Verify that the Cauchy-Riemann equations

The problems are from complex variable class. Please specify the terms that you use if necessary and explain each step of your solution. If there is anything unclear in the problem, please tell me. Thank you very much. 6. Let u and v denote the real and imaginary components of the function f defined by the equations... see at ...continues

7. Solve equations (2) for ux and uy to show that

7. Solve equations (2) for ux and uy to show that ... see attachment

Cauchy-Riemann equations

8. Let a function f (z) = u + i v be differentiable at a nonzero point ... see attachment

Apply the given theorem to verify that each of these functions is entire

1. Apply the given theorem to verify that each of these functions is entire: (a) f (z) = 3x + y + i (3y - x) (b) f (z) = sin x cosh y + i cos x sinh y (c) f (z) = e-y sin x - i e-y cos x (d) f (z) = (z2 - 2) e-x e-iy.

Determine the singular points of the function

4. In each case, determine the singular points of the function and state why the function is analytic everywhere except at those points:... see attachment

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 ...continues