MATLAB An Unsolvable Integral (according to Matlab)

AI Thread Summary
The discussion revolves around the difficulty of solving a specific integral using Matlab, which is expressed as the integral of e^{-t^2} over a rational function. The user encounters a warning in Matlab indicating that an explicit solution cannot be found, while successfully obtaining results from Mathcad and Mathematica, which provide similar numerical answers around 0.088. The user speculates that Mathcad and Mathematica might be utilizing the theorem of residues to evaluate the integral, noting the presence of simple poles in the integrand. Additionally, there is a reference to Gradshtyn & Rytzhik's tables of integrals, with a request for clarification on what "G & R" refers to. The discussion also includes a mention of a related integral provided by Mathematica, highlighting the complexity and potential methods for solving such integrals in different software environments.
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Hello!

I have the following problem: I cannot solve the integral below by the means of Matlab.
<br /> \int_{-\infty}^{\infty} \frac{e^{-t^2}}{\left(2-t\right)^2 + 16} dt<br />​
When I write the following in Matlab
Code:
>> syms t;
>> y = exp(-t^2) / (16 + (2 - t)^2);
>> int(y, t, -inf, inf)
it gives me the output :bugeye:
Code:
Warning: Explicit integral could not be found.
> In sym.int at 58
 
ans =
 
int(exp(-t^2)/(16+(2-t)^2),t = -Inf .. Inf)

I managed to calculate the integral by the means of both Mathcad and Mathematica. Mathcad gave 0.088 as an answer (I had to explicitly specify "Infinite Limit" as a method). Mathematica gave me 0.0880741, I used the NIntegrate function:

<br /> \mbox{NIntegrate}\left[\frac{e^{-t^2}}{\left(2-t\right)^2 + 16}, \left\{t, -\infty, \infty \right\} \right]<br />​

Does anyone have an idea, how I can solve this integral in Matlab? What do Mathcad and Mathematica use in order to solve it?

Thanks!
 
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I think it/they use(s) the theorem of residues.The integrand has simple poles at 2\mp 4i.

Daniel.
 
I didn't find this integral in G & R 5-th edition,CD version

\int_{0}^{\infty} \frac{e^{-ax^{2}+bx+c}}{x^{2}+d^{2}} \ dx

,but this one was

\int_{0}^{\infty} \frac{e^{-\mu^{2}x^{2}}}{x^{2}+b^{2}} \ dx

Daniel.
 
Last edited:
dextercioby said:
G & R 5-th edition,CD version

Could you please explain what "G & R" is?
 
Gradshtyn & Rytzhik,"Tables of Series,Integrals and Products",Academic Press,5-th edition,CD version.

Daniel.
 
Here it is,courtesy of Mathematica,a closely related integral.

\int_{-\infty}^{+\infty} \frac{e^{-x^{2}}}{(2-x)^{2}+4} \ dx =\frac{\sqrt{\pi}}{6}\left[3\sqrt{\pi}\cos 8-12 \ _{1}F_{2}\left(1,\frac{3}{4},\frac{5}{4};-16\right) +64 \ _{1}F_{2}\left(1,\frac{5}{4},\frac{7}{4};-16\right)\right]

Daniel.
 

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