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

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2
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1answer
75 views

Bessel's Differential Equation - textbook queries:

In order to ask this question I must first give some background information as written in my text book: Given Bessel's Differential equation: $$x^2y^{\prime\prime}+xy^{\prime}+(x^2-p^2)y=0$$ ...
0
votes
1answer
32 views

Particular values of the Riemann zeta function.

On the wikipedia, near the bottom of the "Specific Values" section, there is a statement that bothers me. $$\zeta(-13)=\zeta(-1)$$ Firstly, it is well noted that the summations must be evaluated ...
1
vote
1answer
50 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where ...
5
votes
1answer
43 views

Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: ...
0
votes
1answer
42 views

Having trouble evaluting error function integrals

I am trying to evaluate $$I = \int_1^{\infty } \left(\frac{\operatorname{erf}\left(a -b\log (x)\right)}{2 x^2}-\frac{\operatorname{erf}\left(a + b\log (x)\right)}{2 x}\right) \, dx$$ Let $\log (x) = ...
5
votes
1answer
56 views

Anti-treta function in terms of standard special functions

Define treta$^*$ function as $$ \tau(\alpha_1,\alpha_2,\alpha_3) = \iint_{0< x_1< x_2<1} x_1^{\alpha_1-1}(x_2-x_1)^{\alpha_2-1}(1-x_2)^{\alpha_3-1}\, d(x_1,x_2).\tag{1} $$ Similarly to the ...
0
votes
0answers
7 views

Whittaker function integral representation

Let $W_{k,m}(z)$ denote the Whittaker $W$ function, which by definition satisfies $$ W_{k,m}^{''}(z)+\left(-\frac{1}{4}+\frac{k}{z}+\frac{1/4-m^2}{z^2}\right)W_{k,m}(z) =0, $$ where $W_{k,m}^{''}(z)$ ...
0
votes
2answers
58 views

How to Find the Global Minimum and Maximum of this Multivariable Function?

We have the set $$M=\{(x,y,z)\in\mathbb R^3: x^2 + y^2 = z \wedge x+y+z=12\}$$ and the function $$F(x,y,z) = xy+ z^2.$$ How can we find the global maximum and global minimum of F on M and prove ...
3
votes
2answers
412 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
5
votes
2answers
190 views

Deriving the Normalization formula for Associated Legendre functions: Stage $4$ of $4$

The question that follows is the final stage of the previous $3$ stages found here: Stage 1, Stage 2 and Stage 3 which are needed as part of a derivation of the Associated Legendre Functions ...
3
votes
1answer
63 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha ...
4
votes
1answer
40 views

An always increasing function

Suppose I wanted a function $f(x)$ such that the following properties are had. $f(x)$ maps $\mathbb{R}\to\mathbb{R}$. $f(a)>f(b)$ if $a>b$. The function may or may not be continuous, but it ...
1
vote
1answer
383 views

Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
0
votes
0answers
38 views

How to evaluate I(y) = $\int_0^t e^{ax^b} e^{-cx^d} x^f dx$ in terms of special functions?

To put the above in the proper context, I am trying to solve a Bernoulli equation of the second order: $\frac{dy}{dt} = -\frac{A}{p-q}(e^{-pt}-e^{-qt})y-Be^{-rt}y^2$ where constants A, B, p, q, r ...
2
votes
0answers
45 views

Summation Involving Hermite Polynomials

From the generating formula for Hermite polynomials we know that $$ e^{2xz - z^2} = \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n!} \, . $$ The sum $$ \sum_{n=0}^\infty \frac{H_n(x) \, z^n}{n! ...
2
votes
1answer
110 views

Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$ For example $$\int_{- ...
3
votes
0answers
72 views

How to compute the following integral?

Someone has an idea to calculate the following integral $$I_{a,b,\alpha} = \int_{1}^{+\infty} e^{-at} \,(1-t^{-1})^b \log^{\alpha}(1-t^{-1}) \, dt; \quad a,b>0, -1<\alpha<0.$$ Thank you in ...
-1
votes
1answer
50 views

How to compute the integral $I_{\alpha} $?

Someone has an idea to calculate the following integral $$I_{\alpha} = \int_{0}^{+\infty} t^{-\alpha} (1-a)^{t} dt; \quad 0<a,\alpha<1.$$ Thank you in advance
1
vote
1answer
41 views

hypergeometric transformation

I came across the following ${}_3F_2$ hypergeometric polynomial: $$ {}_3F_2\left(\left.\begin{array}{c} 1,1,-n\\ 2, -1-2n \end{array}\right| -x\right) $$ for some large $x > 0$. I am wondering ...
0
votes
0answers
88 views

Analytic Bound on The Riemann Zeta Function

Given the canonical infinite product representation (Weierstrass form) of the gamma function, $$\Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} ...
0
votes
3answers
42 views

Is there a way to combine functions so that you combine their derivatives?

Suppose $y,z$ are functions. What manipulation: "$?$" to the functions would yield the following? (if any) $$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac ...
2
votes
2answers
67 views

Proof to $\int_{0}^{\infty}\sin(t)t^{z-1}\,\mathrm{d}t= \sin\left ( \frac{\pi z}{2} \right )\Gamma(z)$

I tried to check the source of the proof to the equation $$\int_{0}^{\infty}\sin(t)t^{z-1}\,\mathrm{d}t= \sin\left ( \frac{\pi z}{2} \right )\Gamma(z),\qquad -1<\Re z < 1$$ but it only has a ...
0
votes
1answer
10 views

Completeness Relation for Tricomi Confluent Hypergeometric Function

Consider the Kummer differential equation $$ \frac{d}{dz}\left[z^be^{-z}\frac{dw}{dz}\right]=az^{b-1}e^{-z}w,\quad z\in\mathbb{R}. $$ It is an eigenvalue problem of Sturm-Liouville type with weight ...
1
vote
1answer
87 views

What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 ...
0
votes
1answer
18 views

Expressing trigonometric function in terms of integral of Bessel function

I am trying to show that, \begin{align*} \frac{1-\cos x}{x} = \int_{0}^{\pi/2}J_1(x\cos\theta)\,\mathrm{d}\theta \end{align*} I did the following but cannot figure out how to continue. ...
1
vote
0answers
24 views

prove the following problem based on beta function [closed]

$\int_{0}^{1} \frac {x^{m-1}\cdot (1-x)^{n-1}}{(a+x)^{m+n}}dx = \frac{B(m,n)}{a^{n}\cdot (1+a)^{m}}$
3
votes
1answer
110 views

Changes of variables to get an Elliptic Integral of the First Kind

I'm working with a non-linear second order ODE which has an analytical solution in terms of the Jacobi elliptical function $sn(u|k^2)$. The equation is $y''=y(\gamma - \frac{y^2}{2})$ where $\gamma$ ...
5
votes
1answer
112 views

Fundamental period of the Weierstrass $\wp$ elliptic function?

Consider the Weierstrass $\wp$ elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia ...
0
votes
2answers
99 views

Elliptic Integrals of the First Kind

Suppose I have $$F(\phi(x), k) = x$$ where the elliptic integral of the first kind is defined to be $$F(\phi, k) = \int_{0}^{\phi} \frac{1}{\sqrt{1-k^2\sin(\theta)}} \, d\theta $$ How could I invert ...
5
votes
1answer
698 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
1
vote
2answers
70 views

Where does this formula for the volume of a n-dimensional ball come from?

I recently came across the following formula for the volume of an n-dimensional unit ball: $$\frac{\pi^{n/2}}{\Gamma(n/2 + 1)}$$ Why exactly does this formula work?
1
vote
1answer
83 views

How can I show that $\frac{\exp\left({-\frac{tx}{1-t}}\right)}{1-t}$ is the generating function for the Laguerre polynomials

How do I show that $$\frac{\exp\left({-\frac{tx}{1-t}}\right)}{1-t} = \sum_{n=0}^\infty \left( \sum_{k=0}^n \frac{(-1)^k n! x^k }{(k!)^2 (n-k)!} \right ) \cdot \frac{t^n}{n!} $$ So far, I tried ...
3
votes
1answer
107 views

what does the ${_2F_1}\left[\cdot, \cdot, \cdot, \cdot\right]$ function mean?

I was reading the integral tables, where it says: \begin{equation} \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax ...
4
votes
1answer
51 views

How to prove this continued fraction connection between $\gamma$ and $e$?

There is apparently a curious connection between Euler-Mascheroni constant $\gamma$ and $e$ in the form of an infinite series and continued fraction: $$e \gamma=e \sum_{n=1}^{\infty} ...
0
votes
0answers
20 views

Let $\theta(z) = \sum q^{n^2}$, is $\theta(-1/z)$ also a theta function?

I am learning about theta functions. Let $q = e^{2\pi i \, z}$ and $\theta(z) = \sum q^{n^2}$. How does it behave under $\mathrm{SL}_2(\mathbb{Z})$ ? In general we have: $$ \theta\left( - ...
1
vote
0answers
54 views

Can I just make this function up?

The Lambert W function was made to solve the problem $xe^x=k$ for $x$, which is given as $x=W(k)$. Could I just make a function $x=F(k)$ which solves $x\cos(x)=k$? Even though the solution has an ...
15
votes
1answer
282 views

Integrate $\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
3
votes
1answer
701 views

2-increasing functions

I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. The function H, whose ...
11
votes
3answers
662 views

Special Gamma function integral

I'm trying to evaluate this integral $$\int_{0}^{1} \sin (\pi x)\ln (\Gamma (x)) dx$$ and I got to the point, when I need to find $\displaystyle \int_{0}^{\pi } \sin (x)\ln (\sin (x)) dx$ but ...
12
votes
1answer
1k views

a conjectured continued-fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for ...
-1
votes
0answers
30 views

Integral of Legendre polynomial and exponential

I want to evaluate the following integral: $$ I=\int_{-1}^{1} P_{n}(x) ~ e^{ i \alpha x + \beta x^2} ~ \rm{d}x, $$ for real-valued $\alpha$ and $\beta$. Is there some convenient expression that ...
4
votes
0answers
83 views

A conjectured asymptotic expansion of a function related to the sine and cosine integrals

Recall the definitions of the sine and cosine integrals:$$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{si}(x)=-\int_x^\infty\frac{\sin t}t ...
0
votes
0answers
33 views

Calculate number of trials reaching $p_k$ probability for $k$ successes given the $p_t$ probability of each trial success

Basically, I'd like to be able to answer questions in the form of "What is the number of trials needed to have at least $p_k$ probability of at least $k$ successes, given that on each trial the ...
1
vote
0answers
109 views

How to Solve Equation Involving Digamma Function? (EM Algorithm)

I am trying to derive the EM-algorithm of mixtures of negative binomial distribution $Neg\;Bin(r,p)$. I have the updating equations for updating the E-step as well as $p$ and the mixing coefficients ...
1
vote
3answers
71 views

What is the Domain of Gamma Function?

The gamma function is defined as $\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}dt$ for s>0. But then it says that "The gamma function is defined for all complex numbers except the negative integers ...
1
vote
1answer
280 views

Two Questions about Gamma Function Terminology

Gamma function is also known as generalized factorial function . 1. Why does the term "generalized" have been used? 2. Why is the Gamma function called Euler's second integral?
2
votes
0answers
54 views

Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, ...
4
votes
1answer
32 views

Mixed partials of the Beta function B$(a,b)$ at $(1,0^+)$

In this post M.N.C.E gave the equality below $$\frac{\partial ^{5}}{\partial a^{3}\partial b^{2}}\mathrm{B}\left ( 1,0^{+} \right )=\left [ \frac{1}{b}+O\left ( 1 \right ) \right ]\left [ \left ( ...
4
votes
1answer
80 views

How to evaluate $\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x$

How to evaluate the integral below $$\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x~~~~~~(t>0)$$ The WolframAlpha gave me a horrible answer $$\frac{t}{2}e^{-\frac{t^{4}}{2}}\left \{ ...
0
votes
1answer
38 views

The Cauchy product $\sum_{n=1}^\infty \frac{\log n}{e^n}= \left( 1-\frac{1}{e} \right)\sum_{n=1}^\infty\frac{\log n!}{e^n} $

I know that the Cauchy product is defined $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...