Imagine Integrating
When all methods fails, just imagine.
A freshman calculus course will introduce the double integral. There I was in a tutorial class, a freshman facing the problem at hand.
I was recently acquainted with the sign ‘
’, so the double integral didn’t fool me. It was evaluating the inner integral that seems puzzling. We all have seen 
, but integrating 
? Not quite. To that, the teacher said ‘It will save us a lot of trouble by integrating with respect to y first’.
Integrate along x and then along y.
becomes 
and instead of the limits of y from 0 to 
, we use the limits of x from 0 to 
. Our integral is now
Once we know what method to use, the answer is pretty straightforward. So there I was, seeking a challenge for myself.
Reverse the order. y first and then x.
in terms of x. All usual methods would have failed but there remain one last approach: venturing into the imaginary realm.
I must admit that I have not taken any formal lessons in complex variable analysis. My knowledge of the field comes only from reading a book about 
and another textbook you can find in the library section. Still, with that little definition of 
, very interesting results can surface, which we shall quickly see.
Okay, here was my plan. Since the integrand was 
raised to a variable power, if we could ‘sneak’ an 
into the index, we could then split the integrand into cosine and sine terms. It then allowed us to integrate, suspecting that everything will fall into place when we take the real part of the result.
We will integrate by substitution in order to change the variable 
. However, what variable would we pick to replace ‘x’. We needed to pick one, that when squared will leave an 
. I was quick to think 
but notice that by squaring the 
, we’ll get -1. The solution is then to let 
. This way
and 
Substituting into the integral, we get
Now we’re talking. We manage to split the integrand into the usual trigonometry terms which would allow us to integrate accordingly and then take the real part of the result.
I was initially quite amazed with myself. This is a hard problem? You got to be kidding me. The joy was short lived when I notice another obstacle to the problem. What exactly is the real part?
I was quick to overlook the 
outside the integral. While 
is undoubtedly real, 
isn’t necessarily so. Or is 
the real part? To makes matters even more confusing, we are looking for the real part after doing the integrating.
Taking a tour into the imaginary street.
and 
are essentially the same integral problem. While it is clear that the former gives a real result, it is certainly unclear whether the latter gives a real or imaginary result.
Perhaps I should wait until I take my two-term course on complex variable analysis to answer this question. If anything, it does show the intriguing results you can get by using imaginary numbers. Although we could not work out an answer, we could at least approach the problem from another angle. Just like how we could reach a destination by taking another street, the ‘imaginary street’ in this case, which may or may not lead to a dead end. You never know unless you try.
I’m open to any explanations to why the use of imaginary numbers either fails or succeeds. Perhaps this is easy for a freshman math major.
