Residue Calculation for Higher-Order and Simple Poles — A Detailed Example

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In this video, we solve a comprehensive problem involving the Residue Theorem in Complex Analysis.

We analyze the function:
f(z) = e^z / ((z+1)³ * (z-2))

Step-by-step solution visualized with Manim:
1. Identify Poles: Find the singularities and determine their orders (Pole at z=-1 is Order 3, Pole at z=2 is Simple).
2. Calculate Residue at Order 3 Pole: Apply the general formula for residues of order n, involving the second derivative and the quotient rule.
3. Calculate Residue at Simple Pole: Use the limit formula for simple poles to find the second residue.
4. Apply Residue Theorem: Visualize the contour enclosing both poles and calculate the final contour integral ∮ f(z) dz by summing the residues.

This example is perfect for understanding how to handle different types of singularities in one problem.


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