Hi I'm having some trouble with showing a function is analytic at a point. If someone could help me with these questions I'd appreciate it.
(a) Find all points where the function f(z)=(x^3+y^3+3y) + i(y^3-x^3+3y) is differentiable and compute the derivative at those points.
using the cauchy riemann formulae i found that this funtion is differentiable at all points z=x+iy where x^2=y^2+1 or y^2=x^2-1 which, by taking the square root on both sides, is undefined at x=0 and hence the derivative of f at a point z=x+i(sqrt(x^2-1)) is f'(x+i(sqrt(x^2-1)))=3x^2-i3x^2. Is this correct? If not, then where am I going wrong.
I'm having more difficulty with this next question:
(b)Is the function in (a) analytic at any point? Justify your answer clearly.
by the definition:A function f(z) is said to be analytic at a point z0 if there exists some ε-spherical neighborhood of z0 at all points of which f '(z) exists. I'm having a hard time using this definition to show that it is analytic at all points z=x+iy where x^2=y^2+1 or y^2=x^2-1 or am I wrong in thinking that it's analytic at these points? I just don't know how to approach this and reason it out.
Thanks in advance for any help.