Graduate Proofs
Proofs written during graduate study in mathematics.
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Proof A Power Series with a Natural Boundary
The lacunary series z + z^2 + z^4 + ... blows up at every dyadic root of unity, so no arc of the unit circle can be crossed.
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Proof Entire Functions of Polynomial Growth
An entire function with a bounded k-th derivative, or growth at most |z|^k, is a polynomial of degree at most k.
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Proof The Bessel Function of Order Zero
J0 is the only solution of Bessel's equation analytic at the origin; with Bessel's integral and the bound |J0(x)| <= 1.