What is a complex line integral?
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
What is meant by complex integration?
Definition of complex integration : the integration of a function of a complex variable along an open or closed curve in the plane of the complex variable.
What is complex line integral?
Complex line integral In complex analysis, the line integral is defined in terms of multiplication and addition of complex numbers. Suppose U is an open subset of the complex plane C, f : U → C is a function, and is a curve of finite length, parametrized by γ: [a,b] → L, where γ(t) = x(t) + iy(t). The line integral.
How do you find the integral of a complex function?
To introduce the integral of a complex func tion, we start by defining what is meant by the integral of a complex-valued function of a real variable. Let /(f) = u(t) + iv(t) for a < t < b, where u(t) and v(t) are real-valued functions of the real variable t.
What are the different ways of computing a line integral?
Ways of computing a line integral. Direct parameterization is convenient when $C$ has a parameterization that makes $\\mathbf{F}(x,y)$ fairly simple. A natural parameterization of the circular path is given by the angle $\heta$. Because $C$ is a circle centered at the origin, the term $\\sqrt{x^2 + y^2} = r$ is a constant along $C$.
Why is the integrand of a circle centered at the origin?
Because is a circle centered at the origin, the term is a constant along . Hence, the integrand becomes fairly simple, and we choose direct parameterization as our method of integration. Let the parameterization be given by . Because the curve is a circle, we parameterize it with the angle .
How do you find the integrable functions of T?
Let /(f) = u(t) + iv(t) for a < t < b, where u(t) and v(t) are real-valued functions of the real variable t. If u and v are continuous functions on the interval, then from calculus we know that u and v are integrable functions of t.