Unformatted text preview: Û
dzÔ ≤ 2π n+1 2πr = n
2π Ô ı (z - z )n+1 Ô
Corollary 5.5 (Louville's Theorem)
Entire bounded functions are constant.
Proof: S’pose that f is bounded on the entire complex plane, so that |f(z)| ≤ K for some
constant K. We now use the case n = 1 of the above theorem, giving
where r is the radius of an arbitrary circle with center z0. Since r is arbitrarily large, it
must be the case that f'(z0) = 0. Since this is true for every z0 é C , it must be the case
that f(z) = constant. (If f'(z) = 0, then the partial derivatives of u and v must all vanish,
and so u and v are constant.)
ı |z - z |n+1 dz
The integrand is now constant, since |z - z0| = r, the radius of the circles. Therefore, the
integral on the right boils down to
n O dz
≤ Corollary 3 (Fundamental Theorem of Algebra)
Every polynomial function of a complex variable has at least one zero.
is entire. But it is also...
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