Question

Problem 1 Consider the composition f(w(z)) of two complex valued functions of a complex variable, f(w) and w(z), where z = x+iy and w=u+iv. Assume that both functions have continuous partial derivatives. Show that the chain rule can be written in complex form as of _ of ou , of Oz . . of az " dw dz * dw dz and Z of ou , of ou dw dz* dw ƏZ Show as a consequence that if f(w) is analytic in w and w(z) is analytic in z, then f(w(z)) is an analytic function of 2.

Answer 1

Cauchy Riemann equation furiu is anlytic of au , av 2x Hener dis analybees ng 22 i Stete otis analytie them =0:- ☺ in preoblem 1; ze xury wautiv. and +(w (7.)) is two complex valued function. of flo) is analytics in this by @ che = 0;

and if wl?) is analytic me, than 1 00 which implies trab of of most by chain Rulley ; - of.0+ 0w.o 0.-) f is analytic. = f (w (2)) is ad analytic inz .