Show that ∮(z-z1)^-1(z-z2)^-1 d z=0 for a simple closed path...

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational functions into a sum of simpler fractions. In the context of complex integration, this method simplifies the evaluation of integrals by reducing them to sums of terms whose integrals can be handled more straightforwardly, often via the residue theorem or Cauchy’s integral formula.

Residue Theorem

The residue theorem is a central result in complex analysis that links the contour integral of a function to the sum of residues at its poles. It states that if a function is analytic inside and on a closed contour except for a finite number of isolated singularities, then the integral around the contour is 2?i times the sum of the residues at those singularities.

Singularities and Poles

In complex analysis, singularities are points where a function ceases to be analytic. A pole is a type of isolated singularity where the function behaves like a finite-order negative power of (z - z0). Recognizing and classifying poles is crucial for evaluating complex integrals using methods like the residue theorem.

Contour Integration

Contour integration is a method in complex analysis where integrals of complex functions are evaluated along paths (contours) in the complex plane. This technique is useful for functions that are analytic except at isolated singularities, and many powerful theorems, such as Cauchy’s integral theorem and the residue theorem, stem from it.

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