Further Analysis | Part IB, 2001

State Liouville's Theorem. Prove it by considering

z=Rf(z)dz(za)(zb)\int_{|z|=R} \frac{f(z) d z}{(z-a)(z-b)}

and letting RR \rightarrow \infty.

Prove that, if g(z)g(z) is a function analytic on all of C\mathbb{C} with real and imaginary parts u(z)u(z) and v(z)v(z), then either of the conditions:

 (i) u+v0 for all z; or (ii) uv0 for all z\text { (i) } u+v \geqslant 0 \text { for all } z \text {; or (ii) } u v \geqslant 0 \text { for all } z \text {, }

implies that g(z)g(z) is constant.

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