- #1

jimmycricket

- 116

- 2

## Homework Statement

Find a constant k such that the function [itex] v(x,y) = y^3-4xy +kx^2y [/itex] can be the imaginary part of a holomorphic function f on

**C**

## Homework Equations

The Cauchy-Riemann equations: [itex]u_x=v_y[/itex] and [itex]u_y=-v_x[/itex]

## The Attempt at a Solution

So far I have taken the partial derivatives of v w.r.t y and equated it to [itex]u_x[/itex] and then integrated w.r.t x giving:

[itex]u=3xy^2-2x^2y+(k/3)x^3+f(y)[/itex]

Then differentiating w.r.t y to give:

[itex]u_y=6xy-2x^2y+(k/3)x^3y+f'(y)[/itex]

Next I equate this to [itex]-v_x[/itex] giving:

[itex]6xy-2x^2y+(k/3)x^3y+f'(y)=4y-2kxy[/itex]

Now Im not sure which direction to head next or even if this is the correct approach to begin with. Help is greatly appreciated