes
0answers
7 views

References regarding Green's function on a square domain in 2D

I'm trying to obtain the Green's function on a square domain, i.e, I'm trying to solve the following BVP: \begin{cases} \Delta G(\bar x,\bar y) = \delta^2(\bar x - \bar y),& \bar x, \bar y\...
0
votes
0answers
11 views

Help me find Taylor series for the function $ \frac{1}{(-t;q)^2 _ \infty } $ [on hold]

$$ \frac{1}{(-t;q)^2 _ \infty } $$ where $ \frac{1}{(-t;q)_ \infty } $ q-pochhammer function
1
vote
0answers
62 views

Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$?

(This is related to this question.) Define the integral, $$I_n = \int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$$ with polylogarithm $\rm{Li}_n(x)$. Given the Nielsen generalized polylogarithm $S_{n,p}(z)$, $$...
2
votes
0answers
43 views

Integral $\int_0^x \ln^{n}\left(2\sin\frac{t}2\right)\,dt$

Evaluate $$\mathcal{L}_n(x)=\int_0^x \ln^n\left(2\sin\frac{t}2\right)\,dt\qquad n\in\Bbb N_0, x\in[0,\pi/2]$$ In this answer to a question of mine, @TitoPiezasIII claims that $$\frac{(-1)^n}{n!}\...
4
votes
0answers
125 views

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

In this post, the OP asks about the integral, $$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$ I. User DavidH gave a beautiful (albeit long)...
0
votes
0answers
12 views

Compounding interest rate maximization problem [on hold]

I have to solve a Problem where a Person wants to optimize his income by choosing how often to compound while there are cost of doing so. $\max \ \ W \cdot (1 +r/m)^m - mc$ with $W$=initial Wealth, ...
1
vote
1answer
35 views

Chebyshev polynomial property

I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
3
votes
1answer
549 views

Does anyone recognize the function from this picture?

I was playing with the exterior algebra, and stumbled on this interesting function from $\Bbb N^2 \to \Bbb N$, which I'll call $f(x,y)$. This is plotted from $1 \leq x,y \leq 100$: In this picture, ...
1
vote
0answers
19 views

Help with an identity involving PolyLog and Logs.

I am plotting the following expression on Mathematica $$ \Im\left[-96 \text{Li}_3\left(e^{-i t}\right)-48 t^2 \log \left(1-e^{-i t}\right)\right]-96 \Re\left[t \text{Li}_2\left(e^{-i t}\...
3
votes
2answers
131 views

Integral $\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$

Prove that $$\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$$ I was given this integral in my post Request for crazy integrals. I have never seen an integral like this before ...
2
votes
1answer
90 views

Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

I am trying to evaluate $$P=\frac\pi2\sum_{n\geq1}\frac{{2n\choose n}}{4^n n^2}$$ I used the beta function to show that $$P=\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$$ IBP: $$P=\sin^{-1}(x)\...
1
vote
0answers
25 views

Modified Airy functions

The question is quite formal. I recall the definition of Airy function $$Ai(\tau^{2/3}\zeta)=\frac{\tau^{1/3}}{2\pi}\int e^{i(\sigma^3/3+\sigma\zeta)}d\sigma,\quad Ai'(\tau^{2/3}\zeta)=\frac{i\tau^{1/...
1
vote
1answer
45 views

Is Lerch's transcendent a multi-valued function?

Lerch’s Transcendent is defined by $${\Phi\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$$ when $|z|<1$ or $\Re s>1,|z|=1$. If $s$ is not an integer then $|\mathrm{ph}(a)|<\...
-1
votes
0answers
19 views

Special case of the H-function of two variables

Please help me for to simplify the H-function of two variables in the attached image. Is this function simplified as a known function???? enter image description here
-2
votes
0answers
49 views

Legendre's polynomial formula

I have this problem, and I don't know how to solve it. I did several easier problems of this type, but this is difficult, so I ask for help. Here's the problem: If $P_n$ is the Legendre polynomial,...
1
vote
1answer
56 views

Transformations relating 3F2 at z with 3F2's at 1/z

I am searching for some transformations for a 3F2 hypergeometric function which send the argument z to 1/z. I am aware of the one given in NIST book (p. 410, Formula 16.8.8) in the special case q=2 (...
15
votes
2answers
367 views

Integral $T_n=\int_{0}^{\pi/2}x^{n}\ln(1+\tan x)\,dx$

For $n\in\Bbb N_0$, evaluate in closed form $$T_n=\int_{0}^{\pi/2}x^{n}\ln(1+\tan x)\,dx$$ After seeing @mrtaurho's answer to this question, I realized that it would be possible to generalize his ...
2
votes
1answer
31 views

About a solution to ODE $ (c x)^2/2 y''(x) + a x (1 - b x) y'(x) - d y(x) = 0 $

The ordinary differential equation $$ \frac{(c x)^2}{2} y''(x) + a x (1 - b x) y'(x) - d y(x) = 0 $$ should have two independent solutions which are given by a confluent hypergeometric function $U$ ...
0
votes
1answer
48 views

Finding the $m^\text{th}$ term of an expression?

How to find the $m^\text{th}$ term for the following expression: $$ \left.\frac{\partial^m}{\partial s^m}e^{a s^2}\right|_{s=0}$$ Is there any analytical approach? I computed first few terms ...
1
vote
1answer
40 views

Connecting integral involve integer and half-integer Bessel function

Suppose I have that: $$ \int_0^\pi xJ_0(\alpha x)g(x) dx \geq 0 \ \ \ \ (*) $$ then is there anything I can say about: $$ \int_0^\pi xJ_{\frac{-1}{2}}(\alpha x)g(x) dx \ \ (\geq 0?) \ \ \ (**)$$ ...
1
vote
0answers
30 views

Mittag-Leffler function and Fox-Wright function

I find the following identity in many special functions books without proof. This identity is called the Laplace transform of the Mittag-Leffler function with three parameters. The result is in the ...
0
votes
1answer
26 views

What are the poles and zeros of the Euler Beta function?

For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ does the Euler Beta function $B(x,y)$ equal zero? For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ ...
2
votes
2answers
105 views

Computing the integral $x^2\textrm{sech}^2(x)$

I'm trying to compute the integral $$\int_{0}^{\infty}dx \, x^{2}\operatorname{sech}^{2}(x)=\frac{\pi^{2}}{12}.$$ Manually, one obtains, quite naively, $$\int dx \, x^{2}\operatorname{sech}^{2}(x)=\...
0
votes
0answers
48 views

Evaluating the integral x/(e^(x)+1) [duplicate]

I'm trying to evaluate the following integral $$I=\int_{0}^{\infty}dx\frac{x}{e^{x}+1}=\frac{\pi^{2}}{12},$$ where the result comes from Mathematica. Manually, one has $$\int dx\frac{x}{e^{x}+1}=\...
3
votes
1answer
81 views

Prove an transformation formula for Gauss hypergeometric function $_2F_1(a,b;c;z)$

In " Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme, at page 113 is reported this formula: $$_2F_1(a,b;c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\...
2
votes
1answer
30 views

Equation involving difference of beta CDFs

Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $I_p(a,b)$ is the regularized incomplete beta function. Question: Given $\alpha,\beta,$ and $c>0$, what is $p$? ...
0
votes
0answers
37 views

Looking for an unbounded, monotonic, non-symmetric s-shaped function

I am hoping to find a function that corresponds to an s-shaped curve that satisfies the following properties: 1. unbounded 2. it is not symmetric about its inflection point 3. its derivative at the ...
5
votes
2answers
117 views

Find maximum of $\log(1+x)(1-I_x(a,b-a))$

I am struggling to find (or at least set some bounds on) $$\arg \max_x \log(1+x)(1-I_x(a,b-a)),$$ where $I_x(a,b-a)$ is the regularized incomplete beta function, i.e $$I_x(a,b-a) = \frac {\int_0^x ...
0
votes
1answer
29 views

For what constant $c$ does Struve function $\mathbf{H}_1(x) = c$ have infinitely many roots?

I. Sine integral We start with the plot of the sine integral $\rm{Si}(x)$, The median point $1.57$ in fact is $\frac{\pi}2 \approx 1.5708$, so the equation $\rm{Si}(x) = \frac{\pi}2$ has infinitely ...
5
votes
2answers
421 views

What's a good approximation to the zeros of the cosine integral?

I. Logarithmic integral The logarithmic integral $\rm{li}(z)$ has a unique positive zero at $z \approx 1.451363$ called the Ramanujan-Soldner constant. II. Cosine integral The cosine integral $\rm{...
0
votes
0answers
21 views

Special values of Meijer G-Function

We define a function $$ f(k)= G_{3,3}^{3,2}\left(1\left| \begin{array}{c} -1,\frac{2}{k}-2,\frac{1}{k}-1 \\ 0,\frac{1}{k}-2,\frac{2}{k}-2 \\ \end{array} \right.\right) $$ where $G(\cdot)$ denote ...
0
votes
0answers
54 views

Show that $\Gamma(n+\frac{1}{3})\cdot \Gamma(n+\frac{2}{3}) = a\left(\frac{(3n)!}{3^{3n}\cdot n!}\right)$

Show that $\Gamma(n+\frac{1}{3})\cdot \Gamma(n+\frac{2}{3}) = a\left(\frac{(3n)!}{3^{3n}\cdot n!}\right)$. Furthermore, find an expression for $a$. I just can’t seem to equate these, I've tried using ...
0
votes
0answers
28 views

Integration involving x at the power and the gamma function

Suppose a nonnegative function $f$ on the positive real line and $f(x)\neq\Gamma(x)$. Do we know any closed formula of the following integral $$\int_{\mathbb{R}^+} \frac{\alpha^x}{\Gamma(x)}f(x)dx,$$ ...
1
vote
1answer
97 views

Study this improper integral $ \int_0^1 \frac{dt}{\sqrt{t}\,\sqrt{1-t}\,\sqrt{1-\alpha\,\sqrt{1-t}}}$

I'm trying to study the behavior of this improper integral $$ \int_0^1 \frac{dt}{\sqrt{t}\,\sqrt{1-t}\,\sqrt{1-\alpha\,\sqrt{1-t}}}$$ for:$$\alpha>0 $$ While I just can't understand the ...
1
vote
0answers
29 views

Solving for probability from expressions involving incomplete beta function

For a given $\alpha,\beta,a,b>0$, I'm trying to find the value of $p$ that satisfies the following two equations $$\frac{1}{2}\bigg[a\bigg(b-\frac{\alpha}{\alpha+\beta+1}\bigg)+1\bigg] = I_p(\...
0
votes
0answers
13 views

Function of an irregular zigzag line

First time questioner on the maths site, so thanks in advance to anyone kind enough to help me out. For context, I'm an author and I'm trying to find an accurate way for an AI character to refer to an ...
1
vote
1answer
32 views

Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
3
votes
1answer
57 views

Show that $\psi(n)$ has finitely many roots

Define $\psi(n)=\pi(n)-\phi(n)$ where we have the prime counting function and totient function respectively. I'm interested in where $\psi(n)=0$. Specifically is it possible to prove that there are ...
4
votes
0answers
51 views

Why are some Ramanujan $G_n$ and $g_n$ functions highly factorable?

Given the Dedekind eta function $\eta(\tau)$ with $\tau = \sqrt{-n}$. Define the Ramanujan $G_n$ and $g_n$ functions as, $$G_n = 2^{-1/4}\frac{\eta^2(\tau)}{\eta(\tau/2)\,\eta(2\tau)}$$ $$g_n = 2^{-1/...
2
votes
1answer
54 views

Bessel differential equation from integral

It is a relatively well-known fact that $$\int_{0}^{2\pi}e^{-ikr\cos\theta}d\theta=2\pi J_{0}(kr),$$ where $J_{0}$ is the Bessel function of the first kind and order zero. I'm trying to show that this ...
0
votes
0answers
21 views

Why is the unit step function in the end is written as $u(t-t_1-t_0)$ instead of just $u(t-t_0)$?

So this is basically about signals and systems and how to compute the output of the system using only the impulse response and the given input signal- where x(t) is the input signal and h(t) is the ...
1
vote
2answers
70 views

Kummer transform of the confluent hypergeometric function of second kind

I can see the kummer transformation of the confluent hypergeometric function of first kind throught the integral representation. However, I failed to see that for the second kind. More specificially, ...
0
votes
0answers
18 views

The Overlap Integral of Three Associated Legendre Functions

I obtained the following integral in my (physics) research: $ \int_{-1}^{1} \mathsf{P}_{-\frac{1}{2}+i \mu}^{ik_1}(x) \mathsf{P}_{-\frac{1}{2}+i \mu}^{ik_2}(x) \mathsf{P}_{-\frac{1}{2}+i \mu}^{ik_3}(...
0
votes
1answer
11 views

Beta function integral appearing in the $O(n)$-model

While studying the $O(n)$-model I found myself in the need of integrating $$\frac{\int_0^{\pi}\text{d}\theta\sin(\theta)^{n-2}\cos(\theta)^{2r}}{\int_0^\pi\text{d}\theta\sin(\theta)^{n-2}},$$ where $n\...
0
votes
0answers
37 views

How to get inequality ‎$‎0‎\leq ‎\gamma‎_g + ‎log ‎g(1)\leq ‎\frac{g^‎\prime_{-}(1) + g^‎\prime_{+}(1) ‎}{2g(1)}‎$‎?

‎Let ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎be a‎ ‎real‎ ‎function ‎such ‎that ‎‎$\log g(x)$ ‎is ‎concave and ‎$‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ ‎for each ‎$‎w&...
3
votes
1answer
29 views

Associated Laguerre polynomials of half-integer parameters

It appears that associated Laguerre polynomials $L_n^{(\alpha)}(x^2)$ for half integer parameters in $n$ and $\alpha$ may be expressed in terms of an exponential function, an imaginary error function ...
0
votes
0answers
32 views

Closed form for a family of definite integrals involving a Gaussian and error functions.

Let $n\ge 0$ be an integer and let $c \in {\mathbb R}$. Let us define: \begin{eqnarray} {\mathfrak F}^{(A,B)}_{a,b} &:=& \int\limits_A^B \frac{\log(z+a)}{z+b} dz\\ &=& F[B,a,b] - F[A,...
1
vote
0answers
19 views

Asymptotic expansion of the confluent Heun function

Is the asymptotic expansion of the confluent Heun function known?? The confluent Heun's differential equation is given by \begin{equation} y''(z) + \left( \epsilon + \frac{\gamma}{z}+ \frac{\delta}...