Math 752 Spring 2015

Class 3/23

1 Harmonic Functions

Definition 1. G open subset of C. A u : G → C is harmonic if u has continuous second partial derivatives and ∂2u ∂2u + = 0. ∂x2 ∂y2 The left-hand side is called the Laplacian of u, denoted by ∇2u. Let 0 ∈/ G and rewrite the differential equation above in polar coordi- nates: ∂2u 1 ∂u 1 ∂2u + + = 0. ∂r2 r ∂r r2 ∂θ2 Sketch of derivation: Start with

ur = ux cos θ + uy sin θ

uθ = uxr sin θ + uyr cos θ and differentiate again and simplify which eventually leads to 1 1 u + u = u + u − u rr r2 θθ xx yy r r Examples of harmonic functions: iθ n U1(re ) = r cos(nθ), (n ≥ 0) iθ n U2(re ) = r sin(nθ), (n ≥ 1), We note that ∇2u = 0 is linear in u, hence a complex valued function is harmonic if and only if its real and imaginary part are harmonic. Hence, u(reiθ) = r|n|einθ is harmonic for n ∈ Z. A very important example of real harmonic functions comes from an- alytic functions. Let F be analytic and let u =

ux = vy, uy = −vx. Theorem 1. f on a region G is analytic iff

u(x + s, y + t) − u(x, y) = ux(x, y)s + uy(x, y)t + ϕ(s, t),

v(x + s, y + t) − v(x, y) = vx(x, y)s + vy(x, y)t + ψ(s, t), where ϕ(s, t) and ψ(s, t) are o(s, t) as (s, t) → (0, 0), and note now that f(z + s + it) − f(z) u s + u t + i(v s + v t) ϕ(s, t) + iψ(s, t) = x y x y + s + it s + it s + it The C-R equations simplify this to f(z + s + it) − f(z) ϕ(s, t) + iψ(s, t) = u (z) + iv (z) + , s + it x x s + it which gives the claim by letting (s, t) → (0, 0).

Theorem 2. Let u be real valued and harmonic on some disk |z| < r. Then there is a harmonic function v on the disk (called the ) such that f = u + iv is analytic on G. Proof. Wanted: v, infinitely differentiable, so that

vx = −uy, vy = ux, because then vxx + vyy = −uyx + uxy = 0. Consider the form

P dx + Qdy := uydx − uxdy.

We have Py = uyy = −uxx = Qx, hence the form is exact, and since any disk is simply connected, Calculus III gives: there exists a potential function v with vx = P = uy and vy = Q = −ux. Done! Corollary 1. If u : G → C is harmonic, then u is infinitely differentiable. Remark. One can show that the existence of the harmonic conjugate on an G is equivalent to G being simply connected.