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We all know that the complex integral calculus can be useful for computing real integrals. I was wondering if there are any similar example where we can use Green's theorem to compute one-variables integrals.

Now it is clear that if we have an integral $\int_a^b f dx$ on the real line we can view this as a curve integral in the plane of $\int P dx + Q dy$ for infinitely many choices of $P$ and $Q$ where we integrate over the line between $a$ and $b$. We could then integrate this vector field over some other curve, $\gamma$, with the same endpoints and try computing the difference between our original integral and the new one with Green's theorem. It is clear that for random choices of $P$, $Q$ and $\gamma$ we will not have simplified our problem.

But are there any examples where this technique is useful?

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One can make contrived examples, where the integral in one variable is too cumbersome, but Green's theorem gives the answer in a jiffy. Are you looking for examples already existing in literature somewhere? – Aryabhata Jan 15 '11 at 3:21
I would prefer examples where you can imagine a motivation for wanting to compute the integral, other than illustrating Green's theorem. (Which may be because it arises in applied work or other reasons.) Part of the background to my question is that I wonder whether one could motivate students by mentioning this as an example of an application of Green's theorem. – Johan Jan 15 '11 at 14:57
up vote 3 down vote accepted

It is easy to compute $\int_{-\infty}^\infty e^{-x^2/2}dx$ by an elementary argument, but maybe you want the integral $\int_{-\infty}^\infty e^{-x^2/2}\cos(\lambda x)dx$ without having complex analysis (Cauchy's theorem) at your disposal. So you can apply Green's theorem to the vector field $$(P,Q):=e^{(y^2-x^2)/2}\bigl(\cos(x y),\sin(x y)\bigr)$$ and an elongated rectangle $[-a,a]\times[0,\lambda]$; then let $a\to\infty$.

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Johan, isn't $\displaystyle \int_C Fdz=2i\iint_{\Omega}\frac{\partial F}{\partial \bar{z}}dxdy$ true where F=u+iv? I think every example that you mention in your first sentence, is infact, an application of Green's theorem (I didn't check my last sentence.)

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