Subject GRE-Style: Complex Integration

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Complex Integration 3 worked problems on residues, Cauchy integral formula, and Laurent series Key Tools: Cauchy's Integral Formula: . Residue Theorem: where the sum is over singularities inside . The integrand has a simple pole at , which is inside . Write the integrand as where and . By Cauchy's Integral Formula: GRE Speed: Recognize the CIF pattern instantly. If you see , the answer is . Takes 10 seconds. Two Singularities, Two Residues Both (simple pole) and (double pole) are inside . Residue at (double pole): use the formula : Wait -- let's compute more carefully: . Double Pole Formula: For a pole of order 2 at : . Don't forget to differentiate! Question: What is the residue of at ? Laurent Expansion via Taylor Series Write the Taylor series for and divide by : The residue is the coefficient of in the Laurent series: GRE Power Move: For functions of the form , just write out enough terms of the Taylor series, divide by , and read off the coefficient. Memorize: , ,

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