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A holomorphic function equals its average on every circle inside a larger concentric holomorphy disc

Statement

Let $f$ be holomorphic on $D (a, R)$ and let $0 < r < R$ . Then

$f (a) = \frac{1}{2 π} \int_{0}^{2 π} f (a + r \exp (i θ)) d θ .$

Thus a holomorphic function equals its average on every positive-radius circle lying with a larger concentric disc inside its holomorphy domain.

Facts & Assumptions

Given: A function $f$ holomorphic on $D (a, R)$ and a radius $0 < r < R$ .

[L1]

Under these hypotheses, Cauchy's circle formula gives $f (z) = (2 π i)^{- 1} \int_{γ} f (ζ) / (ζ - z) d ζ$ for $∣ z - a ∣ < r$ , where $γ (θ) = a + r \exp (i θ)$ is positively oriented (Cauchy's integral formula on a circle compactly contained in a disc of holomorphy).

[L2]

For a piecewise- $C^{1}$ contour and an integrand continuous on its trace, the complex contour integral equals the parameter integral of the pulled-back integrand multiplied by the contour derivative (For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals).

[L3]

Every holomorphic function is continuous (Complex differentiability at a point implies continuity there).

Proof

technique · direct

1.1L1

Apply [L1] at the centre $z = a$ to obtain $f (a) = (2 π i)^{- 1} \int_{γ} f (ζ) / (ζ - a) d ζ$ .

2.1step 1.1L2L3algebra

By [L3], $f$ is continuous; on the circle the denominator $ζ - a$ is nonzero because its modulus is $r > 0$ , so elementary complex division makes $ζ \mapsto f (ζ) / (ζ - a)$ continuous on the trace. With $ζ = γ (θ) = a + r \exp (i θ)$ , [L2] gives $d ζ = i r \exp (i θ) d θ$ while $ζ - a = r \exp (i θ)$ , so step 1.1 becomes $f (a) = (2 π i)^{- 1} \int_{0}^{2 π} f (a + r \exp (i θ)) i d θ$ .

3.1step 2.1algebra∎

Cancelling the nonzero factor $i$ in step 2.1 yields the stated circular average; the calculation requires $r > 0$ and also covers every constant or zero function.

Depends on

Dependency tree · 2 levels

Sources

Matthias Weber, Complex Analysis, Corollary 2.2.1