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Make a function holomorphic

  • #1
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
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

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
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.
 
Last edited:
  • #3
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
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751

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 Im 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.
 
  • #4
116
2
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?
 
  • #5
Dick
Science Advisor
Homework Helper
26,258
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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).
 

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