The definite Sine Integral
is often computed with contour integration. This requires knowledge of complex analysis. I just realized that it can also be computed very nicely with just real analysis. Instead of considering only this single integral define a function depending on a non-negative real number by
Then will be the integral that we want to compute. It is also clear that , since the term becomes small very rapidly. Let’s first do some hand waving exploration. Assume that is differentiable and that its derivative can be computed by differentiating the integrand (as a function of ). Then we find
This integral can be computed with an explicit anti-derivative for positive .
Filling in the boundaries of integration we find
And again, this equation can be solved explicitly: for some constant . This equation also holds for because was assumed to be differentiable and hence continuous. We also know that tends to for and that means that and in particular .
The hand waving at least results in the correct value for the definite Sine Integral! Instead of rigorously checking all steps above (which might be possible) I take an alternative approach and get rid of the infinity in the integration. For each positive integer define a function by
Then . Since the integrand is differentiable any number of times in both and , there is now no problem in computing the derivative by differentiating the integrand
Integrating this equation and plugging in the correct value for we find
Taking the limit for this results in
The last integral term in this equality tends to zero as and taking this limit for finally result in the value for the definite Sine Integral.
Anyone familiar with the Laplace transform would probably summarize this post as a one liner