Show that is an harmonic function, and find a conjugate harmonic function .

Check that holds for all x and y.

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- Oct 18th 2008, 10:00 AMMaccamanHarmonic Function
Show that is an harmonic function, and find a conjugate harmonic function .

Check that holds for all x and y. - Oct 18th 2008, 02:57 PMmr fantastic
If you know the definition of a harmonic function then you'll realise that all you have to do is a lot of partial differentiation. This is tedious but not intellectually difficult.

I don't help with tedium but can help with intellectual difficulty. Where are you stuck? - Oct 18th 2008, 07:54 PMMaccaman
Here is what I have done...

so

and

therefore

so is a harmonic function.

Now to find a conjugate harmonic function

This is where I get lost.....am I now supposed to differentiate w.r.t x or y?? I've tried both and keep getting bizarre values. How do I get the conjugate function??

As far as the third part goes (with out v)

then I think im supposed to do something like

which enables me to solve for some values A and B??

(Thinking) - Oct 18th 2008, 08:40 PMmr fantastic
To get the harmonic conjugate of u, read this: PlanetMath: harmonic conjugate function

- Oct 18th 2008, 10:37 PMMaccaman
Here is what I have now.....(Wondering)

Therefore

====>

so

therefore ===>

so

is that correct??

Also is my procedure for part 3 from above correct?

(I have to go to work now so I wont be able to write a response for another 6 - 7 hrs)....(Crying) - Oct 19th 2008, 05:19 AMshawsend
I get:

which satisfies the Cauchy-Riemann equations with given above. Also, I use the standard approach to calculating the complex conjugate of which looks intimidating but is easy to calculate for easy functions:

However it's not easy for your function. I used Mathematica to calculate above and I still had to do some manual processing of the data to arrive at my expression for . Is there an easier way to do it?

Also, I don't understand what you're trying to do with that gradient expression. Is that suppose to work for just this particular case? Never seen anything like that for the general case. - Oct 19th 2008, 05:36 AMMaccaman
Could've made a stupid mistake..........I was in a rush since I had to go to work......I'll double check it and see what I get....

- Oct 19th 2008, 05:57 AMMaccaman
You are right, and so am I ,almost............I just made a little typo......(or at least i think im almost right....im going to be embarrassed if i claim to be almost right and then i turn out to be wrong)

I should have written

( the typo is the - y^3 which I didnt include).

If you differentiate both your answer and mine with respect to x, you end up with

Now I'll try and show you what I was trying to do with the gradient expression (next post).. - Oct 19th 2008, 06:39 AMMaccaman
Shawsend, this is what I was trying to do with the gradient expression.....this answers the question, does it not? (Wondering)

similarly,

So

Therefore holds for all x and y.