Thread: How to show that this function is entire

Assume that f(z) is entire. Show that e^f(z) is also entire.

I did the following:

f(z) = u(x,y) + iv(x,y) = u + iv

let w(z) = e^(f(z)) = e^(u + iv) = (e^u)(cos(v) + isin(v)) = g(x,y) + ih(x,y)

but I couldn't figure out what to do... I'd be very glad if you can help me...

Prove that $\displaystyle e^{f(z)}$ is holomorphic in $\displaystyle \mathbb{C}$ using the sufficient conditions related with the Cauchy Riemann equations.

Originally Posted by FernandoRevilla

Prove that $\displaystyle e^{f(z)}$ is holomorphic in $\displaystyle \mathbb{C}$ using the sufficient conditions related with the Cauchy Riemann equations.

Yes, but I couldn't come to a conclusion...

By definition, an entire function is a holomorphic function whose domain is the whole complex plane.