Unable to show that functions are harmonic.

In summary, if the functions u(x,y) and v(x,y) have continuous second partial derivatives and satisfy the Cauchy-Riemann equations, they are harmonic functions. This is because the Cauchy-Riemann equations can be differentiated once more to show that u(x,y) and v(x,y) satisfy the Laplace equation, which is a defining property for harmonic functions.
  • #1
Kruum
220
0

Homework Statement



If the functions [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex] have continuous second partial derivatives and they satisfy the Cauchy-Riemann equations. Show that [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex] are harmonic functions.

Homework Equations



The Cauchy-Riemann equations are given: [tex]\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}[/tex] and [tex]\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}[/tex]

And functions are harmonic if [tex]\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} = 0[/tex] (Laplace equation)

The Attempt at a Solution



I've been stuck with this one for couple of hours now, and I really can't get much out of it. The only thing that's gone through my head is to differentiate the Cauchy-Riemann equations once more and trying to arrange the terms so that the Laplace equation is satisfied. But to no avail.
 
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  • #2
Are you forgetting that u,x,y=u,y,x (the commas indicate partial derivatives). I.e. the derivative is equal regardless of the order of differentiation.
 
  • #3
Dick said:
Are you forgetting that u,x,y=u,y,x (the commas indicate partial derivatives). I.e. the derivative is equal regardless of the order of differentiation.

No, I'm not. But you're tip doesn't ring a bell.

Edit: I'll take that back! Got it now! Thanks for the help! How can I be so stupid...
 
  • #4
Differentiate the first equation with respect to x and the second one with respect to y. The first equation has u,xx in it and the second one has -u,yy. And they are equal to the same thing. So they are equal to each other.
 
  • #5
Dick said:
Differentiate the first equation with respect to x and the second one with respect to y. The first equation has u,xx in it and the second one has -u,yy. And they are equal to the same thing. So they are equal to each other.

I solved it just before you replied. I can't believe I couldn't see that. Thanks a bunch!
 

Related to Unable to show that functions are harmonic.

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

A function is considered harmonic if it satisfies Laplace's equation, which means that it has continuous second-order partial derivatives and the sum of its second-order partial derivatives equals zero at every point.

2. How can you show that a function is harmonic?

To show that a function is harmonic, you can use the definition of harmonicity and check if the function satisfies Laplace's equation. You can also use the Laplace operator (∇²) to determine if a function is harmonic.

3. Why is it important to show that a function is harmonic?

It is important to show that a function is harmonic because it is a fundamental property in many areas of mathematics and physics. Harmonic functions play a crucial role in the study of potential theory, electrostatics, fluid dynamics, and other fields.

4. What are some common techniques for proving that a function is harmonic?

Some common techniques for proving that a function is harmonic include using the definition of harmonicity, using the Laplace operator, and applying the mean value property. Other techniques may involve using complex analysis or properties of harmonic conjugates.

5. Can a function be harmonic in certain regions and not in others?

Yes, a function can be harmonic in certain regions and not in others. For example, a function may satisfy Laplace's equation in one region but not in another due to boundary conditions or discontinuities. It is important to carefully consider the domain of a function when determining its harmonicity.

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