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 < ... < tn = b and considering the expression
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:
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).
Read more about this topic: Line Integral
Famous quotes containing the words complex, line and/or integral:
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (18601904)
“As for conforming outwardly, and living your own life inwardly, I do not think much of that. Let not your right hand know what your left hand does in that line of business. It will prove a failure.... It is a greater strain than any soul can long endure. When you get God to pulling one way, and the devil the other, each having his feet well braced,to say nothing of the conscience sawing transversely,almost any timber will give way.”
—Henry David Thoreau (18171862)
“Self-centeredness is a natural outgrowth of one of the toddlers major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, whats his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of himan integral pieceis being torn from him.”
—Lawrence Balter (20th century)