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Zeros and poles
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<p>In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a> (a branch of mathematics), a pole is a certain type of <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singularity</a> of a <a href="/facts/Complex-valued_function/hgCoQlH0">complex-valued function</a> of a <a href="/facts/Complex_number/2w7mqI7e">complex</a> variable. It is the simplest type of non-<a href="/facts/Removable_singularity/6MPFDNha">removable singularity</a> of such a function (see <a href="/facts/Essential_singularity/KMb1R292">essential singularity</a>). Technically, a point <i>z</i>0 is a pole of a function f if it is a <a href="/facts/Zero_of_a_function/mlK3dhoT">zero</a> of the function 1/<i>f</i> and 1/<i>f</i> is <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> (i.e. <a href="/facts/Complex_differentiable/kVzthtEf">complex differentiable</a>) in some <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of <i>z</i>0. </p><p>A function f is <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic</a> in an <a href="/facts/Open_set/QfTCIqMu">open set</a> U if for every point z of U there is a neighborhood of z in which at least one of f and 1/<i>f</i> is holomorphic. </p><p>If f is meromorphic in U, then a zero of f is a pole of 1/<i>f</i>, and a pole of f is a zero of 1/<i>f</i>. This induces a duality between <i>zeros</i> and <i>poles</i>, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> plus the <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a>, then the sum of the <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicities</a> of its poles equals the sum of the multiplicities of its zeros. </p>