Contour Integral of 1/z with respect to z along the Unit Circle  Complex Variables  Summary and Q&A
TL;DR
The video provides a stepbystep solution for integrating 1/z along the unit circle in the counterclockwise direction.
Key Insights
 ❓ The formula for integrating a function along a contour involves parametrizing the contour and evaluating the definite integral with the function and its derivative.
 🧡 The unit circle can be represented using the function W(t) = e^it, where t ranges from 0 to 2π.
 🤪 The derivative of e^it is i * e^it, which is used in the integration formula to solve for the integral of 1/z along the unit circle.
Transcript
hey everyone and this problem we have to integrate 1 over Z with respect to the Z along the unit circle and the counter clockwise direction so the formula for an integral like this is as follows so if you have the integral of f of Z with respect to Z along a contour see this is equal to the definite integral from A to B of F of let's say W of T tim... Read More
Questions & Answers
Q: What is the formula for integrating a function along a contour?
The formula states that the integral of a function f(z) with respect to z along a contour C is equal to the definite integral from A to B of f(W(t)) * W'(t) dt, where W(t) provides a parametrization of the contour.
Q: How is the unit circle defined?
The unit circle is a circle with its center at the origin and a radius of 1. It can be represented using the function W(t) = e^it, where t ranges from 0 to 2π.
Q: What is the derivative of e^it?
The derivative of e^it is i * e^it. When applying it to the parametrization of the unit circle, W'(t) becomes i * e^it.
Q: What is the simplified solution to the integral of 1/z along the unit circle?
The final answer is 2πi. By substituting the function 1/z and its derivative into the integration formula and evaluating the definite integral from 0 to 2π, the result simplifies to 2πi.
Summary & Key Takeaways

The video explains the formula for integrating a function along a contour, which involves parametrizing the contour and taking the derivative of the parametrization.

The unit circle is described as a circle centered at the origin with a radius of 1, and every point on the unit circle can be represented using the function e^it, where t ranges from 0 to 2π.

By plugging the function and its derivative into the integration formula and simplifying, the final answer for the integral of 1/z along the unit circle is found to be 2πi.