ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

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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

1.1L1

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 $∣ z ∣ = 1$ .

1.2L2L4

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.

2.1step 1.1step 1.2L3∎

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)