Week 10: Analytic Continuation

Hello everyone and welcome back to my blog!

This week I showed my off site mentor an example of a “proof” that relies on assigning a divergent series a finite value and explained how I planned to tweak it to be correct. The problem I was coming up against was the final step of this proof which was to show that because an analytic function f(x) (I had defined as the integral from 0 to infinity of e^(-t)cos(xt)dt) is equal to 1/(1+x^2) for all x between -1 and 1, then it is equal to this for all values of x. I explained that I thought the best way to approach this is by considering the analytic function f(x)-1/(1+x^2) which must be 0 for all x between -1 and 1 and to prove from there that it is 0 everywhere. He recommended a different approach and explained the process by which functions over the complex plane are typically analytically continued.

This approach involves taking an analytic function which is only defined for a circle of input values on the complex plane and using the Cauchy integral formula (which makes a statement about analytic functions) to expand the possible input values to a larger circle on the complex plane. By repeating this process, the domain of input values for which the function is defined grows until every point on the complex plane is covered.

At any rate, by analytically continuing the function f(x) to take the values 1/(1+x^2), I have managed to tweak a few more proofs originally dependent on divergent sums.