Cauchy-Riemann on holomorphic sets

Question

I am given $e^{z^2+3z+4}$ 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.

so I let $z=x+iy$

which gives me $^{(x+iy)^2+3(x+iy)+4}$

when simplified I get $^{x^2+2xyi-y^2+3x+3iy+4}$

this simplifies to $^{x^2-y^2+3x+4}e^{2xyi+3iy}$

which simplifies to $^{x^2-y^2+3x+4}(\cos(2xy+3y)+i\sin(2xy+3y))$

and if I distribute then $e^{x^2-y^2+3x+4}(\cos(2xy+3y)+e^{x^2-y^2+3x+4}i\sin(2xy+3y)$

I don't know where to go from this point, help would be appreciated

Answer 1

$HINT$: If $f(z)$ is holomorphic in domain $D$ then so is $e^{f(z)}$.
Can you see that why $f(z)$ is holomorphic on the whole complex plane $\mathbb C$?

Answer 2

Instead of proving that this particular function is holomorphic everywhere, it would be more advantageous to prove (rigorously) that compositions of holomorphic functions are holomorphic, and then to prove that $e^z$ and complex polynomials are holomorphic, at which point your question is just a particular example of a composition of exponential and polynomial function.