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D. R. Wilton ECE Dept. ECE 6382 Power Series Representations 8/24/10
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Geometric Series Consider 1 1
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Geometric Series, cont’d Consider
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Geometric Series, cont’d Consider Factor out the largest term!
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Uniform Convergence Consider
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Uniform Convergence, cont’d Consider
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Uniform Convergence, cont’d Consider N1N1 N8N8 N6N6 N4N4 N2N2 1 1 Key Point: Term-by-term integration of a series is allowed over any region where it is uniformly convergent.
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Taylor Series Expansion of an Analytic Function
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Taylor Series Expansion of an Analytic Function, Cont’d
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The Laurent Series Expansion Consider This generalizes the concept of a Taylor series, to include cases where the function is analytic in an annulus. z0z0 a b or Converges for z Key point: The point z 0 about which the expansion is made is arbitrary, but determines the region of convergence of the Laurent or Taylor series. zaza zbzb
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The Laurent Series Expansion, cont’d Consider Examples: z0z0 a b This is particularly useful for functions that have poles. z But the expansion point z 0 does not have to be at a singularity, nor must the singularity be a simple pole: y x branch cut pole zbzb zaza
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The Laurent Series Expansion, cont’d Consider z0z0 a b Theorem: The Laurent series expansion in the annulus region is unique. (So it doesn’t matter how we get it; once we obtain it by valid steps, it must be correct.) Hence Example:
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The Laurent Series Expansion, cont’d Consider We next develop a general method for constructing the coefficients of the Laurent series. z0z0 a b C Note: If f ( z ) is analytic at z 0, the integrand is analytic for negative values of n. Hence, all coefficients for negative n become zero (by Cauchy’s theorem). Final result: (This is the same formula as for the Taylor series, but with negative n allowed.)
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Consider The Laurent Series Expansion, cont’d Pond, island, & bridge
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Consider Contributions from the paths c 1 and c 2 cancel! The Laurent Series Expansion, cont’d Pond, island, & bridge
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The Laurent Series Expansion, cont’d Consider
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Examples of Taylor and Laurent Series Expansions Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Examples of Taylor and Laurent Series Expansions,cont’d Consider
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Summary of Methods for Generating Taylor and Laurent Series Expansions Consider
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