evol_w10lv said:Homework Statement
We need to calculate this complex integral as line integral:
Homework Equations
The Attempt at a Solution
This is correct, I guess:
But not sure about this part:
Are dx, dy, x, y chages correct or there is other method to use?
NIntegrate[(z^2 + z*Conjugate[z])*I*Exp[I*t] /. z -> Exp[I*t], {t, Pi, 0}]A complex integral is an integral that involves complex-valued functions. It can be thought of as a generalization of the real-valued integrals that are commonly used in calculus. Complex integrals are used to calculate the area under a curve in the complex plane, and they are also used in many areas of physics and engineering.
A line integral is a type of integral that is used to calculate the total value of a function along a given curve or path. It involves breaking the curve into small segments and taking the sum of the function values at each segment. Line integrals are commonly used in vector calculus to calculate work, flow, and other physical quantities.
To calculate a complex integral as a line integral, you first need to parameterize the curve or path that the integral will be taken over. This involves expressing the curve in terms of a single parameter, such as t. Then, you can use the fundamental theorem of calculus to convert the complex integral into a line integral, which can then be solved using standard techniques from vector calculus.
Complex integrals as line integrals have many applications in mathematics, physics, and engineering. Some common examples include calculating work done by a force field, calculating fluid flow in a pipe, and solving problems in electromagnetism and quantum mechanics.
Yes, there are several special techniques that can be used to calculate complex integrals as line integrals. These include using Cauchy's integral formula, the residue theorem, and contour integration. These techniques can be particularly useful when dealing with complex functions that have singularities or when the path of integration is not a standard shape, such as a circle or rectangle.