In Reply to: Complex Analyis posted by Mike Bateman on February 02, 2000 at 16:58:54:
>Hi Professor Tao, I was wondering if you could clear a problem up >for me. In your lecture notes you have cosh(w)-sinh(w)=1. Should'nt >this be in terms of z instead of w?
I'm not sure where exactly in the notes you are referring to.
The hyperbolic cosh and sinh functions satisfy the identity
cosh(w)^2 - sinh(w)^2 = 1
for all complex numbers w. Since this identity works for all w,
I can use any other letter or quantity in place of w. So
I could also write
cosh(z)^2 - sinh(z)^2 = 1
or (say)
cosh(2z+e^w)^2 - sinh(2z+e^2)^2 = 1
and these would all be valid identities.
Terry