Finding the Harmonic Conjugate of \phi

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In summary, the conversation discusses the properties of analytic functions and their relationship to harmonic functions. The harmonic conjugate of a given function is shown to be related to its real and imaginary parts, and can be found using the Cauchy-Riemann equations. Alternatively, the Cauchy-Riemann equations can be used to show that the partial derivatives of the product of the real and imaginary parts are equal to zero, thus proving the function is harmonic.
  • #1
jonmondalson
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Homework Statement


If [itex]w = u(x,y)+iy(x,y)[/itex] is an analytic function then
[itex]\phi(x,y) = u(x,y)v(x,y)[/itex]
is harmonic, where u and v are the real and imaginary parts of w.
What is the harmonic conjugate of [itex]\phi[/itex]?

Homework Equations


So I know for analytic functions the Cauchy-Riemann equations:
[itex]\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}[/itex] and
[itex]\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}[/itex]

And for a harmonic function:
[itex]\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0[/itex] and
[itex]\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2}=0[/itex]

The Attempt at a Solution


I tried to find a function [itex]\Phi[/itex] that would satisfy:
[itex]\frac{\partial \Phi}{\partial dy}=\frac{\partial \phi}{\partial dx}[/itex] and
[itex]\frac{\partial \Phi}{\partial dx}=-\frac{\partial \phi}{\partial dy}[/itex]

for which I obtained:
[itex]\Phi = \int \frac{\partial u}{\partial x} v + \frac{\partial v}{\partial x} u \partial y[/itex] and
[itex]\Phi = \int \frac{\partial u}{\partial y} v + \frac{\partial v}{\partial y} u \partial x[/itex][/quote]

But I have no idea if this is correct, or relevant to finding the harmonic conjugate.

Any help is greatly appreciated.
 
Last edited:
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  • #2
consider w2
 
  • #3
lurflurf said:
consider w2

Thanks for this lurflurf. So I get
[itex]w^2 = u^2 - v^2[/itex]

I really have no idea what to do with this, could you kindly give me another little hint?
 
Last edited:
  • #4
That is not w2 that is Re[w2] what about Im[w2]?
alternatively try to write
(uv)x=uxv+uvx
(uv)y=uyv+uvy
in terms of oposite partials via Cauchy-Riemann equations
 
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  • #5
I'm confused by this. I'm trying to understand, but I don't know how to get any expression for the conjugate.
 
  • #6
If cannot guess the conjugate what you started doing with integrals would also work, the next step is to use the Cauchy-Riemann equations to that your integrals and partials are with respect to the same variables thus giving an answer without partials.
 
  • #7
May I make a suggestion: If f=u+v is analytic, then u and v are harmonic and so satisfy Laplace's equation each of them. That is, u_xx+u_yy=0, and v_xx+v_yy=0 and also if f is analytic then u and v satisfy the Cauchy Riemann equations. Well there you go: For:

[tex]g=uv[/tex]

compute:

[tex]\frac{\partial^2 g}{\partial x^2}+\frac{\partial^2 g}{\partial y^2}[/tex]

using the chain rule, and use what I've just said to show it's zero.
 

What is the Harmonic Conjugate of \phi?

The Harmonic Conjugate of \phi is a mathematical concept used in complex analysis and electrical engineering. It is the function that, when combined with \phi, forms a harmonic function, meaning that it satisfies Laplace's equation. In other words, the Harmonic Conjugate is the function that complements \phi to create a function that is both analytic and satisfies the Cauchy-Riemann equations.

Why is Finding the Harmonic Conjugate of \phi important?

Finding the Harmonic Conjugate of \phi is important because it allows for the creation of harmonic functions, which have many applications in mathematics and physics. For example, they are used to model heat flow, fluid flow, and electromagnetic fields. In electrical engineering, harmonic functions are used to describe alternating currents and voltages.

How do you find the Harmonic Conjugate of \phi?

To find the Harmonic Conjugate of \phi, you can use the Cauchy-Riemann equations, which relate the real and imaginary parts of an analytic function. By solving these equations, you can determine the Harmonic Conjugate of \phi. Another method is to use the method of images, which involves reflecting a point across a boundary to create a harmonic function.

What are some properties of the Harmonic Conjugate of \phi?

The Harmonic Conjugate of \phi has several important properties, including being orthogonal to \phi, meaning that the angle between the two functions is always 90 degrees. It also has the same level curves as \phi, meaning that they intersect at right angles. Additionally, the real and imaginary parts of the Harmonic Conjugate have a relationship similar to that of the gradient and curl in vector calculus.

What are some applications of the Harmonic Conjugate of \phi?

As mentioned earlier, the Harmonic Conjugate has many applications in mathematics and physics. In addition to modeling physical phenomena, it is also used in signal processing, image processing, and data analysis. It is also used in the study of conformal mappings, which are transformations that preserve angles between curves.

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