What Constant Makes a Function Holomorphic?

[jimmycricket]

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

 

[Dick]

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

You are working too hard. Can you show that Cauchy-Riemann implies ##v_{xx}+v_{yy}=0##? You don't really need to find u.

 

[LCKurtz]

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-\color{red}{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 I am not sure which direction to head next or even if this is the correct approach to begin with. Help is greatly appreciated

Your method is OK and will solve the problem. But you have mistakes in your work. Hard to point out your errors since you omitted some steps. In particular, the term in red is incorrect.

 

[jimmycricket]

Dick and LCKurtz thanks both for replying. Dick: following your method I get [itex]v_{xx}+v_{yy}=2ky+6y=0 \rightarrow k=-3[/itex]
This method is showing that the function is harmonic for k=-3. Is this sufficient for answering the question of which k makes the function holomorphic?

 

[Dick]

Dick and LCKurtz thanks both for replying. Dick: following your method I get [itex]v_{xx}+v_{yy}=2ky+6y=0 \rightarrow k=-3[/itex]
This method is showing that the function is harmonic for k=-3. Is this sufficient for answering the question of which k makes the function holomorphic?

Yes, if the function is holomorphic then v needs to be harmonic. So the only possible value is k=(-3).