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To any product of linear terms corresponds a pole expansion of its  logarithmic derivative.  So,  it's good to discuss both types of expansions together.

(2021-07-08)   Wallis product   (Wallis, 1655)
A result obtained by  John Wallis (1619-1703)  from  Wallis integrals:

p	=	2		2		4		4		6		6		8		8			=	Õ k ≥ 1		2k		2k	=	Õ k ≥ 1		1
2		1		3		3		5		5		7		7		9						2k-1		2k+1				1-1/4k2

Wallis integrals (Numericana)   |   Wallis product (1655)   |   John Wallis (1616-1703)
 
Wallis product formula (5:42)  by  Jens Fehlau  (Flammable Maths, 2019-05-28).
The Wallis product for pi, proved geometrically (25:26)  by  Grant Sanderson (3blue1brown, 2018-04-20).
Amazing formula for pi: the Wallis product (11:56)  by  Presh Talwalkar  (MindYourDecisions, 2016-10-12).

(2021-07-10)   Euler's product formula for sine.
Now best expressed with the  normalized sinc function  (sine cardinal).

sinc  z	=	sin p z	=	Õ k ≥ 1	( 1 - z2/k2 )
		p z

Note that the  Wallis product  is readily obtained from this,  with  z = ½.

The  unnormalized sine cardinal  is preferrably called  sampling function  and abbreviated   Sa x = (sin x)/x.  At zero,  the value of either flavor of sine cardinal is defined by continuity to be  1,  which makes it an  entire function.

Proof :  

Euler's product formula for sine  by  Robin Whitty (Theorem of the Day #247).
 
Infinite Factorization of sin(x)  (Mathematics StackExchange, 2013-02-31).
 
Euler's sine product from Mittag-Leffler's Pole Expansion of cotg (11:36)  by  Jens Fehlau  (2019-04-29).

(2021-07-18)   Factorization of the reciprocal gamma function.
The reciprocal gamma function is an entire  function.

As such,  it's often used as a  precursor  in the numerical computation of the  Gamma function.

g (z)	=	1	=   z  exp ( g z )	Õ k ≥ 1	( 1 + z / k ) exp ( -z/k )
		G (z)

g  is the  Euler-Mascheroni constant  (0.57721566490153286060651209...).

Reciprocal gamma function   |   Karl Weierstrass (1815-1897)

(2021-07-12)   Weierstrass factorization theorem   (1876)
Factorization of an entire  function.

Weierstrass factorization theorem   |   Karl Weierstrass (1815-1897)
 
Genus and Order of an Entire Function  by  Bob Gardner.

(2021-07-17)   Euler's Partial Fraction Expansion of Cotangent
The  cotangent  function may be abbreviated  ctg,  cotg,  cotan  or  cot.

p  cotg p z	=	1	+	å k ≥ 1	(	1	+	1	)
		z				z - k		z + k

That's the  logarithmic derivative  of   Euler's factorization  of  sin p x.

How did Euler prove the partial-fraction expaansion of cotangent  (Math Stack Exchange, 2016-07-05).
 
Euler's proof and Herglotz's Proof  by  Daniel Glasscock   |   Gustav Herglotz (1881-1953; PhD 1900).
 
Cotangent's Expansion Derivation using Fourier series (14:35)  by  Jens Fehlau  (2019-04-25).

(2021-07-17)   Mittag-Leffler's expansion theorem   (1876)
Expanding a  meromorphic function  about its poles.

Mittag-Leffler's theorem (1876)   |   Proof   |   Gösta Mittag-Leffler (1846-1927)
 
Pole expansion using Mittag-Leffler's theorem  (Math Stack Exchange, 2019-10-23).

(2021-07-18)   Pole expansions of some meromorphic functions

cotg z	=	cos z	=	1	+	å k ≥ 1	(	1	+	1	)
		sin z		z				z - k p		z + k p

Replace  z  by  p/2-z  to obtain:

tg z   =	sin z	=	1	+	å k ≥ 1	(	1	-	1	)
	cos z		p/2 - z				(k+½)p - z		(k-½)p + z

Let's pair the successive terms differently  without  affecting the sum:

Partial fraction decomposition
 
Mittag-Leffler's theorem (1876)   |   Gösta Mittag-Leffler (1846-1927)