Why can't I get this integration to work?

Question

Kendra Hamilton 2023-10-6

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2029934-why-can-t-i-get-this-integration-to-work

评论： Kendra Hamilton 2023-10-6

采纳的回答： Walter Roberson

在 MATLAB Online 中打开

Hi, I'm trying to integrate the function I(z) from -infinity to infinity. This is my code:

syms P w0 r n lamda z

Pi = sym(pi);

zR = Pi*w0^2*n/lamda;

I(z) = 2*P*zR^2/(Pi*w0^2)*1/(zR^2+z^2)*exp(-2*r^2*zR^2/(w0^2*zR^2+w0^2*z^2));

I1 = int(I, z, -inf, inf)

I1 =

but it just prints

I1 = int((2*P*n^2*w0^2*pi*exp(-(2*n^2*r^2*w0^4*pi^2)/(lamda^2*(w0^2*z^2 + (n^2*w0^6*pi^2)/lamda^2))))/(lamda^2*(z^2 + (n^2*w0^4*pi^2)/lamda^2)), z, -Inf, Inf)

Can someone tell me what I'm doing wrong?

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Answer 1

Walter Roberson 2023-10-6

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2029934-why-can-t-i-get-this-integration-to-work#answer_1327209

移动：Walter Roberson 2023-10-6

在 MATLAB Online 中打开

syms P w0 r n lamda z real

Pi = sym(pi);

zR = Pi*w0^2*n/lamda;

I(z) = 2*P*zR^2/(Pi*w0^2)*1/(zR^2+z^2)*exp(-2*r^2*zR^2/(w0^2*zR^2+w0^2*z^2));

IE = expand(I)

IE(z) =

I1 = int(IE, z, -inf, inf)

I1 =

Now look at the form of that integral. Can we find a simpler integral that has the same general form but still cannot be integrated? Yes.

syms B C x

F = exp(-B/(x^2+C))

F =

int(F, x, -inf, inf)

ans =

A key point for this is that if you hardcode B as 1 then MATLAB can proceed, and likewise if you hard-code C as 0. But the combination of the two is more than MATLAB can deal with.

Maple is not able to deal with this integral either

Your actual integral is worse than this.

Your basic problem, then, is that there is no closed form integral for that expression that can be found by MATLAB (or Maple either)