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
154 views

How does one calculate the amount of time required for computation?

For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by $ ...
1
vote
1answer
167 views

Hankel curve formula for gamma function proof

I recently read a proof for the following formula which I don't understand completely. For $Re(z)>0$: $\Gamma(z)=\frac{1}{e^{2\pi iz}-1}\int_{C_\delta}e^{-\zeta}\zeta^{z-1}d\zeta$ , where ...
11
votes
1answer
495 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
0
votes
1answer
57 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 ...
2
votes
1answer
86 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 ...
0
votes
0answers
21 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 ...
2
votes
1answer
244 views

Integration and $\gamma(n,x)$

It isn't hard to prove that: $$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$ Or put in a different way: $$\int_0^x e^{-t} \frac{t^n}{n!} dt = ...
3
votes
1answer
122 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 ...
6
votes
5answers
4k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
6
votes
2answers
171 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
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0answers
13 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 ...
0
votes
0answers
42 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
1answer
90 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: $$ ...
2
votes
1answer
27 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 $x\ge 0$ ? ...
0
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0answers
22 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
132 views

Integral Representation of Bessel Function (K)

There is an integral representation for the modified Bessel function of the second (or third depending on who you talk to) kind (denoted $K_\nu$) that says: $$K_\nu(z) = ...
3
votes
2answers
199 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.
3
votes
0answers
44 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( ...
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 $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$
1
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1answer
77 views

Define Beta function in terms of recursive relation?

The Beta function has recursive relation: $$ Beta(x,y) = Beta(x, y+1) + Beta(x+1, y) \! $$ Is there some least redundant condition, if added, will make the recursive relation an equivalent ...
1
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2answers
379 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
10
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1answer
142 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 ...
12
votes
1answer
186 views

On Continuous Replicative Functions

Knuth, in The Art of Computer Programming Vol. 1, defines a replicative function as a function $f$ such that $$f(x)+f\left(x+\frac{1}{n}\right)+\cdots +f\left(x+\frac{n-1}{n}\right)=f(nx)$$ whenever ...
4
votes
1answer
69 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 $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
3
votes
1answer
90 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
1
vote
1answer
56 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 ...
5
votes
3answers
115 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 ...
8
votes
1answer
148 views

Algebraic structures whose Hilbert-Poincaré series are special functions

Are there good examples of algebraic structures whose Hilbert-Poincaré series are that of special functions? I'm particularly interested in cases where complex analytic reasoning about those series ...
1
vote
1answer
47 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 ...
3
votes
1answer
72 views

Expressing upper incomplete gamma function of half-integer order in terms of gamma function?

N. M. Temme, "Special Functions" (Wiley 1996) gives the following expression that expresses the upper incomplete gamma function in terms of the ordinary gamma function, for integer orders: $$ ...
2
votes
1answer
26 views

Inequality with Bessel Functions of the first kind

How can be proven the following inequality: $$\int_0^1dx|J_k(x)'J_k(x)|\lt\frac{1}{2}\int_0^1dx|J_k(x)'^2|$$ where obviously: $J_k(x)'=\frac{d}{dx}J(k,x)$? Thanks.
26
votes
9answers
7k views

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ ...
4
votes
1answer
64 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 ...
3
votes
0answers
70 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( ...
2
votes
0answers
43 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) - ...
6
votes
0answers
223 views

Integral $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful (yes Beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 ...
18
votes
1answer
626 views

Integral $\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) ...
3
votes
0answers
28 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 ...
0
votes
1answer
25 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
54 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.
1
vote
1answer
36 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 ...
6
votes
2answers
136 views

Infinite Series :$ \sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2}$

Prove that: $$\sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2} = \frac{-\pi^{\frac{3}{2}}}{12}\left( \pi^2+6\gamma(1-2\log ...
2
votes
1answer
134 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...
9
votes
1answer
191 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
4
votes
0answers
165 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
0
votes
0answers
47 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
18 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
16 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
15 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
51 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 ...