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RESIDUE THEOREM

RESIDUE THEOREM. Presented By Osman Toufiq Ist Year Ist SEM. RESIDUE THEOREM Calculus of residues. Suppose an analytic function f ( z ) has an isolated singularity at z 0 . Consider a contour integral enclosing z 0. z 0.

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RESIDUE THEOREM

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  1. RESIDUE THEOREM Presented By Osman Toufiq Ist Year Ist SEM

  2. RESIDUE THEOREM Calculus of residues Suppose an analytic function f (z) has an isolated singularity at z0. Consider a contour integral enclosing z0 . z0 The coefficient a-1=Res f (z0) in the Laurent expansion is called the residue of f (z) at z = z0. If the contour encloses multiple isolated singularities, we have the residue theorem: z0 z1 Contour integral =2pi ×Sum of the residuesat the enclosed singular points

  3. Residue formula: To find a residue, we need to do the Laurent expansion and pick up the coefficient a-1. However, in many cases we have a useful residue formula:

  4. Residue at infinity: Stereographic projection: Residue at infinity: Suppose f (z) has only isolated singularities, then its residue at infinity is defined as Another way to prove it is to use Cauchy’s integral theorem. The contour integral for a small loop in an analytic region is One other equivalent way to calculate the residue at infinity is By this definition a function may be analytic at infinity but still has a residue there.

  5. z+ z- C C r=1 r=1

  6. z+ z- C r=1

  7. z+ z- C z0 r=1

  8. Question: How about Answer: We can go the lower half of the complex plane using a clockwise contour.

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