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EspañolPOLE (COMPLEX ANALYSIS)
In complex analysis, a 'pole' of a holomorphic function is a certain type of singularity that behaves like the singularity 1/''z''''n'' at ''z'' = 0. This means that, in particular, a pole of the function ''f''(''z'') is a point ''z'' = ''a'' such that ''f''(''z'') approaches infinity uniformly as ''z'' approaches ''a''.
Formally, suppose ''U'' is an open subset of the complex plane 'C', ''a'' is an element of ''U'' and ''f'' : ''U'' − {''a''} → 'C' is a holomorphic function. If there exists a holomorphic function ''g'' : ''U'' → 'C' and a nonnegative integer ''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'. A pole of order 0 is a removable singularity.
From above several equivalent characterizations can be deduced:
If ''n'' is the order of pole ''a'', then necessarily ''g''(''a'') ≠ 0 for the function ''g'' in the above expression. So we can put
:
for some ''h'' that is holomorphic in an open neighborhood of ''a'' and has a zero of order ''n'' at ''a''. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.
Also, by the holomorphy of ''g'', ''f'' can be expressed as:
This is a Laurent series with finite ''principal part''. The holomorphic function ∑''k''≥0''ak'' (''z - a'')''k'' (on ''U'') is called the ''regular part'' of ''f''. So 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.
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.
★ Zero (complex analysis)
★ Residue (complex analysis)
★ Electronic filter
★ Control theory#Stability
★
★ Module for Zeros and Poles by John H. Mathews
| Contents |
| Definition |
| Remarks |
| See also |
| External links |
Definition
Formally, suppose ''U'' is an open subset of the complex plane 'C', ''a'' is an element of ''U'' and ''f'' : ''U'' − {''a''} → 'C' is a holomorphic function. If there exists a holomorphic function ''g'' : ''U'' → 'C' and a nonnegative integer ''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'. A pole of order 0 is a removable singularity.
From above several equivalent characterizations can be deduced:
If ''n'' is the order of pole ''a'', then necessarily ''g''(''a'') ≠ 0 for the function ''g'' in the above expression. So we can put
:
for some ''h'' that is holomorphic in an open neighborhood of ''a'' and has a zero of order ''n'' at ''a''. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.
Also, by the holomorphy of ''g'', ''f'' can be expressed as:
This is a Laurent series with finite ''principal part''. The holomorphic function ∑''k''≥0''ak'' (''z - a'')''k'' (on ''U'') is called the ''regular part'' of ''f''. So 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.
Remarks
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
★ Zero (complex analysis)
★ Residue (complex analysis)
★ Electronic filter
★ Control theory#Stability
External links
★
★ Module for Zeros and Poles by John H. Mathews
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