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Complex line integrals are linear in the integrand
Statement
For continuous on the trace of a rectifiable contour and ,
Facts & Assumptions
Given: A rectifiable contour, continuous , and complex scalars .
The complex line integral is the stated combination of four real Riemann–Stieltjes integrals (The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral).
Riemann–Stieltjes integrals are linear in the integrand and integrator whenever the displayed integrals exist (Linearity and interval additivity of the Riemann–Stieltjes integral).
Proof
Expand the real and imaginary parts of and apply [L2] to each component integral in [L1].
Recombining the real and imaginary identities gives the displayed complex linearity. Zero scalars and singleton paths are included.
Depends on
Used by
- Cauchy's theorem for a null-homologous cycle Corollary
- The index of a cycle is locally constant off its trace and vanishes far from it Corollary
- Integration over a complex chain and the index of a chain Definition
- Dixon's gluing traced on the boundary cycle of an annulus Example
- The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration Example
- Cauchy-kernel contour integrals may be differentiated by a direct difference-quotient estimate Lemma
- Dixon's glued function is entire and vanishes at infinity Lemma
- Tagged sums approximate a contour integral within oscillation times length Lemma
- Vanishing integrals around triangles construct a primitive for a continuous function on a star-shaped domain Proposition
- A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc Theorem
- Cauchy's integral formula for a null-homologous cycle Theorem
- Cauchy's integral formula on a circle compactly contained in a disc of holomorphy Theorem
- Chain integration and the index are additive in the chain, and reverse with it Theorem
- Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain Theorem
- Laurent coefficients are given by contour integrals and are unique Theorem
- Laurent expansion on an annulus Theorem
- The winding number is constant on each connected component of the complement of the trace Theorem
Dependency tree · two levels
13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)