Complex Line Integrals

• Published in: A Guide to Complex Variables pp. 19 - 32
• Format: Book Chapter
• Language: English
• Published: Washington DC The Mathematical Association of America 2011
• Subjects: holomorphicity and the complex derivative; integrals on curves; curve; FundamentalTheoremof Calculus along curves; properties of; continuously differentiable function; complex line integral; counterclockwise; independence of parametrization; simple, closed; Morera's theorem; Cauchy integral formula; conformal; deformability of curves; Fundamental Theorem of Calculus; closed; Cauchy integral theorem; conformality; Goursat's theorem; characterization of holomorphicity in terms of; complex differentiability; general form
• DOI: 10.5948/UPO9780883859148.003

REAL AND COMPLEX LINE INTEGRALS In this section we shall recast the line integral from calculus in complex notation. The result will be the complex line integral. The complex line integral is essential to the Cauchy theory, which we develop below, and that in turn is key to the argument principle and many of the other central ideas of the subject. CURVES It is convenient to think of a curve as a continuous function γ from a closed interval [a, b] ⊆ ℝ into ℝ2 ≈ ℂ. We sometimes let denote the image of the mapping. Thus Often we follow the custom of referring to either the function or the image with the single symbol γ. It will be clear from context what is meant. Refer to Figure 2.1. It is often convenient to write For example, γ(t) = (cos t, sin t) = cos t + i sin t, t ∈ [0, 2π], describes the unit circle in the plane. The circle is traversed counterclockwise as t increases from 0 to 2π. See Figure 2.1. CLOSED CURVES The curve γ: [a, b] → ℂ is called closed if γ(a) = γ(b). It is called simple, closed (or Jordan) if the restriction of γ to the interval [a, b), which is commonly written γ|[a, b), is one-to-one and γ(a) = γ(b) (Figure 2.2).