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tylerc1991

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## 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.