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D. R. Wilton ECE Dept.

ECE 6382. Pole and Product Expansions, Series Summation. D. R. Wilton ECE Dept. 8/24/10. Pole Expansion of Meromorphic Functions. Note that a pole at the origin is not allowed!.

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D. R. Wilton ECE Dept.

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  1. ECE 6382 Pole and Product Expansions, Series Summation D. R. Wilton ECE Dept. 8/24/10

  2. Pole Expansion of Meromorphic Functions Note that a pole at the origin is not allowed! 1Historical note: It is often claimed that friction between Mittag-Leffler and Alfred Nobel resulted in there being no Nobel Prize in mathematics. However, it seems this is not likely the case; see, for example, www.snopes.com/science/nobel.asp

  3. Proof of Mittag-Leffler Theorem

  4. Extended Form of the Mittag-Leffler Theorem

  5. Example: Pole Expansion of cot z

  6. Example: Pole Expansion of cot z (cont.)

  7. Example: Pole Expansion of cot z (cont.) • Actually, it isn’t necessary that the paths CN be circular; indeed it is simpler in this case to estimate the maximum value on a sequence of square paths of increasing size that pass between the poles

  8. Example: Pole Expansion of cot z (cont.) coth (x) ―

  9. Example: Pole Expansion of cot z (cont.)

  10. Other Pole Expansions • The Mittag-Leffler theorem generalizes the partial fraction representation of a rational function to meromorphic functions

  11. Infinite Product Expansion of Entire Functions

  12. Product Expansion Formula

  13. Useful Product Expansions • Product expansions generalize for entire functions the factorization of the numerator and denominator polynomials of a rational function into products of their roots

  14. The Argument Principle

  15. The Argument Principle (cont.)

  16. y C x … … -3 -2 -1 0 1 2 3 Summation of Series

  17. Summation of Series, cont’d

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