1
$\begingroup$

Let $f(z)=e^x + ie^{2y}$ where z=x+iy is a complex variable defined in the whole complex plane.

a)Where does f'(z) exist? b) Where is f(z) analytic?

Answer:

a) I used the Cauchy Riemann to test whether the function is holomorphic. i got $x=\log2 + 2y$

b) I am not sure how to check if f(z) is analytic??????

  • 1
    I think you just answered your own question: f'(z) exists if and only if x = log 2 + 2y.2012-12-04
  • 0
    If $f'(z)$ exists (aka is holomorphic at $z$), the $f$ is analytic at $z$. This is one of the nicest properties of $\mathbb{C}$, but the standard proof involves the Cauchy Integral Theorem.2012-12-04

1 Answers 1