(a) Recall (Sec. 5) that if z = x + iy, then

By formally applying the chain rule in calculus to a function F(x, y) of two real variables, derive the expression

(b) Define the operator

suggested by part (a), to show that if the first-order partial derivatives of the real and imaginary components of a function f (z) = u(x, y) + iv(x, y) satisfy the Cauchy-Riemann equations, then

Thus derive the complex form ˆ‚f/ˆ‚ = 0 of the Cauchy-Riemann equations.

2i = 2( x + iay). a 2 2e I fe