Hi, I'm having a little bit of trouble comprehending the attached question.

First of all

show that

g(z) =f(z¯)¯¯¯¯¯¯

Is the [f(z bar)]bar simply not the same as f(z) ? as it is the conjugate of the conjugate.

could someone give me a pointer/hint as to how I'd go about showing g(z) is analytic on U bar? C-R equations come to mind.

Finally, isn't the last part of the question a given? How is one supposed to show that, if they have already define g(z) to equal (f(z bar))bar

thanks in advance.

edit:

Another quick q,

where is sin(abs(z)^2) differentiable for over the complex plane? from what i see, sin(abs(z)^2) is defined for all C? Am i wrong?

Same with sin(cosh(z)), this function is defined for all C, correct?