QUIZ 2.6B

Theorem 2.6 If f = f(z) is analytic, then in any formula for f, x and y can only occur in the combination x + iy.

Proof   We note that x = (1/2)(z + ), { 1 }. Hence if w = f(z) = u + iv, we can regard u, v as functions of z, .  Now, w is a function of z alone { 2 } . So

+ i ( { 3 } ) = 0  { 4 } ux = vy,  uy = -vx. 

equating real, imaginary parts to zero.
Hence f analytic the Cauchy-Riemann equations hold as required.

Match the above boxes 1, 2, 3, 4 with the selections
(a)  y = 1/2i (z - ), (b)   (c)  (d) 

My solutions:

1.   ;  2.   ;  3.   ;  4.   .   Check