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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??????

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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. –  dirty derwin Dec 4 '12 at 2:30
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1 Answer

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.

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