Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


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??????

share|cite|improve this question
I think you just answered your own question: f'(z) exists if and only if x = log 2 + 2y. – echoone Dec 4 '12 at 2:28
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. – cderwin Dec 4 '12 at 2:30

Were this function to be analytic, it would agree with the exponential function on the whole real line. Since that subset of the complex plane has a limit point, the only analytic extension off the line is the exponential. Your function can never be analytic on any open subset of the complex plane.

share|cite|improve this answer
More precisely, it would agree with $e^x+i$ on the real line. – ˈjuː.zɚ79365 Jun 17 '13 at 15:11

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.