Math Is Fun Forum

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Harmonic Function

Hi, as my exam is round the corner, I just wish to check if my concept is right. Say if

is analytic on a domain D, am I right to say that

is harmonic on domain D?

Since if

is analytic on domain D it will means that

and

are both harmonic on D and hence satisfy Laplace's equation:

Let

then

which also satisfy the Laplace's equation.

So

is harmonic on domain D?

Last edited by coffeeking (2009-04-30 17:35:32)

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Harmonic Function

could anyone confirm this for me? Thanks.

In addition what about u-v, uv, u/v? Is Laplace equation always the method to check for harmonic?

Thanks in advance.

Avon

Member

Registered: 2007-06-28

Posts: 80

Re: Harmonic Function

By definition, a harmonic function is a function whose Laplacian is 0.

Your proof that the sum of two harmonic functions is harmonic is fine, and is easily adapted to prove that the difference of two harmonic functions is harmonic.

Consider the function u(x,y) = x
u is harmonic but u^2 is not so the product of harmonic functions is not necessarily harmonic.
I'm sure you can find an example to show that the quotient of harmonic functions need not be harmonic.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Harmonic Function

Thanks Avon;

I needed that too.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Harmonic Function

Thanks