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 is holomorphic in using the sufficient conditions related with the Cauchy Riemann equations.

Originally Posted by FernandoRevilla

Prove that is holomorphic in 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.