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. This shows that a function becomes infinite at the poles (left). On the right, the Gamma function does not have poles, it just increases quickly.]] Formally, suppose ''U'' is an ''n'' such that : for all ''z'' in ''U'' − {''a''}, then ''a'' is called a pole of ''f''. If ''n'' is chosen as small as possible, then ''n'' is called the '''order of the pole'''. A pole of order 1 is called a '''simple pole'''. Equivalently, ''a'' is a pole of order ''n''≥ 0 for a function ''f'' if there exists an open neighbourhood ''U'' of ''a'' such that ''f'' : ''U'' - {''a''} → C is holomorphic and the limit : exists and is different from 0. The point ''a'' is a pole of order ''n'' of ''f'' if and only if all the terms the Laurent Series expansion of ''f'' around ''a'' below degree −''n'' vanishes and the term in degree −''n'' is not zero. A pole of order 0 is a Removable Singularity . In this case the Limit lim''z''→''a'' ''f''(''z'') exists as a complex number. If the order is bigger than 0, then lim''z''→''a'' ''f''(''z'') = ∞. If the first derivative of a function ''f'' has a simple pole at ''a'', then ''a'' is a Branch Point of ''f''. (The converse need not be true). A non-removable singularity that is not a pole or a Branch Point is called an Essential Singularity . A holomorphic function whose only singularities are poles is called Meromorphic . SEE ALSO |
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