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

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1answer
74 views

Evaluating an integral by substitution and special functions [duplicate]

How can I evaluate this integral? $$\int_{0}^{1} \frac{dx}{\sqrt{{1+x^4} }}$$ I tried using the substitution $x=\mathrm{e}^{-u}$ but I got nowhere.
2
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1answer
83 views

If $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ holds for $0<z<1$, then also for $0<\operatorname{Re}(z)<1$

In Special Functions p. 10, it has proven that $$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},$$ for $0<z<1$. Then it says that this equality implies for $0<\operatorname{Re}(z)<1$. I do ...
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1answer
59 views

Show some properties of the Digamma Function

Let $\psi(z)$ denote the Digamma function, $\psi(z)=\frac{d}{dz}\ln \Gamma(z)=\frac{\Gamma'(z)}{\Gamma(z)}$. I am meant to show the following properties of $\psi$: $\psi$ is meromorphic in $\mathbb{...
4
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1answer
66 views

Integral representation of Bessel function $K_v(y) = \frac{1}{2} \int_{0}^{\infty} t^{v-1} \text{exp}(-\frac{1}{2}y(t+t^{-1}))\text{d}t$.

How does one find the following representation of the bessel function $K_v(y)$: $$K_v(y) = \frac{1}{2} \int_{0}^{\infty} t^{v-1} \exp \left(-\frac{1}{2}y\left(t+t^{-1}\right) \right)\,\mathrm{d}t.$$ I ...
5
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1answer
120 views

An elliptic integral?

I ran into an integral a little while ago that looks like an elliptic integral of the first kind, however I am having trouble seeing how it can be put into the standard form. I've tried messing ...
3
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5answers
87 views

What are some functions that respect the following criteria? : $f(1/x) = f(x)$ and $\int_{0}^{+\infty} f(x) dx = 1$

I'm looking for some functions that respect these six criteria: $f$ is defined on $[0 ; +\infty[$ $f$ is differentiable everywhere in $[0 ; +\infty[$ $f(0) = 0$ $\lim\limits_{x \to +\infty} f(x) = 0$...
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4answers
64 views

integrating this infinite gaussian integral

How does one integrate $\int_{-\infty}^{+\infty}x e^{-\lambda ( x-a )^2 }dx $ where $\lambda$ is a positive constant. My integral tables are not returning anything useable. The best it return is ...
2
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1answer
25 views

Normalisation of Bessel functions

I've done the integration by parts and obtained $$ \frac{-1}{\alpha^2} \int z^2 J J'$$ but I have no idea how to use Bessel's equation to simplify this as it only appears to get far more complicated....
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1answer
63 views

Solving differential equation $y''(x)+Q(x)y(x)=0$ [closed]

How to solve the following differential equation $$y''(x)+Q(x)y(x)=0$$ And how to find exact solution $y(x)$ in terms of special functions?
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0answers
22 views

Division of half-integer order legendre functions of the second kind with different arguments

I'm in search of a formula for: $\frac{Q_{n-\frac{1}{2}}(\chi_1)}{Q_{n-\frac{1}{2}}(\chi_2)}= ??$ where I am hoping the result to be a function of $\frac{\chi_1}{\chi_2}$. Does anyone know of such ...
2
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0answers
47 views

Asymptotic behavior of zeros of a function

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ ...
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1answer
31 views

Shift of dirac delta function involving a sphere

Alright, I'm clueless on how to kickstart this question. The idea of the dirac delta function by itself is understandable, at least at my current level. But once the question starts throwing in ...
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2answers
41 views

Prove $\frac{-4 \sqrt x + 2 e^x \sqrt x + \sqrt \pi \operatorname{erfi}\sqrt x}{2 \sqrt x}\leq \frac{e^{3x}-3x-1}{3x}$.

From Wikipedia, the imaginary error function, denoted erfi, is defined as $$\operatorname{erfi}(x) = \frac{2}{\sqrt\pi} \int_0^xe^{t^2}\,\mathrm dt.$$ Prove that $$\frac{-4 \sqrt x + 2 e^x \sqrt x + \...
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1answer
27 views

Evaluating a difficult 3-dimension dirac delta

Currently doing problem 1.48 of "Introduction to electrodynamics by David Griffith" I've read the examples, the theory and understood but come the exercise the author has a terrible habit of dishing ...
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1answer
26 views

Recursive function including Bessel functions

I was wondering if anybody knows how to solve (numerically) the following recursive equation (found in http://dx.doi.org/10.1109/3.250392): $$E^{o}_{k}=\sum^{\infty}_{q=-\infty}J_{q-k}(2m)E^{o}_q,$$ ...
3
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0answers
49 views

Expansion of some singular kernel with the help of Bessel and Neumann spherical harmonic functions

With the following notations: $j_n$: spherical Bessel functions, $y_n$: spherical Neumann function, $P_n$: Legendre polynomial, $r$, $\rho$, $\theta$, $\lambda$ arbitrary complex, $R=\sqrt{r^2+\rho^...
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1answer
57 views

Perturbation of the Upper Incomplete Gamma Function

The Upper Incomplete Gamma function, for $t \in \mathbb{R}$, is defined as: \begin{equation} \Gamma(α,β)=\int_{β}^{\infty}t^{α-1}e^{-t}dt \end{equation} For the problem which I am studying it takes ...
3
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5answers
121 views

When actually $f(g(x))=g(f(x))$ holds?

We can see that if $f(x)=g(x)=x$ then $f(g(x))=g(f(x))$. I would like to see other examples of functions $f(x)$ and $g(x)$ such that $f(g(x))=g(f(x))$. P.S. By definition we also must have $D_{f\...
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1answer
167 views

May I know how this integral was evaluated using hypergeometric function?

I can not solve the following integral using the hypergeometric function: $$\int_a^b (\sin x)^{(1/n)}dx$$ Wolframalpha showed the following result. but I do not understand how Wolframalpha came ...
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1answer
39 views

Can you justify the existence of a $x_{*}$ solving $\mbox{li}(x_{*})=\mbox{erf}(x_{*})?$

Can you justify the existence of a $x_{*}$ solving $$\DeclareMathOperator{\li}{li}\DeclareMathOperator{\erf}{erf}\li(x_{*})=\erf(x_{*})?$$ Here $\li(x)$ is a special function, the so called ...
2
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1answer
36 views

How to prove this relation?

Is the relation $$\lim_{x\rightarrow 1}\frac{Q_n^m(x)}{P_n^m(x)}=\frac{\pi}{2}\cot m\pi$$ correct? Here P and Q are the associated Legendre polynomials of the first and second kind respectively. Does ...
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1answer
38 views

Simple form of LegendreQ function

for any n is positive integer LegendreP function can be expressed as $\displaystyle P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}\left[(x^2-1)^n\right]$. Let $\displaystyle q_n(x)=Q_n(x)-P_n(x)\log\left(\...
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0answers
56 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 ...
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0answers
22 views

Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
0
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1answer
40 views

Convergence of series

Is this series convergent? $$S_{N}=\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}}$$ where $c_{n}^{N}$ is coefficient of $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. $$T_{N}(x)=\sum_{n=0}^{N}c_{n}^{N}...
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0answers
24 views

Refences to Sturm–Liouville theory with a singular weight function.

For $\alpha,\beta$, nonpositive integers at least one of which is non-zero, define $\omega\colon (-1,1)\to \mathbb{R}$ as $\omega(x) = (1-x)^\alpha(1+x)^\beta$. Then $\omega$ blows at at $x=-1$ or $x=...
5
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1answer
149 views

Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
0
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1answer
23 views

Numerical integration of $E_1(x)$

I want to solve the following integral for $\gamma_0$: $$\int_{\gamma_0}^\infty \frac{1}{t}e^{-at} dt = c$$ for the specific values $a = 0.01$ and $c = 12.1$. As I understand, this is a variant of ...
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0answers
15 views

inverting a complicated function.

Is it ever possible to rewrite a function, such as $$ x - A\sqrt{y(x)} + B\tanh\left(\sqrt{y(x)}\right) +C =0 $$ in terms of $y(x)$. By invert, I mean, optimistically, express using something like ...
2
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1answer
34 views

How can I show that this Jacobi polynomial can be expressed as the sum of these two Legendre polynomials?

Let $n\in \mathbb{N}^+$ be a positive integer. Let $L_n\colon \mathbb{R}\to \mathbb{R}$ be the $n$'th order Legendre polynomial. Let $J_n^{(\alpha,\beta)}\colon \mathbb{R} \to \mathbb{R}$ be the $n$'...
2
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2answers
19 views

Identifying a function

I am reading a piece of a physic paper where a function is mentioned without being given a name or reference - I guess it is a canonical one and that I should be familiar with. The expression goes as:...
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0answers
34 views

Show that $J_n(x)$ satisfies Bessel equation $ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $

Here is the definition of the Bessel function I am starting with a definition as an integral. $$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$ Essentially we have computed ...
0
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1answer
51 views

The derivatives of Riemann xi function

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
4
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0answers
110 views

Integral of combination of power, exponential, and confluent hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x \end{...
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2answers
249 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
2
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1answer
94 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
3
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1answer
36 views

A formula for length of representation of a number in a “base” without zeros

If you had 2 items the sequence would go like this: $$1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5, \ldots$$ This is $\lfloor\log_2(n+2)\rfloor$. What if I ...
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1answer
66 views

Function with infinite maxima and minima [closed]

Can you please give an example of a function with an infinite number of maxima and minima occurring in any finite time interval? Edit: This question came to me as I was reading on the dirichlet ...
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0answers
8 views

Giving two examples of functions with some properties.

This is a question from a list. Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties: $f(x)=0 \Leftrightarrow 0\leq x\leq 1$; $g(x)=x$ if $|x|\leq 1$, ...
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1answer
69 views

Book recommendation on special functions

I am currently studying real analysis from rudin and really like the chapter on special functions. But Rudin does not give much knowledge about those topics. Reading the references I found book by ...
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1answer
10 views

Legendre functions - Derivation of the recursion relation

From the following: $$\sum_0^\infty [ n(n-1)a_nx^{n-2} - n(n-1)a_n x^n -2na_nx^n + l(l+1)a_n x^n ] = 0$$ (a) I'm trying to get to: $$\sum_0^\infty [ (n+2)(n+1)a_{n+2} - [n(n+1) + l(l+1)]a_n]x^n = 0$...
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1answer
68 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
1
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1answer
33 views

About Beta function $B(\alpha,r\alpha +1)‎\rightarrow‎ 0$

I want to show that $$B(\alpha,r\alpha +1)‎\rightarrow‎ 0$$ when $r‎\rightarrow‎ \infty$ and $0< \alpha <1$. with thanks
4
votes
1answer
106 views

Integral with incomplete gamma function

I am trying to solve this integral: \begin{equation} \frac{1}{c^{b}}\int_{0}^{\infty} x^{n}\, e^{-a x}\, \gamma(b,c(-d+x)) \ \mathrm{d}x \end{equation} where, $n>0$ is an integer, and $a$, $b$, $...
0
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0answers
14 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
1
vote
1answer
69 views

Integrals involving whittaker functions.

I want to compute the following integrals: $$ \int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy $$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind. I already know that ...
3
votes
1answer
54 views

Computing the limit of an integral

Consider the following integral $$ \int_{-\infty}^{\infty}f(t) K(\frac{a-t}{h})dt $$ where (1) $h>0$, $a \in \mathbb{R}$ (2) $f:\mathbb{R}\rightarrow[0,\infty)$ is such that $\int_{-\infty}^{\...
0
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1answer
55 views

integral include lower incomplete gamma

I am trying to calculate the following integral: $$ \int_0^{\infty}e^{-\beta x}\gamma(\alpha,\theta x)dx $$ where all parameters are positive. Any help , Thanks!
4
votes
1answer
92 views

Calculate (or estimate) $S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$.

Let $x\in\mathbb R$, $x>1$ and $$S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$$ where $\zeta(x)$ is the Riemann zeta function. Calculate (or estimate) $S(x)$.
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1answer
39 views

How to prove this limit of Airy Function.

I have no idea how to prove this limit $$\lim_{x\rightarrow \infty }\exp\left ( \frac{2x^{3/2}}{3} \right )\sqrt[4]{x}\mathrm{Ai}\left ( x \right )=\frac{1}{2\sqrt{\pi }}$$ where $\mathrm{Ai}(x)$...