Make a function holomorphic

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In summary: However, you still need to check that this value of k actually works, i.e. that v satisfies the Cauchy-Riemann equations.
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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
 
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  • #2
jimmycricket said:

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.
 
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  • #3
jimmycricket said:

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.
 
  • #4
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
jimmycricket said:
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).
 

1. What does it mean for a function to be holomorphic?

A holomorphic function is a complex-valued function that is differentiable at every point in its domain, meaning it has a well-defined derivative at every point. This property is often referred to as "complex differentiability."

2. Can a function be holomorphic on a subset of its domain?

No, a function must be holomorphic on its entire domain in order to be considered holomorphic. If a function is not holomorphic on any point in its domain, then it is not holomorphic at all.

3. How can I determine if a function is holomorphic?

To determine if a function is holomorphic, you can use the Cauchy-Riemann equations. These equations state that a function f(x+iy) is holomorphic if and only if its real and imaginary parts satisfy the partial differential equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x, where u(x,y) is the real part of f(x+iy) and v(x,y) is the imaginary part of f(x+iy).

4. Can a function be holomorphic and still have singularities?

Yes, a function can be holomorphic and still have singularities. A singularity is a point in the complex plane where a function is not defined or is not differentiable. A holomorphic function can have removable, essential, or pole singularities, but it must be differentiable at all other points in its domain.

5. Are all analytic functions also holomorphic?

Yes, all analytic functions are also holomorphic. An analytic function is one that can be represented by a convergent power series in its domain. Since a convergent power series is differentiable term by term, an analytic function is also differentiable at every point in its domain and therefore, holomorphic.

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