402 Contour Integral
An integral of a complex function along a closed curve that is used to evaluate complex function values through path integrals.
definition [d] (Contour Integral = Complex Line Integral) The integral of a complex function \(f(z)\) along a contour \(C\) in the complex plane, parametrized by \(z(t)\) for \(a \leq t \leq b\):
- \(\displaystyle \int_{C} f(z)\, dz = \int_{a}^{b} f\!\bigl(z(t)\bigr)\, z'(t)\, dt\) .
where
- \(C\) is a contour.
- \(f\) is a complex-valued function.
- \(z(t)\) is a parametrization of \(C\).
- \(z'(t) = dz/dt\).
Note:
- a contour is a path in the complex plane.
- \(C\) is also written \(\Gamma\).
definition [d] (Contour Integral = Complex Line Integral) The limit of Riemann sums of a complex function \(f\) along a contour \(C\) partitioned by points \(z_{0},\ldots,z_{n}\) with sample points \(\zeta_{j}\) on each arc:
- \(\displaystyle \int_{C} f(z)\, dz = \lim \sum_{j=1}^{n} f(\zeta_{j})\,(z_{j} - z_{j-1})\) ,
where the limit is taken as the mesh of the partition tends to zero.
where
- \(C\) is a contour in the complex plane.
- \(f\) is a complex-valued function defined on \(C\).
definition [d] (Contour Integral = Complex Line Integral) The integral of a complex function along a path \(C\) in the complex plane:
- \(\displaystyle \int_{C} f(z)\, dz = \int_{a}^{b} f\!\bigl(z(t)\bigr)\, z'(t)\, dt\) .
For a closed contour, the integral is often written \(\displaystyle \oint_{C} f(z)\, dz\).
where
- \(C\) is a contour; \(z(t)\) for \(a \leq t \leq b\) parametrizes \(C\).
- \(f\) is a complex-valued function.
402.1 References
- Complex-analysis sources in the notebook (e.g. Arfken; Riley–Hobson–Bence). — parametric and Riemann-sum definitions of the contour / complex line integral.
- Stewart, J. Calculus. — does not use the term “contour integral”; closest notion is the line / path integral (see Line Integral).
0 lebesgue integral 1 lebesgue measurable 2 borel measurable 1 simple function 2 measurable set 3 measurable space 3 measure space