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Continuous integrands have complex and absolute line integrals along every rectifiable path

Statement

Let $γ : [a, b] \to C$ be rectifiable and let $f$ be continuous on its trace. Then the complex line integral The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral and the absolute line integral The absolute line integral over a rectifiable path using its arc-length function both exist.

Facts & Assumptions

Given: A rectifiable $γ = x + i y$ and a continuous $f = u + i v$ on its trace.

[L1]

A complex path is rectifiable if and only if its two coordinates have bounded variation, by Complex path length equals coordinate length and piecewise speed integral, transporting A path in $R^{n}$ is rectifiable exactly when every coordinate has bounded variation to its coordinate path.

[L2]

The arc-length function $s_{γ} = s_{Φ \circ γ}$ defined in The absolute line integral over a rectifiable path using its arc-length function is continuous and nondecreasing by the Euclidean assertion applied to the coordinate path (The arc-length function is continuous and nondecreasing, with increments equal to subpath lengths; it is strictly increasing exactly when no nondegenerate subpath is constant).

[L3]

If a real integrand is continuous and a real integrator has bounded variation, then its Riemann–Stieltjes integral exists (A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator).

Proof

technique · direct

1.1L1L3

By [L1], $x$ and $y$ have bounded variation. The four real functions $u \circ γ$ and $v \circ γ$ are continuous, so [L3] gives all four Stieltjes integrals in the complex definition.

1.2L2L3

The function $∣ f \circ γ ∣$ is continuous, and [L2] makes $s_{γ}$ a bounded-variation integrator, so [L3] gives the absolute integral.

2.1step 1.1step 1.2∎

Thus both definitions are well-defined. On a singleton or constant path the relevant integrators are constant and every integral is $0$ .

Continuous integrands have complex and absolute line integrals along every rectifiable path

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources