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

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-1
votes
1answer
12 views

Limit through a figure

If a circular arc of radius 1 subtends an angle of x radians . The centre of the circle is o and the point c is the intersection of two tangents lines at a and b . Now let T(x) be the area of the ...
0
votes
1answer
17 views

Limit of trig functions

We have to evaluate $$\lim_{x\to 2} \frac{\cos^x a +\sin^x a -1}{x-2}.$$ I am working on it for hours I tried using series , replacing $\cos a$ by $t$ and $\sin a$ by $\sqrt{1-t^2}$ but not got any ...
1
vote
0answers
22 views

Discontinous and differentiable

If we have a two functions $f:[-1/2, 2] \to \mathbb{R}$ $g: [-1/2,2] \to \mathbb{R}$ $f(x) = [x^2-3]$ where $[ \cdot ]$ denotes greatest integer $g(x) = |x| f(x) + |4x-7| f(x)$ Now we have to ...
0
votes
0answers
7 views

Plot a Floquet solution to Mathieu equation

In wikipedia https://en.wikipedia.org/wiki/Mathieu_function#Floquet_solution I want to know how the Floquet solution is plotted. One way I am thinking is to write Floquet solution in terms of the ...
0
votes
1answer
19 views

Limit of a function containing $\Psi(x)$

The taylor series expansion of the function $$f(x)=\ln(1+x)$$ around zero is: $$f(x)=\sum_{k=1}^\infty\dfrac{(-1)^{(k+1)}}{k}x^k$$ Putting $x=1$ we have the alternating series: ...
0
votes
1answer
16 views

Find a recurrence relationship for the following :

Find a recurrence relationhip for $a_{n}$: $a_{n}=\dfrac {2n+1}{2}\int^{1}_{-1}f\left( x\right) P_{n}\left( x\right) dx$ Where $f\left( x\right)= e^{-x}$ I have done it many times and keep ...
0
votes
1answer
14 views

How is the following integral related to confluent hypergeometric functions?

I am solving an integral that appears in a physics paper. $$ -\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg] $$ The paper does not give the full solution, it only gives ...
0
votes
1answer
39 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
38 views

hyperbolic sum and elliptic integral 2

I try to show that $$\sum _{k=1}^{\infty } k^{36} \text{sech}(\pi k)=\frac{41222060339517702122347079671259045}{137438953472}+\frac{i \left(\psi _{e^{\pi }}^{(36)}\left(1-\frac{i}{2}\right)-\psi ...
1
vote
1answer
56 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
49 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
0answers
13 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
1answer
45 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) = ...
0
votes
2answers
63 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 ...
2
votes
1answer
81 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: $$x^2y^{\prime\prime}+xy^{\prime}+(x^2-p^2)y=0$$ ...
4
votes
1answer
41 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 ...
3
votes
1answer
66 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 ...
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! ...
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 ...
5
votes
1answer
57 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
1answer
52 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
0
votes
3answers
43 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 ...
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]

$\int_{0}^{1} \frac {x^{m-1}\cdot (1-x)^{n-1}}{(a+x)^{m+n}}dx = \frac{B(m,n)}{a^{n}\cdot (1+a)^{m}}$
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?
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 ...
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
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} ...
11
votes
3answers
664 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 ...
-1
votes
0answers
31 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
84 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 ...
5
votes
2answers
191 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 ...
2
votes
0answers
55 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 ( ...
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 $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...
0
votes
0answers
22 views

Definite integral involving algebraic, exponential, and product of two Meijer's G function

I am having trouble with calculating the following integral: \begin{equation} I = \int_{0}^{\infty}x\exp({-\beta x})\large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} ...
1
vote
0answers
37 views

Integrals with erf^N

Can anyone help with integral of type. In general, what to do if erf is in power higher than 1? $$g(S|S<L)=\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{+\infty} \left [ \frac{1}{\sqrt{2 \pi ...
4
votes
3answers
120 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $

I have to prove that given $\gamma=0.577216\ldots$, the Euler-Mascheroni constant, and $\pi=3.14159\ldots$, we have: $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $$
1
vote
1answer
16 views

sum of integral parts of real number and fraction

For any real x and positive integer n ,show that [x] + [x +1/n] + [x + 2/n] + .... + [x + n-1/n] = [nx] I have used the fact that x-1 < [x] <= x,for all terms and added,but not able to get ...
3
votes
1answer
65 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
1
vote
1answer
20 views

Bessel Function Integral with sin argument

I would like to find if possible a solution (closed form) or approximation for the following integral: $$\int_{\pi/2}^{\pi}\int_{\pi/2}^{\pi}J_{0}\left(\alpha \sin\theta_{k}\right)J_{0}\left(\alpha ...
1
vote
1answer
31 views

How to proof a basis ${\psi_a}$ is complete?

Why $$\int\text{d}a\psi^*_a(y)\psi_a(x)=\delta(y-x)$$ shows the basis is complete? Even, how is $\delta$ defined? I mean, the most consistent definition. I really dislike the definition by 0 and ...
0
votes
1answer
32 views

Infinite sum of Hermite polynomials with same order, but different argument

I am looking for any possible simplification of the following sum for positive reals $\alpha,\beta$ and positive integer $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ ...
0
votes
2answers
28 views

Laplace transform of the square wave to solve PDE

Solve $$y'' + 3y' +2y = r(t)$$ given $y(0)=0$ and $y'(0) = 0$ where $r(t)$ is the square wave, $$r(t) = u(t-1) - u(t-2)$$ I'm just going to type out the answer as I read it and tell you which ...
0
votes
0answers
15 views

An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
0
votes
0answers
20 views

Fourier transform of Si[$x^2 + y^2$]; Energy integrals involving sin integral functions

Problem Statement I'm trying to prove( or disprove ) the following identity \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\Big[\text{Si}[x_1^2 + y^2]- \text{Si}[x_2^2 + ...
0
votes
1answer
21 views

Understanding a text in a book about the estimation

Now $$e_n-e_0=\sum_{k=0}^{n-1}\left [ -\frac{1}{12(k+x)^2}+\mathcal{O}\left ( \frac{1}{(k+x)^3} \right ) \right ]; \tag{*}$$ therefore, $\lim_{n\to\infty}e_n-e_0=K_1(x)$ exists. Set ...
3
votes
0answers
184 views

A generalization of additive function over $\mathbb R$

Let $f:\mathbb R\to\mathbb R$ be a continuous function and $r\ge0$ a fixed value such that for all $x,y\in\mathbb R$ $$|f(x)+f(y)-f(x+y)|\le r$$ Show there exist $a\in\mathbb R$ and a function ...