# Solving Analytic complex variable function. (Cauchy Riemann)?

• Feb 16th 2008, 07:39 AM
jalal0
Solving Analytic complex variable function. (Cauchy Riemann)?
Hello,
Can someone please show me how to answer these 2 questions:

1) Given that f(z) = u(x,y) + iv(x,y) is analytic, and u - v = (x-y)(x^2 + 4xy + y^2), determine the u(x,y) and v(x,y)

2) For f(z) = u(x,y) + iv(x,y) which is analytic, and its real part is u(x,y) = x^2 + Ay^2.
a) Determine the constant A in u(x,y).
b) Given f(0) = 0, determine the function f (z)

Thanks a load in advance. Please, I am really struggling with these problems.
• Feb 16th 2008, 08:21 AM
Plato
Because f(z) is analytic we know that $u_x = v_y \,\& \,u_y = - v_x$.
Now skipping the tedium in doing the partials, from the given $\left\{ \begin{gathered}
u_x - v_x = 3x^2 + 6xy - 3y^2 \hfill \\ u_y - v_y = 3x^2 - 6xy - 3y^2 \hfill \\ \end{gathered} \right.$

From those to we can get $2u_y = 6x^2 - 6y^2$ now solve for u.
Repeat this process and solve for v and adjust the answers.