Graduate Proofs
Proofs written during graduate study in mathematics.
-
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.
-
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.
-
Proof Euler's Identity
Two proofs of e^(iπ) + 1 = 0: from the power series for exp, sin and cos, and from the circle group.
-
Proof Irrational Square Roots
The square root of n is rational only for perfect squares: a descent proof with no primes, and one by Euclid's lemma.
-
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.
-
Proof The Golden Ratio is Irrational
Three proofs that phi is irrational: descent from phi = 1 + 1/phi, the rational root test, and its endless continued fraction.
-
Proof The Reflective Property of the Ellipse
The normal to an ellipse bisects the focal angle: a gradient proof and Heron's shortest-path proof.
-
Proof Three Tangents to a Parabola
Two tangents to any parabola are cut by a third in ratios summing to one, via similarity to y = x^2.
-
Proof Varignon's Theorem
The midpoints of any quadrilateral form a parallelogram with half its area, by one line of vector algebra.