Show that the composite function G (z) = g (2z – 2 + i) is analytic in the half plane

Show that the composite function G (z) = g (2z – 2 + i) is analytic in the half plane x > 1, with derivative .... see attachment

Prove that f (z) must be constant throughout D if

7. Let a function f (z) be a analytic in a domain D. Prove that f (z) must be constant throughout D if (a) f (z) is real-valued for all z in D (b) | f (z) | is constant throughout D. (Question also included in attachment)

Show that u (x, y) is harmonic in some domain and find a harmonic conjugate v (x, y)

1. Show that u (x, y) is harmonic in some domain and find a harmonic conjugate v (x, y) when (a) u (x, y) = 2x (1 – y) (b) u (x, y) = 2x – x3 + 3xy2 (c) u (x, y) = sinh x•sin y (d) u (x, y) = y / (x2 + y2) (Question is also included in attachment)

Complex variables

2. Show that if v and V are harmonic conjugates of u in a domain D, then v (x, y) and V (x, y) can differ at most by an additive constant.

Complex variables

Please see the attached file for full problem description. --- 3. Use Cauchy-Riemann equations and the given theorem to show that the function  f (z) = e z is not analytic anywhere. Theorem: Suppose that f (z) = u (x, y) + i v (x, y) and that f ΄(z) exists at a point z0 = x0 + i y0. Then the first-order partial derivatives of u and v must exist at (x0, y0), and they must satisfy the Cauchy-Riemann equations ux = vy, uy = - vx there. Also, f ΄(z0) can be written f ΄(z0) = ux + i vx, where these partial derivatives are to be evaluated at (x0, y0).