1796 CHAPTER 57. INFINITE PRODUCTS
while the residue at n is1
α2−n2 .
Therefore
0 = limN→∞
IN = limN→∞
2πi
[N
∑n=−N
1α2−n2 −
π cotπα
α
]which establishes the following formula of Mittag Leffler.
limN→∞
N
∑n=−N
1α2−n2 =
π cotπα
α.
Writing this in a slightly nicer form, you obtain 57.2.15.This is a very interesting formula. This will be used to factor sin(πz) . The zeros of
this function are at the integers. Therefore, considering 57.2.13 you can pick pn = 1 in theWeierstrass factorization formula. Therefore, by Corollary 57.2.2 there exists an analyticfunction g(z) such that
sin(πz) = zeg(z)∞
∏n=1
(1− z
zn
)ez/zn (57.2.16)
where the zn are the nonzero integers. Remember you can permute the factors in theseproducts. Therefore, this can be written more conveniently as
sin(πz) = zeg(z)∞
∏n=1
(1−( z
n
)2)
and it is necessary to find g(z) . Differentiating both sides of 57.2.16
π cos(πz) = eg(z)∞
∏n=1
(1−( z
n
)2)+ zg′ (z)eg(z)
∞
∏n=1
(1−( z
n
)2)
+zeg(z)∞
∑n=1−(
2zn2
)∏k ̸=n
(1−( z
k
)2)
Now divide both sides by sin(πz) to obtain
π cot(πz) =1z+g′ (z)−
∞
∑n=1
2z/n2
(1− z2/n2)
=1z+g′ (z)+
∞
∑n=1
2zz2−n2 .
By 57.2.15, this yields g′ (z) = 0 for z not an integer and so g(z) = c, a constant. So far thisyields
sin(πz) = zec∞
∏n=1
(1−( z
n
)2)