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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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