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Weierstrass' ''elementary factors'' have these properties and serve the same purpose as the factors <math> (z-c_n) </math> above.
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Consider the functions of the form <math display="inline">\exp\left(-\tfrac{z^{n+1}}{n+1}\right)</math> for <math>n \in \mathbb{N}</math>. At <math>z=0</math>, they evaluate to <math>1</math> and have a flat slope at order up to <math>n</math>. Right after <math>z=1</math>, they sharply fall to some small positive value. In contrast, consider the function <math>1-z</math> which has no flat slope but, at <math>z=1</math>, evaluates to exactly zero. Also note that for {{math|{{abs|''z''}} < 1}},
:<math>(1-z) = \exp(\ln(1-z)) = \exp \left( -\tfrac{z^1}{1} - \tfrac{z^2}{2} - \tfrac{z^3}{3} + \cdots \right).</math>
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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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