MATLAB CODE of an equation including Bessel's function

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Md Jahid Hasan Sagor
编辑： Torsten 2022-8-18
I am stuck coding the below equation. Would you please give me some guidance? I know f and beta. I need to know gamma(f,beta). Here, alpha are the zeros of the bessel function J1(x). How can I approximate the solution as it goes to infinity.
Please, would anyone kindly help me, how can I approach it?

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Torsten
Torsten 2022-8-17
编辑：Torsten 2022-8-17
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Torsten
Torsten 2022-8-18
编辑：Torsten 2022-8-18
In my case, alpha_k are the zeros of J1(x). I understand that I need to use your function J0ROOTS to find out zeros of J1(x). My question is, will the no. Zeros be equal to the no. of Summation? For example: if I find out first 10 zeros, should I sum up k=1 to 10?
You have an infinite sum. The terms of this sum can be evaluated up to the value of k for which you have alpha_k, the k-th zero of J1(x). So you can only sum up to the k which you choose for the number of zeros supplied for J1.
You used sum(function,2) in the for loop. Does it help me to calculate K=1 to inf?
In the code, the sum over the columns of the matrix
2*besselj(0,a_k.*r)./(a_k.^2.*besselj(1,a_k)).*exp(-a_k.*t(i))
gives L(r,t(i)) , summed up from k=1 to k = n, not k = Inf, as a column vector with r = (0:0.01:1).'
Choose n big enough such that the difference between the finite and infinite sum becomes small enough for your purpose. Practically, test what comes out if you vary n.

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