MATH 396 PROBLEMS 12

IGOR KRIZ

Regular problems:

1. Using calculus of variations, find all possible extrema of

∫_{a}^{b}(ϕ′(x))^{2}dx

over the space of all functions ϕ : [a, b] → ℝ with fixed ϕ(a), ϕ(b).

2. If the action of a variation problem is F (x, y, z) = zf(y), prove that the Euler equation is satisfied by every (smooth) function. What does that say about

𝔉(ϕ) = ∫_{a}^{b} F (x, ϕ(x), ϕ′(x))dx

in this case? Can you prove that conclusion directly (without using Euler equation)?

3. Let f be a function holomorphic in B_{ϵ}(a) = {z| |z − a| < ϵ}. Prove that if f(a) = 0, then f(z) = (z − a)g(z) where g is holomorphic in B_{ϵ}(a). [Use the Taylor formula.]

4. Let f be a holomorphic function, and let L be the circle with radius 1 and center 0, oriented counter-clockwise. Let z be inside L, z ≠ 0. Using the Cauchy formula, evaluate

\frac{1}{2πi} ∫_{L} \frac{f(ζ)ζ}{ζ − z} dζ.

Challenge problem:

5. Given numbers p, q, find an interval [a, b] and a function g : [a, b] → ℝ, g(a) = p, g(b) = q, such that the length of the graph of g is 1 and the area under the graph of g is maximal possible.

(a) This problem is slightly different than the typical variation problem we considered, because you don’t know the interval [a, b]. Therefore, parametrize the graph of g not by (x, g(x)), but by (ϕ(t), ψ(t)) (ϕ, ψ : [0, 1] → ℝ) where the length of the curve increases at a unit length (i.e. (ϕ′)^{2} + (ψ′)^{2} = 1). The derivative of the area under the graph by t is ϕ′(t)ψ(t). Write a variation problem (Euler equation) for ψ.

(b) To solve the Euler equation, use the trick we used in class:

u′ = v ⇒ (u^{2})′ = 2uv.