Evaluating Integrals Using Cauchy's Formula/Theorem

• e(ho0n3
In summary, to evaluate the given integral using Cauchy's Formula or Theorem, we first identify that the integrand is not defined for z=2 and z=-2, making it impossible to apply either formula. However, we can manipulate the integrand to express it as g(z)/(z-(-2)), where g(z) is analytic inside and on the given curve. This allows us to apply Cauchy's Formula and evaluate the integral.
e(ho0n3
[SOLVED] Evaluating Integrals Using Cauchy's Formula/Theorem

Homework Statement

Evaluate the given integral using Cauchy's Formula or Theorem.

$$\int_{|z+1|=2} \frac{z^2}{4 - z^2} \, dz$$

Homework Equations

Cauchy's Theorem. Suppose that f is analytic on a domain D. Let $\gamma$ be a piecewise smooth simple closed curve in D whose inside $\Omega$ also lies in D. Then

$$\int_\gamma f(z) \, dz = 0$$

Cauchy's Formula. Suppose that f is analytic on a domain D and $\gamma$ is a piecewise smooth, positively oriented simple closed curve D whose inside $\Omega$ also lies in D. Then

$$f(z) = \frac{1}{2\pi i} \int_\gamma \frac{f(\zeta)}{\zeta - z} \, d\zeta$$

for all $z \in \Omega$.

The Attempt at a Solution

For z equal to 2 and -2, the integrand is not defined and hence is not analytic in any domain that includes these points. The problem is that the curve |z + 1| = 2 contains the point -2 inside it so I can't apply either Cauchy's Theorem or Formula. What can I do?

hint

z^2/(4 - z^2) = z^2/((2 - z)(2 + z)) = (z^2/(z-2))/(z+2) = g(z)/(z - (-2)), where g(z) = z^2/(z-2) is analytic inside and on your circle

Ah! Thanks for the tip. That really helps.

1. What is Cauchy's Formula/Theorem?

Cauchy's Formula/Theorem is a mathematical theorem that allows for the evaluation of certain integrals using complex analysis. It is named after the French mathematician Augustin-Louis Cauchy.

2. How does Cauchy's Formula/Theorem work?

Cauchy's Formula/Theorem states that if a function is analytic in a closed region and continuous on its boundary, then the value of the integral of the function around the closed region is equal to the sum of its values at all points within the region.

3. What types of integrals can be evaluated using Cauchy's Formula/Theorem?

Cauchy's Formula/Theorem can be applied to integrals that are difficult or impossible to evaluate using traditional methods, such as those containing trigonometric functions or complicated algebraic expressions.

4. Are there any limitations to using Cauchy's Formula/Theorem?

Cauchy's Formula/Theorem can only be applied to integrals that satisfy the conditions mentioned in question 2. Additionally, the function being integrated must be analytic, meaning it can be represented by a power series.

5. How is Cauchy's Formula/Theorem used in real-world applications?

Cauchy's Formula/Theorem has various applications in physics, engineering, and other fields where complex integrals arise. It is used to solve problems related to electric circuits, fluid dynamics, and quantum mechanics, among others.

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