Show that is an harmonic function, and find a conjugate harmonic function .
Check that holds for all x and y.
Here is what I have done...
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??
Here is what I have now.....
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)....
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.
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)..