Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adapted
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complex and absolute line integrals are invariant under increasing continuous reparametrization

Statement

Let ϕ:[c,d]→[a,b] be a strictly increasing continuous bijection, let γ:[a,b]→C be rectifiable, and let f be continuous on the trace of γ. Then ∫γ∘ϕf dz=∫γf dz,∫γ∘ϕ∣f∣ ∣dz∣=∫γ∣f∣ ∣dz∣. For singleton source and target intervals the same identities hold by the zero-integral convention.

Facts & Assumptions

Given: A rectifiable contour, a continuous integrand, and a reparametrization ϕ as in the Statement.

[L1]

Under a strictly increasing continuous bijection between nondegenerate compact intervals, the real Riemann–Stieltjes change-of-variable formula holds (Change of variable for the Riemann–Stieltjes integral).

[L2]

Complex arc length is invariant under continuous surjective monotone reparametrization, including singleton cases (Complex path length is invariant under monotone surjective reparametrization), by coordinate transport of Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal. Applied to every restriction this gives sγ∘ϕ(t)=sγ(ϕ(t)).

[L3]

Published piecewise-C1 line integrals are invariant under orientation-preserving reparametrization and change sign under orientation reversal (Scalar line integrals are parametrization-independent; vector line integrals retain orientation and change sign when it reverses).

[L4]

The complex integral is the combination of four component Riemann–Stieltjes integrals, and the absolute integral is the Riemann–Stieltjes integral against arc length (The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral, The absolute line integral over a rectifiable path using its arc-length function).

Proof

technique · cases
1.1assume-case nondegenerateL1L4

Assume first that both intervals are nondegenerate. Apply [L1] to each of the four component integrals in [L4]; their recombination is unchanged.

1.2assume-case singletonalgebra

If both intervals are singletons, both complex and absolute integrals are 0 by definition.

2.1step 1.1L1L2L4

For the absolute integral in [L4], [L2] identifies the reparametrized arc-length integrator, and [L1] gives the same Stieltjes integral.

3.1step 1.1step 2.1step 1.2L3cases-exhaustive∎

The cases exhaust the Statement and prove both identities. On piecewise-C1 contours apply [L3] to the coordinate path and coordinate vector fields; it gives exactly the increasing identity; decreasing reparametrization is excluded and instead changes the complex integral's sign.

Depends on

Dependency tree · 2 levels

Sources