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

2. Pole Expansion of Meromorphic Functions Note that a pole at the origin is not allowed! 1 Historical 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.aspwww.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. Actually, it isn’t necessary that the paths C N 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. Summation of Series x y 123 … … -3-20 C

17. Summation of Series, cont’d