Cauchy - Riemann Function in terms of Z

[KleZMeR]

Homework Statement

I found the function V, which is the conjugate harmonic function for U(x,y)=sin(x)cosh(y). I am attaching my work. It turns out to be a two-term function with trig factors. I am then to write F(Z) in terms of Z, but is plugging in x, and y, in terms of Z into my trig functions good enough? I think there's some simplification that can take place, i.e. Euler, as I started, but I am just wondering if there is a specific direction I should take to simplify before I crunch the math? There are many directions and this is my first problem like this. Any help would be appreciated.

Homework Equations

The Attempt at a Solution

 

[KleZMeR]

Ok, I think I simplified it in terms of Z, if anyone disagrees please let me know! I used cosh(y)=cos(i*y), and i*sinh(x)=sin(i*x) , and another often-used trig sub.

 

[Dick]

Ok, I think I simplified it in terms of Z, if anyone disagrees please let me know! I used cosh(y)=cos(i*y), and i*sinh(x)=sin(i*x) , and another often-used trig sub.

The harmonic conjugate of a function U is supposed the be the imaginary part of an analytic function where U is the real part. sin(z*) is NOT analytic. You have a sign problem and from your posted photos I can't tell where it came from. Try and show your steps in TeX. Or find it yourself.

 

[KleZMeR]

Thanks Dick, yes it was a sign problem.