1. Complex differentiation

Hey, I have this problem from a tute i was given,
Its all in the picture,

For part a i just showed the derivatives exist and that they were continuous, then used laplace to show that they are harmonic du/dx - du/dy = 0,
so that work, then to find v i just found the harmonic conjugate and i got -3y^2x+x^3 + c = v
so u+iv = y^3 - 3x^2y +i(3y^2x+x^3 + c )
Is that the complete solution to the question? I feel like I am missing somthing, more of a proof or somthing.

Part b i showed in the picture, i think the first part seems ok, then showing for elsewhere i just let z0 be any point, and tried to find if there were any points besides zero where the limit exists but lim z0/delz is infinity, so is that sufficient to say that that its only differentiable at zero?
We did somthing different aswell, let it be z= u+iv, then f(z) = u^2 + ivu cause Re(z) = u, then let these u^2 and vu be two new variables and applied C-R using implicite differentiation and found somthing different, that it is differentiable when Re(z) = constant or Im(z) = 0

Am i on the right track at all?

2. Re: Complex differentiation

Originally Posted by Daniiel
Hey, I have this problem from a tute i was given,
Its all in the picture,

For part a i just showed the derivatives exist and that they were continuous, then used laplace to show that they are harmonic du/dx - du/dy = 0,
so that work, then to find v i just found the harmonic conjugate and i got -3y^2x+x^3 + c = v
so u+iv = y^3 - 3x^2y +i(3y^2x+x^3 + c ) Note the missing minus sign.
Is that the complete solution to the question? I feel like I am missing something, more of a proof or something.
If z = x+iy and f(z) = u+iv, then I think they want you to express f(z) as a function of z (rather than x and y). With the change of sign that I indicated, you have $\displaystyle f(z) = ix^3 - 3x^2y - 3ixy^2 + y^3.$ Can you see how to write that as a function of z? [Think: Binomial theorem.]

Originally Posted by Daniiel
Part b i showed in the picture, i think the first part seems ok, then showing for elsewhere i just let z0 be any point, and tried to find if there were any points besides zero where the limit exists but lim z0/delz is infinity, so is that sufficient to say that that its only differentiable at zero?
We did somthing different as well, let it be z= u+iv, then f(z) = u^2 + ivu cause Re(z) = u, then let these u^2 and vu be two new variables and applied C-R using implicite differentiation and found somthing different, that it is differentiable when Re(z) = constant or Im(z) = 0

Am i on the right track at all?
The first part of b) is ok. For the second part, you are right to use the C–R equations, but you are getting the notation a bit mixed. If z = x+iy then $\displaystyle f(z) = x^2 + ixy.$ So you should take $\displaystyle u = x^2$ and $\displaystyle v = xy.$ Then apply the C–R equations.

3. Re: Complex differentiation

oh sweet thanks very much, so in terms of z for a, f(z)= iz^3 +ic right?
So for part b, because both C-R equations don't hold it is not differentiable elsewhere? only if x=y=0