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The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration

Example

On the positively oriented unit circle $γ$ , $\int_{γ} \frac{\exp z}{z} d z = 2 π i .$

Facts & Assumptions

Given: The positively oriented unit circle.

[L1]

The complex exponential is the series $\exp z = \sum_{n \geq 0} z^{n} / n!$ (The complex exponential by its power series), and this series converges absolutely for every complex $z$ (The complex exponential series converges absolutely for every complex argument).

[L2]

Complex line integrals are linear in the integrand (Complex line integrals are linear in the integrand).

[L3]

Uniform convergence on a fixed contour permits passage of the limit through the line integral (A uniformly convergent sequence of continuous integrands on a fixed contour permits passage of the limit through the complex line integral).

[L4]

On a positive circle, the integral of $(z - a)^{m}$ is $0$ for integer $m \neq - 1$ and $2 π i$ for $m = - 1$ (On a positively oriented circle about a, the integral of (z-a)^m is zero for every integer m except -1, and is 2 pi i for m=-1).

Verification

technique · direct

1.1

On $∣ z ∣ = 1$ , the exponential tail is bounded by the convergent numerical series $\sum 1 / n!$ , so the partial sums converge uniformly; division by $z$ preserves the bound because $∣ z ∣ = 1$ .

1.2

By [L2] and [L4], integrating the finite sum $\sum_{n = 0}^{N} z^{n - 1} / n!$ gives $2 π i$ from the $n = 0$ term and $0$ from every $n \geq 1$ term.

L2L4

2.1

Apply [L3] to the uniform convergence in step 1.1 and pass to the limit in step 1.2. The circle excludes $z = 0$ , so division is defined.

step 1.1step 1.2L3∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 124 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4, §1.3 (standard reference, not scraped)
R. Howell and J. Mathews, Complex Analysis, §6.2 (standard reference, not scraped)