Complex Integration | Two Singularities Inside the Contour

#complexanalysis #cauchyintegral #calculus #math #maths #mathematics #manim #visualization #contourintegration #cauchyintegralformula #integral #formula #complex #analysis #python #pythonprogramming #solved #examples #two #poles #engineering #residue In this video, we evaluate a complex contour integral with two singularities using the Cauchy Integral Formula. We solve the integral: I = ∮ dz / (z^2 + 16) over the circle |z| = 5. Step-by-step solution visualized with Manim: 1. Factor the denominator: z^2 + 16 = (z - 4i)(z + 4i). 2. Identify singularities: z = 4i and z = -4i. Both lie inside the contour |z|=5. 3. Apply the Deformation of Contours principle to split the integral into two parts. 4. Apply Cauchy's Integral Formula to each singularity separately. 5. Sum the results to get the final answer (Result is 0). This example demonstrates how to handle multiply connected domains in Complex Analysis.    • Theorem: Every Convergent Sequence is Boun...      • Solving Differential Equation xy\,dx + (x+...      • Does it Converge or Diverge? Integral of 1...      • Integral of 1/x^2: Power Rule with Negativ...      • Complex Analysis: Finding the Laurent Seri...      • Metric Spaces: Definition, Axioms, and Exa...