Suppose we have a general integral of the form
and we know how to expand in a series
It will usually be enough for the series to converge in some average sense to our original function.
Because polynomials and trigonometric functions tend to be particularly easy to integrate, this technique is often effective if one uses power series expansions or Fourier series expansions.
undergraduate calculus, undergraduate real analysis
Suppose we want to calculate the following integral on the unit disc of the complex plane
where and is the Lebesgue measure in the plane. Using the geometric series expansion
and polar coordinates we can rewrite the integral in the form
There are two or three tricks in the above lines worth mentioning. First we used polar coordinates as a means of decoupling the radial and angular variable. This becomes totally apparent in the last line where we end up with a product of integrals, one involving only the radial and one involving only the angular variable. This was combined with the interchange integrals or sums trick to bring the summation operators outside the integrals. Finally observe that we used the square and rearrange trick by writing
In this case we didn't have to square first since our expression was already squared.
To finish the calculation observe that is equal to whenever and otherwise because of the orthogonality of the exponentials . Thus we have
where we have used the Taylor series expansion for . Observe that the calculation of the integral reduced to calculating integrals of power functions since we expanded our function in a power series.
Let us now look at a slightly more complicated variant of the integral in Example 1. Here we have an extra logarithmic term under the integral sign
Following exactly the same steps as in Example 1 we end up with the expression
Now in Example 1 we ended up with the integral which is trivial to calculate exactly. Here we have to deal with the integral which does not look so trivial. However, we can once again expand the function in the power series and calculate
One can now simplify this last sum by observing that it is in fact a telescoping sum:
Thus, ignoring numerical constants, our original integral can be written in the form
One can probably look up the latter series in a table and discover that in fact
However, there is a Tricki way to see that quite fast. This uses essentially the Divide and Conquer trick so we will describe this calculation in an Example therein.
(prove exponential integrability of function based on the norms of the function.)