Evaluating Real Integrals with Complex Analysis

In this section, we will explore a collection of techniques  for integrating real integrals in terms of complex integrals.  This is among the most useful applications of complex analysis.  Often, in engineering or scientific endeavors, we can express behavior in terms of an integral.  It may, however, be very difficult extract a value from that integral.  We can apply numerical techniques, but integrals that emerge from physical situations can be cumbersome and require huge amounts of computational power to obtain sufficient accuracy.  


Sometimes, we may simply apply a complex substitution or view our real integral as part of a complex contour and obtain an exact value with relative ease!  Since the examples we consider here are for pedagogical purposes for the most part, we will see few examples of the full power of the Cauchy residue formula.  

Consider the example below.  We see that this runs through a full period of the sine and cosine function which suggests we appeal to a remarkably familiar substitution at this point that yield a tractable contour integral.


Complex techniques can simplify all sorts of similar real integrals.  Other examples include the Cauchy Principal Value of infinite integrals.  Classical Riemann integration is not well-defined for integrals over unbounded intervals.  Moreover, it can be very difficult or even impossible to obtain exact values for such integrals using the techies of integration that are learned at undergraduate level.  If we integrate an odd function -a to a, the result should always be zero.  We extend this to the infinite case to obtain a Cauchy Principal value.  Indeed, more generally, we may often apply the residue theorem to problematic portions of a real integral.  Consider the following example.