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 attachment

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