Using Cauchy-Riemann equations on holomorphic sets

Question

I am given $\frac{1}{e^z}$ and asked to state the subsets of $\mathbb C$ on which the following function is holomorphic, and to calculate the derivative on its domains of holomorphicity. My first step is to put this in Cauchy-Riemann form giving me $\frac{1}{e^{x+iy}}$ but I don't know where to go after performing the substitution.

Answer 1

You have $\frac1{e^z}=e^{-z}$. Since the exponential function is holomorphic, as well as the function $z\mapsto-z$, your function is holomorphic: it is the composition of two holomorphic functions.

Or you can use the fact that, if $x,y\in\Bbb R$, then$$\frac1{e^{x+yi}}=e^{-x-yi}=e^{-x}\bigl(\cos(y)-i\sin(y)\bigr).$$Now, if you define$$u(x,y)=e^{-x}\cos(y)\quad\text{and}\quad v(x,y)=-e^{-x}\sin(y),$$then$$u_x(x,y)=-e^{-x}\cos(y)=v_y(x,y)$$and$$u_y(x,y)=-e^{-x}\sin(y)=-v_x(x,y).$$

Answer 2

Alternative: The required subset of differentiability (or analyticity) is given by the subset $S=\{z\in\mathbb C \lvert \partial f/\partial\overline z=0\}$.
In your case $f(z)=e^{-z}$ and $\partial f/\partial\overline z=0$ holds for all $z\in\mathbb C$ so $S=\mathbb C$ is the set of analyticity as it is open and non-empty.