Since the integration is performed in the xy-plane,we have so that . The path of integration C consists of lines say DA,AB,BE,ED,where A,B,E,D are (a,0),(a,b),(-a,-b),(-a,0) respectively.
On DA,y=0,dy=0,x varies from -a to a, On AB,x=a,dx=0,y varies from a to -a, On BE,y=b,dy=0 and x varies from a to -a. and on ED,x=-a,dx=0 and y varies from b to 0..
Using (2),LHS of (1)
Now, on simplification.
Hence the theorm is verified.