Prove. Complex form of Green’s theorem. Let B(z, ) be continuous and have continuous partial derivatives in a region R and on its boundary C. Then
where dA represents the element of area dxdy and z and are complex conjugate coordinates z = x + iy and = x - iy.
Proof. Let B(z, ) = P(x, y) + iQ(x, y) [i.e. P(x, y) + iQ(x, y) is the function obtained by substituting z = x + iy and = x - iy into B(z, ) ]. Given a region R with boundary C and a function B(z, ), continuous and with continuous partial derivatives in R , the integral around C is given by
Applying Green’s theorem to the right member we get
so we can write 2) as