Homework Help: Holomorphic functions

I'm not too sure how to show this. Perhaps if I show d(e^z)/dz = e^z then does this conclude that e^z is holomorphic on all of C?

 

I suppose I should be using the cauchy riemann equations then?

I was told not to use limits here so I didn't want to use the base definition of holomorphic-ness.

 

Because in my notes he mentioned the power series

 

i tried to use C-R, but I'm unable to split it up into

U(x,y) or V(x,y)

using z=x+iy into cosh(z)

 

e^xe^iy

= e^x(cosy + i.siny)

=e^xcosy + i.e^xsiny

The C-R satisfy this and so it is holomorphic on all of C? so this would immediatly imply cosh(z) is entire?

And I would check if it were true for e^-z as well.

 

ah ye that would make more sense if i did it directly, thanks!