Line Integral - Complex Line Integral

Complex Line Integral

The line integral is a fundamental tool in complex analysis. Suppose U is an open subset of the complex plane C, f : U → C is a function, and is a rectifiable curve parametrized by γ : → L, γ(t)=x(t)+iy(t). The line integral

may be defined by subdividing the interval into a = t₀ < t₁ < ... < t_n = b and considering the expression

$\sum_{k=1}^{n} f(\gamma(t_k)) =\sum_{k=1}^{n} f(\gamma_k) \Delta\gamma_k=S_{n}.$

The integral is then the limit of this Riemann sum as the lengths of the subdivision intervals approach zero.

If the parametrization is continuously differentiable, the line integral can be evaluated as an integral of a function of a real variable:

$\int_L f(z)\,dz =\int_a^b f(\gamma(t))\,\gamma\,'(t)\,dt.$

When is a closed curve, that is, its initial and final points coincide, the notation

is often used for the line integral of f along . A closed curve line integral is sometimes referred to as a cyclic integral in engineering applications.

The line integrals of complex functions can be evaluated using a number of techniques: the integral may be split into real and imaginary parts reducing the problem to that of evaluating two real-valued line integrals, the Cauchy integral formula may be used in other circumstances. If the line integral is a closed curve in a region where the function is analytic and containing no singularities, then the value of the integral is simply zero; this is a consequence of the Cauchy integral theorem. The residue theorem allows contour integrals to be used in the complex plane to find integrals of real-valued functions of a real variable (see residue theorem for an example).

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