Integration of the Green's function along a cylindrical surface

Question

Julian Beckmann 2020-5-24

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https://ww2.mathworks.cn/matlabcentral/answers/532393-integration-of-the-green-s-function-along-a-cylindrical-surface

回答： Raunak Gupta 2020-5-29

I am trying to integrate following expression:

, along a cylindrical surface. I was planning to use the cylindrical expansion of the Green's function and the symbolic maths toolbox of matlab. Consequently, I produced the following code:

syms rho rhop phi phip z zp k m 
GreenCylindrical = (4/pi) * int(cos(k * (zp - z)) * (((1/2) * besseli(0,k * rho) * besselk(0,k * rhop)) + ...
    symsum(cos(m * (phip - phi)) * besseli(m,k * rho) * besselk(m,k * rhop),m,1,inf)),k,0,inf);
phiS = int(int(cos(phi) * GreenCylindrical,z,-inf,inf),phi,0,2*pi);

But when I run this code, it seems that Matlab does not compute the integral at all, because I obtain the following result:

phiS =
 
int(int((5734161139222659*cos(phi)*int(cos(k*(z - zp))*(symsum(cos(m*(phi - phip))*besseli(m, k*rho)*besselk(m, k*rhop), m, 1, Inf) + ...
    (besseli(0, k*rho)*besselk(0, k*rhop))/2), k, 0, Inf))/4503599627370496, z, -Inf, Inf), phi, 0, 2*pi)
 

I could expect that there is problem with the fact, that the function is dependent on six variables, but I am only integrating along two of them. However, I could not find a way to get it working yet. Thank in you in advance for your help

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Answer 1

Raunak Gupta 2020-5-29

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Hi,

While integrating the following function with respect to two variable the overall function still comes out dependent on four variables. There are some tips mentioned in the documentation of int which you may use to get the solution but still without simplifying the expression it will be hard to get the closed form solution with the help of int because it depends on the fact that antiderivative of function should exist in the first place.

Maybe you can somehow simply the expression which is not dependent on the variables of integration, so those terms can be treated as constant and thus you can get a closed form solution.