Math Problem Statement

Using Cauchy’s integral formula, evaluate ∫ 𝒛+𝟏 𝒛 𝟐+𝟐𝒛+𝟒 𝒅𝒛 𝒄 where C is the circle |𝒛 + 𝟏 + 𝒊| = 𝟐

Solution

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Contour Integration
Singularities
Residues

Formulas

Cauchy's Integral Formula: ∫_C f(z)/(z - a)^n dz = 2πi f^(n-1)(a)/(n-1)!
Quadratic expression factorization: z^2 + 2z + 4 = (z + 1)^2 + 3

Theorems

Cauchy's Integral Formula
Residue Theorem

Suitable Grade Level

Undergraduate (Complex Analysis)