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

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,

Proof of Mittag-Leffler Theorem

Extended Form of the Mittag-Leffler Theorem

Example: Pole Expansion of cot z

Example: Pole Expansion of cot z (cont.)

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

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

Example: Pole Expansion of cot z (cont.)

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

Infinite Product Expansion of Entire Functions

Product Expansion Formula

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

The Argument Principle

The Argument Principle (cont.)

Summation of Series x y 123 … … C

Summation of Series, cont’d