MATLAB CODE of an equation including Bessel's function

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?

Answer 1

[Torsten]

Here is a very similar problem solved:

https://de.mathworks.com/matlabcentral/answers/1777535-infinite-sum-with-bessel-s-function?s_tid=srchtitle

[Md Jahid Hasan Sagor]

Thank you so much for your kind help @Torsten. I have some questions on this.

1.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?

2. You used sum(function,2) in the for loop. Does it help me to calculate K=1 to inf?

Thank you so much again

[Torsten]

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.

[Md Jahid Hasan Sagor]

I got it. Thank you so much for your help.