I'm not too sure how to show this. Perhaps if I show d(ez)/dz = ez then does this conclude that ez is holomorphic on all of C?
Sure, use Cauchy-Riemann. That's easy enough. Then you'd want to show e^(-z) is also holomorphic. Do you know the sum of holomorphic functions is also holomorphic? If not then you could just directly show cosh(z) is holomorphic using CR.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.
Sure, check e^(-z) as well. It's almost the same thing. I'm a little puzzled why you can split e^(z) and e^(-z) up and not cosh(z). You already said cosh(z)=(e^z+e^(-z))/2.exeiy
= ex(cosy + i.siny)
=excosy + i.exsiny
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.