The discussion revolves around the properties of holomorphic functions in complex analysis, specifically focusing on the function e^z and its implications for related functions like cosh(z). Participants are exploring the criteria for holomorphicity without relying on limits, instead considering the Cauchy-Riemann equations and power series representations.
The conversation is ongoing, with participants providing guidance on using the Cauchy-Riemann equations and discussing the implications of holomorphic functions. There is a recognition of the relationship between e^z and cosh(z), but no consensus has been reached on the best approach to demonstrate holomorphicity.
Participants have mentioned constraints such as avoiding limits and the need to adhere to specific definitions of holomorphicity. There is also an acknowledgment of prior knowledge regarding power series and the properties of holomorphic functions.
Dick said: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?
Firepanda said: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.
Firepanda said: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)
Dick said:How about e^z. Can you split that up?
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.Firepanda said: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.
Dick said: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.