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 Incoming links: Composition of harmonic conjugate pairs

f_{xx} + f_{yy} = 0

(f, g)

f + i g

is holomorphic; the corresponding Cauchy-Riemann equations are

f_{x} = g_{y}

and

f_{y} = - g_{x}

P = p (u, v)

and

Q = q (u, v)

, and assume the inner pair's image lies in the outer pair's domain. Their equations are

u_{x} = v_{y}

u_{y} = - v_{x}

p_{u} = q_{v}

and

p_{v} = - q_{u}

. The chain rule gives

P_{x} = p_{u} u_{x} + p_{v} v_{x} = p_{u} u_{x} - q_{u} v_{x} = Q_{y},

P_{y} = p_{u} u_{y} + p_{v} v_{y} = - p_{u} v_{x} - q_{u} u_{x} = - Q_{x} .

Thus the composite satisfies the Cauchy-Riemann equations. Its components are

C^{2}

, and differentiating gives

P_{xx} + P_{yy} = Q_{y x} - Q_{x y} = 0

and similarly

Q_{xx} + Q_{yy} = 0

. The composite pair is harmonic conjugate, including when the inner map has a critical point.

Assume the image of the domain of

f

lies in the domain where

g

is holomorphic. The complex chain rule gives

(g \circ f)^{'} (z) = g^{'} (f (z)) f^{'} (z),

so the composition is holomorphic, including at points where

f^{'} = 0

S (x, y) = s (u (x, y), v (x, y))

Δ S = s_{uu} ∣\nabla u ∣^{2} + 2 s_{uv} \nabla u \cdot \nabla v + s_{vv} ∣\nabla v ∣^{2} + s_{u} Δ u + s_{v} Δ v .

∣\nabla u ∣^{2} = ∣\nabla v ∣^{2}

; parts (i) and (ii) remove the last three terms. Since

s

is harmonic in its own variables,

Δ S = ∣\nabla u ∣^{2} (s_{uu} + s_{vv}) = 0.