Yes, that seems to be the case.Woolyabyss said:Homework Statement
Use Cauchy’s integral formula to evaluate the integral along γ(t) of (z/(z+9)^2)dz
where γ(t) = 2i + 4e^it , 0 ≤ t ≤ 2π.
Homework Equations
Cauchy's integral formula
The Attempt at a Solution
I was just wondering is the integral not just zero by Cauchy's theorem since (z/(z+9)^2) is holomorphic inside the circle defined by γ(t) ( the singularity at -9 is outside the circle ).
Cauchy's integral formula is a powerful mathematical tool used to evaluate integrals that involve complex functions. It states that the value of a complex integral around a closed contour is equal to the sum of the function values at all points inside the contour, multiplied by 2πi.
Cauchy's integral formula is often used in complex analysis and calculus to evaluate integrals that would otherwise be difficult or impossible to solve. It is commonly used in physics, engineering, and other fields that involve complex functions and contour integrals.
The steps for using Cauchy's integral formula are: 1) Identify the contour of integration, 2) Determine the function to be integrated, 3) Ensure that the function is analytic within the contour, 4) Evaluate the function at all points inside the contour, and 5) Multiply the sum of the function values by 2πi.
Some common mistakes when using Cauchy's integral formula include using a contour that is not closed or not analytic, not correctly identifying and evaluating the function at all points inside the contour, and incorrectly multiplying the function values by 2πi. It is important to carefully check all the steps and ensure that the formula is being applied correctly.
Yes, Cauchy's integral formula can be used to evaluate integrals with real functions, as long as the contour of integration is closed and the function is analytic within the contour. It is often used to evaluate real integrals that involve complex functions by converting them into complex integrals and using Cauchy's integral formula.