57.2. FACTORING A GIVEN ANALYTIC FUNCTION 1795

Corollary 57.2.2 Let f be analytic on C, f has a zero of order m at 0, and let the otherzeros of f be {zk} , listed according to order. (Thus if z is a zero of order l, it will be listedl times in the list, {zk} .) Also let

∞

∑n=1

(r|zn|

)1+pn

< ∞ (57.2.13)

for any choice of r > 0. Then there exists an entire function, g such that

f (z) = zmeg(z)∞

∏n=1

Epn

(zzn

). (57.2.14)

Proof: Since f has a zero of order m at 0, it follows from Theorem 51.5.3 that {zk} can-not have a limit point in C and so you can apply Theorem 57.2.1 to the function, f (z)/zm

which has a removable singularity at 0. This proves the corollary.

57.2.1 Factoring Some Special Analytic FunctionsFactoring a polynomial is in general a hard task. It is true it is easy to prove the factorsexist but finding them is another matter. Corollary 57.2.2 gives the existence of factors of acertain form but it does not tell how to find them. This should not be surprising. You can’texpect things to get easier when you go from polynomials to analytic functions. Neverthe-less, it is possible to factor some popular analytic functions. These factorizations are basedon the following Mitag-Leffler expansions. By an auspicious choice of the contour and themethod of residues it is possible to obtain a very interesting formula for cotπz .

Example 57.2.3 Let γN be the contour which goes from −N− 12 −Ni horizontally to N +

12 −Ni and from there, vertically to N + 1

2 +Ni and then horizontally to −N− 12 +Ni and

finally vertically to −N− 12 −Ni. Thus the contour is a large rectangle and the direction of

integration is in the counter clockwise direction. Consider the integral

IN ≡∫

γN

π cosπzsinπz(α2− z2)

dz

where α ∈ R is not an integer. This will be used to verify the formula of Mittag-Leffler,

1α+

∞

∑n=1

2α

α2−n2 = π cotπα. (57.2.15)

First you show that cotπz is bounded on this contour. This is easy using the formulafor cot(z) = eiz+e−iz

eiz−e−iz . Therefore, IN → 0 as N→ ∞ because the integrand is of order 1/N2

while the diameter of γN is of order N. Next you compute the residues of the integrand at±α and at n where |n| < N + 1

2 for n an integer. These are the only singularities of theintegrand in this contour and therefore, using the residue theorem, you can evaluate IN byusing these. You can calculate these residues and find that the residue at ±α is

−π cosπα

2α sinπα