Proving Entirety of conj(f(conj(z))) for an Entire Function f

[tylerc1991]

Homework Statement

Show that if a function f(z) = u(x,y) +iv(x,y) is entire, then the function conj(f(conj(z))) is entire.

Homework Equations

(i) The Cauchy-Riemann (CR) equations hold for functions that are entire: u_x = v_y and u_y = -v_x

(ii) conj(_) is the conjugate (i.e. there is a conjugate bar over f and over z)

The Attempt at a Solution

Since f is entire, CR is satisfied:
so u_x = v_y and u_y = -v_x

this implies:
u_x = -(-v_y) and -u_y = -(-v_x)

this implies:
CR is satisfied for a function g(z) = u(x,-y) - iv(x,-y)

but g(z) = conj(f(conj(z)))

this implies:
CR is satisfied for conj(f(conj(z)))

Since f is entire, the partial derivatives are continuous

this implies:
conj(f(conj(z))) is entire.

 

[Dick]

You've got the right idea. The presentation is a little confusing. Might be best if you say conj(f(conj(z))=u1(x,y)+i*v1(x,y) where u1(x,y)=u(x,-y) and v1(x,y)=(-v(x,-y)) and show u1 and v1 satisfy CR.

 

[tylerc1991]

OK I can reword it. But there are no holes in the proof? Thank you very much for your help!

 

[Dick]

It's all there. I just have to read between the lines a bit to figure out u_x=(-(-v_y)) is important. Do rewrite with different symbols for the real and imaginary parts of conj(f(conj(z)).