ai calculator learanadeAI

Residue Theorem Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions
Get Solution
Example
Created on 2024-06-20Asked by Madison Young (Solvelet student)
Evaluate the integral ∫∣z∣=2ezz(z−1) dz \int_{|z|=2} \frac{e^z}{z(z-1)} \, dz using the residue theorem.

Solution

To evaluate the integral ∫∣z∣=2ezz(z−1) dz\int_{|z|=2} \frac{e^z}{z(z-1)} \, dz using the residue theorem: 1. **Identify the singularities inside the contour ∣z∣=2|z| = 2:** The singularities are at z=0z = 0 and z=1z = 1. 2. **Calculate the residues at the singularities:** - **Residue at z=0z = 0:** Res(ezz(z−1),0)=lim⁡z→0z⋅ezz(z−1)=lim⁡z→0ezz−1=e00−1=−1. \text{Res}\left(\frac{e^z}{z(z-1)}, 0\right) = \lim_{z \to 0} z \cdot \frac{e^z}{z(z-1)} = \lim_{z \to 0} \frac{e^z}{z-1} = \frac{e^0}{0-1} = -1. - **Residue at z=1z = 1:** Res(ezz(z−1),1)=lim⁡z→1(z−1)⋅ezz(z−1)=lim⁡z→1ezz=e11=e. \text{Res}\left(\frac{e^z}{z(z-1)}, 1\right) = \lim_{z \to 1} (z-1) \cdot \frac{e^z}{z(z-1)} = \lim_{z \to 1} \frac{e^z}{z} = \frac{e^1}{1} = e. 3. **Apply the residue theorem:** ∫∣z∣=2ezz(z−1) dz=2πi(Res(ezz(z−1),0)+Res(ezz(z−1),1))=2πi(−1+e). \int_{|z|=2} \frac{e^z}{z(z-1)} \, dz = 2\pi i \left( \text{Res}\left(\frac{e^z}{z(z-1)}, 0\right) + \text{Res}\left(\frac{e^z}{z(z-1)}, 1\right) \right) = 2\pi i (-1 + e). Therefore, the integral evaluates to 2πi(e−1)2\pi i (e - 1). Solved on Solvelet with Basic AI Model
Some of the related questions asked by Mia Jackson on Solvelet
1. Use the residue theorem to evaluate the contour integral ∮Cz3z2+1dz \oint_C \frac{z^3}{z^2 + 1} dz , where C C is the unit circle centered at the origin.2. Calculate the residue of the function f(z)=z3z2+1 f(z) = \frac{z^3}{z^2 + 1} at the point z=i z = i .,
DefinitionThe residual theorem is a powerful method in complex analysis through which the value of the contour integral needs to be calculated. That is, if f is analytic inside and on a simple closed contour C, possibly excluding isolated singularities, then the integral of f around C is equal to 2πi times the sum of the residues of f at these singularities. Example: Consider ∮C​z2+1ez​dz, where C is the unit circle.
Related Calculators
Applications of Derivatives CalculatorConformal Mapping CalculatorEquations with Square Roots CalculatorBoundary Conditions Calculator
Need topic explanation ? Get video explanation
@Copyright Solvelet 2024Privacy PolicyTerms and Condition