Complex analysis - holomorphic functions

Cauchy Riemann equations should be a good starting point...

I’ve just done that and I’ve come to the conclusion that f is holomorphic for all x and y. Does that sound right?

Seems likely; I've not actually done any calculations, but looking at it I think you can fairly easily write it as a polynomial in z = (x+iy), and so it's differentiable.

I think I’ve done a) correctly then. Any advice with b)? I’ve got that it was holomorphic on 0, pi/2, etc, but I don’t know how to differentiate it

If you look at the derivation for the Cauchy-Riemann equations, you'll see various expressions for the complex derivative that involve the terms in the Cauchy Riemann equations. (Because the Cauchy-Riemann derivation is basically "find two different expressions that must both equal the complex derivative, and deduce that they must be equal). Just use one of those.

[I could give you a simple "do this" formula, but you shouldn't need me to if you understand the derivation.

I was going to give a longer explanation, but TSR lost my post and I can't face doing it again.

If you need more help, find a derivation of the CR equations online that you understand (or at least is equivalent to your lectures), post a link to it, and I will try to give more help using that as a reference].

I don't know how to differentiate the solution at all, i'm looking at it, and its just stumping me. I've done the cauchy riemann equations for the first part, to find that its holomorphic on 0, p/2, etc, but I don't how to actually differentiate the function

Have you not covered the derivation of the C-R equations? I.e. where they come from?

The technique used in the derivation shows that $f'(z_0) = u_x(x_0, y_0) + \mathbf{i} v_x (x_0, y_0)$ when we have $f(x+iy) = u(x,y) + \mathbf{i} v(x,y)$ and $z=x+\mathbf{i}y$.

No, I definitely haven't covered that. Should I have already done it? My lecturer basically told me to google anything we get stuck on