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→Elementary factors: Better with displaystyle |
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For {{math|{{abs|''z''}} < 1}} and <math>n>0</math>, one may express it as
<math display="inline">\displaystyle E_n(z)=\exp\left(-\tfrac{z^{n+1}}{n+1}\sum_{k=0}^\infty\tfrac{z^k}{1+k/(n+1)}\right)</math> and one can read off how those properties are enforced.
The utility of the elementary factors <math display="inline">E_n(z)</math> lies in the following lemma:<ref name="rudin"/>
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The case given by the fundamental theorem of algebra is incorporated here. If the sequence <math>\{a_n\}</math> is finite then we can take <math>p_n = 0</math>, <math>m=0</math> and <math>e^{g(z)}=c</math> to obtain <math>\, f(z) = c\,{\displaystyle\prod}_n (z-a_n)</math>.
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The trigonometric functions [[sine]] and [[cosine]] have the factorizations
<math display=block>\sin \pi z = \pi z \prod_{n\neq 0} \left(1-\frac{z}{n}\right)e^{z/n} = \pi z\prod_{n=1}^\infty \left(1-\left(\frac{z}{n}\right)^2\right)</math>
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while the [[gamma function]] <math>\Gamma</math> has factorization
<math display=block>\frac{1}{\Gamma (z)}=e^{\gamma z}z\prod_{n=1}^{\infty }\left ( 1+\frac{z}{n} \right )e^{-z/n},</math>
where <math>\gamma</math> is the [[Euler–Mascheroni constant]].{{citation needed|date=April 2019}} The cosine identity can be seen as special case of
<math display=block>\frac{1}{\Gamma(s-z)\Gamma(s+z)} = \frac{1}{\Gamma(s)^2}\prod_{n=0}^\infty \left( 1 - \left(\frac{z}{n+s} \right)^2 \right) </math>
for <math>s=\tfrac{1}{2}</math>.{{citation needed|date=April 2019}}
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