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

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1
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1answer
79 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
5
votes
1answer
339 views

Generalized Legendre differential equation

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ which is ...
1
vote
1answer
47 views

Satisfaction of Bessel equation by any other function.

Is it possible that any function $y(x)$ other than Bessel group of functions, satisfy Bessel's equation? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
2
votes
0answers
43 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
3
votes
0answers
33 views

Hypergeometric function with negative $b$ and $a>c>0$

Recall the definition of the hypergeometric function $$_2F_1(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{n!(c )_n}x^n$$ where $(a)_n$ is defined to be $a(a+1)\cdots(a+n-1)$. We suppose that none ...
0
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2answers
74 views

Basic question on complex integration

I have a very basic question on complex integration. How is the definite integral $$ \int_{z_1}^{z_2}{f(z)dz} $$ $z \in \Bbb{C}$ to be interpreted in the absence of a specific path over which ...
2
votes
1answer
35 views

composite function problem

If I have the following expression: $$g(f(x))-g(x)=1,$$ it is possible to express $f(x)$ in terms of the $g(x)$: $$f(x)=g^{-1}(1+g(x)).$$ Is it possible to express $g(x)$ in terms of $f(x)$?
1
vote
0answers
27 views

Identity involving the hypergeometric function

Let $n$ be an integers greater than one and $p,q$ be real numbers.How do I prove the following identity: \begin{equation} F_{2,1}\left[ \begin{array}{cc} \frac{3}{2} - n && 2-n \\ & ...
1
vote
1answer
70 views

Copulas and their properties

I am working with the following copula, and have a few questions about it: $C(x,y) = xy + \theta (1-x)(1-y)xy$ Here $\theta \in [-1,1]$ and $x,y \in [0,1]$ First, I am trying to show this copula is ...
0
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0answers
32 views

Integrating products of Hankel and Riccati Bessel functions

I want to do the integral: $$ \int_0^\infty dr h_l^+(kr)\hat j_l(kr) $$ where $h_l^+$ is the type 1 Hankel function, $\hat j_l$ is the type 1 Riccati-Bessel function. I would like a algebraic ...
1
vote
1answer
29 views

Simplify $\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$

I would like to rewrite this integral $$\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$$ (where $a>\mathbf{R^+}$ and $J_{\frac{3}{2}}$ is the bessel function ...
3
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1answer
152 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
4
votes
0answers
65 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
0
votes
0answers
44 views

How to get the asymptotic formula of generalized Bessel function?

How to get the asymptotic formula of generalized Bessel function? $$J_{\nu}^{(\mu)}(z)=\frac{2}{\sqrt{\pi}\Gamma(\nu+1-1/\mu)}\Big(\frac{z}{2}\Big)^{\mu \nu/2} \int_{0}^{1} ...
2
votes
0answers
38 views

In the space $L^2 [0,1]$ to solve for all values ​​of the complex parameters $\lambda$ and $b:$ [closed]

In the space $L^2 [0,1]$ to find a solution of the integral equation for all values ​​of the complex parameters $\lambda$ and $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$
0
votes
0answers
24 views

Solving non-linear, but separable and autonomous, matrix ODE $H'(z) = A H(z)^k + S$

Start with the non-linear scalar ODE: $H'(z) = A H(z)^k + S$. You can separate and integrate this to find something like: $$z + C_1 = \int_0^{H}\frac{1}{S - A q^k}dq$$ From this, you can use the ...
3
votes
1answer
131 views

How to reproduce the Mathematica solution for $\int(\cos x)^{\frac23}dx$?

I entered this integration problem to Mathematica Online Integrator an got a solution I would never have been able to find manually. $$\int\root 3 \of{\cos(x)^2}\,dx=\frac{(-3\cos(x)\root 3 ...
4
votes
1answer
82 views

Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$

Question: Prove that $$\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk=\frac{1}{16\pi}\Gamma^4\left( \frac{1}{4}\right)$$ My attempt Start by the transformation $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
1
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1answer
65 views

Continuous non differentiable functions :)

I was searching for functions like Weierstrass (continuous but differentiable nowhere), but I haven't found any. If you could tell me some that would be great. Also, I would like to find some ...
10
votes
1answer
174 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
5
votes
3answers
125 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
1
vote
1answer
52 views

Equailty involving Elliptic integrals and hypergeometric function

How to prove the following $$\,_2F_1\left(-1/2,-1/2,1,k^2 \right)=\frac{2}{\pi}\left(2E+(k^2-1)K \right)$$ where we define The complete integral of first kind $$K=K(k) = \int^1_0 ...
4
votes
1answer
147 views

Infinite sum of Bessel Functions

I came across the following sum in my work involving the infinite sum of a product of Bessel functions. Does anyone have any idea of how to express this in a simpler form? 'a' and 'b' are positive ...
6
votes
1answer
79 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
2
votes
1answer
148 views

Integral involving the Spherical Bessel Function of the First Kind

How can I prove the equation below using Spherical Bessel Function Recurrence Relation? (where $ j_{n}(x) $ means Spherical Bessel function of first kind) Definition using BesselJ function: $$ ...
4
votes
1answer
51 views

integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$

I arrived at the following result $$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$ where the exponential integral $E(z)$ is defined ...
6
votes
2answers
179 views

How prove this $p(x)>0$ if $p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$

let the polynomials $$p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$$ and such $$a_{0}+\sum_{a_{i}<0}(1-\dfrac{i}{n})\binom{n}{i}a_{i}>0$$ and ...
0
votes
1answer
69 views

Reduction of DEs to Bessel equation

A question in my textbook asks me to write down the general solution to: $\frac{d}{dx}(x^2\frac{dR(x)}{dx}) + [k^2x^2 - n(n+1)]R(x) = 0$ in terms of Bessel functions. Now two similar questions ...
3
votes
2answers
84 views

Prove: $\sin (3\pi/2 - x) = -\cos(x)$

I know the sine of $3\pi/2$ is $-1$. So i plug it in the function making it. $\sin(-1-x) = -\cos x$. However, I don't know where to go from here.
5
votes
1answer
126 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
1
vote
1answer
41 views

BesselJ function on negative real numbers

I have evaluated Bessel$J_v(x)$ with some real $v$ and negative real $x$ in MATHEMATICA. I cannot understand how the result is complex (non-real). I look at the series definition of BesselJ and I ...
0
votes
0answers
51 views

Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
0
votes
1answer
28 views

Solutions in terms of Bessel functions

I came across this question that asked to express solutions to: $$x^2y'' + xy' + (4x^2 - v^2)y = 0,\quad 0\le x<\infty$$ in terms of Bessel functions subject to the boundary conditions y(x) is ...
0
votes
1answer
23 views

Integral Bessel recurrence relation

I want to show that $\int x^vJ_{v-1}(x)dx = x^vJ_v(x) + C$. Now I know the recurrence relations of the Bessel equation/function and the one I need to use is $x^vJ_v(x) = x^vJ_{v-1}(x)$ I'm just ...
0
votes
1answer
16 views

What is the value of bessel functions at 0?

I would like to know what the value of the bessel functions of the first kind and the modified bessel of the first kind is at 0. I think for order 0 they are 1 and for orders greater than 0 they are ...
3
votes
0answers
54 views

Bessel J function problem

Let $\xi_{0k}$ be the k-th positive zero of $J_{0}$ Bessel function. Determine the coefficients $c_k$, so that $1 = \sum^{\infty}_{k=1} c_kJ_0(\frac{x \xi_{0k}}{2})$. I don't see what to do, is ...
0
votes
1answer
21 views

Graph of a Sine Function Increased on One Side

If we want something looks like this sine wave, what is a function that will satisfy this: Further, how can we make this go both ways, meaning either 1) a sine wave staying the same at the bottom ...
0
votes
1answer
56 views

A definite integral in terms of Meijer G-function

I am trying to find some relevant functional identities involving Meijer G-functions in order to prove $$ \int_0^\infty\frac{\log(x+1)}{x}\mathrm{e}^{-zx}\,\mathrm{d}x = G^{3,1}_{2,3}\left(z \middle| ...
0
votes
0answers
22 views

Numerical evaluation of an infinite 3D sum of cosine?

Consider the following function: $$f\left(x, y, z\right) = \sum_{\left(n, m, l\right)\in \mathbb{N}_*^3}e^{-\alpha\left(n^2+m^2+l^2\right)}\frac{\cos\left(\omega nx\right)\cos\left(\omega ...
1
vote
2answers
62 views

Determining if any general funtion u(x,y) makes f(z)=u(x,y)+iv(x,y) analytical

I have a question about Complex Analytical functions. I have some homework that asks: let $f(z) = u(x,y) + iv(x,y)$. Indicate the following functions for which u(x,y) may be analytic: $6(x^2-y^2)$ | ...
3
votes
2answers
200 views

Evaluation of another definite integral

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought. , where $r$ is any positive integer while $\psi$ and $\nu$ are positive real numbers.
0
votes
0answers
27 views

how to solve this product of bessel integral?

$$W = \int_0^z r J_7(ar) J_0(br) dr$$ i would like to solve this integration. Any information regrading this helps me a lot. Thank you.
1
vote
0answers
37 views

How to solve this infinte series related to bessel integral?

$$\int_0^z t^{\rho} J_{\mu}(at) J_{\nu}(bt) dt = \frac{(\frac 1 2 a z)^{\mu} (\frac 1 2 bz)^{\nu} z^{\rho + 1}}{\Gamma(\mu + 1)\Gamma(\nu + 1)} \times \sum_{k = 0}^{\infty} \frac{(-1)^k (\frac 1 2 a ...
1
vote
1answer
40 views

Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
2
votes
1answer
51 views

Upper bound on the modified Bessel function of the first kind

I am looking for an upper bound tighter than $e^x$ for the modified Bessel function of the first kind. Does there exist $\epsilon\in (0,1)$ such that $I_0(x)\le e^{x^{1-\epsilon}}$ for all $x\ge 0$ ? ...
2
votes
1answer
49 views

Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$?

Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$? Thanks ahead:) What I have tried: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
1
vote
1answer
87 views

Calculate the residue of $\cot\pi z$ at poles $z=n$

I'm having trouble calculating the residue of $f(z) =\cot\pi z$. The function has a simple pole for every integer n, and i'm, trying to find the residue at n. I know that by the residue theorem: ...
0
votes
1answer
161 views

Bessel function radius of convergence

Assuming the series representation of the Bessel function $J_v(x)$ I was given that the radius of convergence was ∞. I've tried using the ratio test but I dunno how the gamma function would ...
3
votes
1answer
56 views

Is there a simpler form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$?

Is there a simpler (i.e. manifestly real) form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$ or $\Im \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$, or more generally for $\frac{\Gamma(1/2-ia)}{\Gamma(1-ia)}$ with ...
0
votes
1answer
30 views

Radius of Convergence ratio test

using the ratio test for the following sum from n = 0 to infinity of $$ \sum_{m=0}^{+\infty}\frac{(-1)^m}{(m!)^2} x^{2m +10} $$ I need to find the radius of convergence. I managed to get up to ...