Homework Help: Holomorphic functions

1. Mar 8, 2010

Firepanda

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?

Last edited: Mar 8, 2010
2. Mar 8, 2010

Dick

Why does d/dz(e^z)=e^z show e^z is holomorphic? Or at least, why are you saying you aren't sure this shows it's holomorphic?

3. Mar 8, 2010

Firepanda

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.

4. Mar 8, 2010

Firepanda

Because in my notes he mentioned the power series

Last edited: Mar 8, 2010
5. Mar 8, 2010

Dick

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.

6. Mar 8, 2010

Firepanda

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)

7. Mar 8, 2010

Dick

How about e^z? Can you split that up?

8. Mar 8, 2010

Firepanda

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.

9. Mar 8, 2010

Dick

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.

10. Mar 8, 2010

Firepanda

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