Normal probability integral using complex integration

Question

The usual method for proving that the $N(0,1)$ probability density integrates to $1$ involves squaring the integral and transforming to polar coordinates. I remember seeing it done using complex integration techniques, but could not find this method despite extensive search. I dont know enough about complex integration to do it myself. Can someone provide a detailed explanation, or a reference to where this is done.

Answer 1

Here is an approach in German (but easy to understand nonetheless):

See Exercise $5$ of the following series

http://www2.math.ethz.ch/education/bachelor/lectures/hs2014/math/fkt_theorie/s07.pdf

Here the solution:

http://www2.math.ethz.ch/education/bachelor/lectures/hs2014/math/fkt_theorie/l07.pdf

Remark: if I remember correctly there is also an approach with a rectangle, but I cannot seem to find the reference.