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42 views

Riemann surface of $K_0(z)$

The question concerns the modified Bessel function of the second kind of order zero ($K_0(z)$). How does its Riemann surface looks like? How can one evaluate its values on the "other" sheet(s)?
1
vote
1answer
79 views

Expression for infinite sum of two Bessel functions and a power

I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers): $$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} ...
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46 views

Bessel function stable recurrence relation

I want to compute hypergeometric 1f1 function using I can't use direct computation of Bessel functions due to complexity. I want to use BesselJ recurrence relation: But forward recursion is ...
4
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2answers
156 views

Integrals of two Bessel functions of the first kind and a modified bessel function of the second kind

I'm searching for a suitable (hopefully simple enough) solution to the following form of integral: $$\int_0^\infty \mathrm{d}x~x^n J_\nu(a x) J_\nu(b x) K_\mu(c x) $$ Where $n$, $\nu$, and $\mu$ are ...
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0answers
63 views

Existence of BVP solutions for non-linear Bessel-like equation

The basic problem: we have the differential equation $$ g''(r) + \frac{2}{r} g'(r) - \frac{2}{r^2} g(r) = F(g(r)) $$ in which $F(g)$ satisfies $F(0)=F(1)= 0$ and $F(x) > 0$ for $0<x<1$. ...
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2answers
186 views

Indefinite integration of multiplication of two Bessel function

I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case? $$\int_{a}^{\infty}J_2(bx)J_1(cx)\,dx$$ Thanks in ...
2
votes
1answer
109 views

The approximation of first-ordered modified Bessel function of the second kind

After analysing the outage probability of a single relay selection system, I got to the following form: $P = 1 + \sum\limits_{k = 1}^K {\left( \begin{array}{l} K\\ k \end{array} \right){{\left( { - 1} ...
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votes
2answers
327 views

$\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral: $\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$ ($\mathrm{BesselJ}[n,x]$ is ...
1
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0answers
123 views

Simplify this expression with modified Bessel functions of the second kind

I'm interested in the function $$\frac{K^{(0,2)}(0,x)}{K(0,x)}$$ where the numerator is the modified Bessel function of the second kind twice differentiated with respect to $\alpha$ and taken at ...