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''of a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606401.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606402.png" /> of the extended complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606403.png" />-plane''
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''of a meromorphic function  $  w = f ( z) $
 +
at a point $  a $
 +
of the extended complex $  z $-
 +
plane''
  
 
The residue
 
The residue
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606404.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm res} _ {a} \
 +
 
 +
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
 
 +
$$
 +
 
 +
of the logarithmic derivative  $  f ^ { \prime } ( z) / f ( z) $
 +
at the point  $  a $.
 +
Representing the function  $  \mathop{\rm ln}  f ( z) $
 +
in a neighbourhood  $  V ( a) $
 +
of a point  $  a \neq \infty $
 +
in the form  $  \mathop{\rm ln}  f ( z) = A  \mathop{\rm ln} ( z - a ) + \phi ( z) $,
 +
where  $  \phi ( z) $
 +
is a regular function in  $  V ( a) $,
 +
one obtains
 +
 
 +
$$
 +
\mathop{\rm res} _ {a} \
 +
 
 +
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
  = A .
 +
$$
 +
 
 +
The corresponding formulas for the case  $  a = \infty $
 +
have the form
 +
 
 +
$$
 +
\mathop{\rm ln}  f ( z)  = A  \mathop{\rm ln}
 +
\left (
 +
\frac{1}{z}
 +
\right ) + \phi ( z) ,
 +
$$
 +
 
 +
$$
 +
\mathop{\rm res} _  \infty 
 +
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
  = A .
 +
$$
 +
 
 +
If  $  a $
 +
is a zero or a pole of  $  f ( z) $
 +
of multiplicity  $  m $,
 +
then the logarithmic residue of  $  f ( z) $
 +
at  $  a $
 +
is equal to  $  m $
 +
or  $  - m $,
 +
respectively; at all other points the logarithmic residue is zero.
 +
 
 +
If  $  f ( z) $
 +
is a meromorphic function in a domain  $  D $
 +
and  $  \Gamma $
 +
is a rectifiable Jordan curve situated in  $  D $
 +
and not passing through the zeros or poles of  $  f ( z) $,
 +
then the logarithmic residue of  $  f ( z) $
 +
with respect to the contour  $  \Gamma $
 +
is the integral
 +
 
 +
$$ \tag{1 }
 +
 
 +
\frac{1}{2 \pi i }
  
of the logarithmic derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606405.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606406.png" />. Representing the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606407.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606408.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l0606409.png" /> in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064011.png" /> is a regular function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064012.png" />, one obtains
+
\int\limits _  \Gamma
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064013.png" /></td> </tr></table>
+
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
\
 +
d z  = N - P ,
 +
$$
  
The corresponding formulas for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064014.png" /> have the form
+
where  $  N $
 +
is the number of zeros and  $  P $
 +
is the number of poles of  $  f ( z) $
 +
inside  $  \Gamma $(
 +
taking account of multiplicity). The geometrical meaning of (1) is that as  $  \Gamma $
 +
is traversed in the positive sense, the vector  $  w = f ( z) $
 +
performs  $  N - P $
 +
rotations about the origin  $  w = 0 $
 +
of the  $  w $-
 +
plane (see [[Argument, principle of the|Argument, principle of the]]). In particular, if  $  f ( z) $
 +
is regular in  $  D $,
 +
that is,  $  P = 0 $,
 +
then from (1) one obtains a formula for the calculation of the index of the point  $  w = 0 $
 +
with respect to the image  $  \Gamma  ^ {*} = f ( \Gamma ) $
 +
of  $  \Gamma $
 +
by means of the logarithmic residue:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064015.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
\mathop{\rm ind} _ {0}  \Gamma  ^ {*}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064016.png" /></td> </tr></table>
+
\frac{1}{2 \pi i }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064017.png" /> is a zero or a pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064018.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064019.png" />, then the logarithmic residue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064020.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064021.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064023.png" />, respectively; at all other points the logarithmic residue is zero.
+
\int\limits _  \Gamma
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064024.png" /> is a meromorphic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064026.png" /> is a rectifiable Jordan curve situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064027.png" /> and not passing through the zeros or poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064028.png" />, then the logarithmic residue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064029.png" /> with respect to the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064030.png" /> is the integral
+
\frac{f ^ { \prime } ( z) }{f ( z) }
 +
\
 +
d z .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
Formula (2) leads to a generalization of the concept of a logarithmic residue to regular functions of several complex variables in a domain  $  D $
 +
of the complex space  $  \mathbf C  ^ {n} $,
 +
$  n \geq  1 $.
 +
Let  $  w = f ( z) = ( f _ {1} \dots f _ {n} ) : D \rightarrow \mathbf C  ^ {n} $
 +
be a [[Holomorphic mapping|holomorphic mapping]] such that the [[Jacobian|Jacobian]]  $  J _ {f} ( z) \not\equiv 0 $
 +
and the set of zeros  $  E = f ^ { - 1 } ( 0) $
 +
is isolated in  $  D $.
 +
Then for any domain  $  G \subset  \overline{G}\; \subset  D $
 +
bounded by a simple closed surface  $  \Gamma $
 +
not passing through the zeros of  $  f $
 +
one has a formula for the index of the point  $  w = 0 $
 +
with respect to the image  $  \Gamma  ^ {*} = f ( \Gamma ) $:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064032.png" /> is the number of zeros and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064033.png" /> is the number of poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064034.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064035.png" /> (taking account of multiplicity). The geometrical meaning of (1) is that as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064036.png" /> is traversed in the positive sense, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064037.png" /> performs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064038.png" /> rotations about the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064039.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064040.png" />-plane (see [[Argument, principle of the|Argument, principle of the]]). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064041.png" /> is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064042.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064043.png" />, then from (1) one obtains a formula for the calculation of the index of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064044.png" /> with respect to the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064046.png" /> by means of the logarithmic residue:
+
$$ \tag{3 }
 +
\mathop{\rm ind} _ {0}  \Gamma  ^ {*}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{1}{( 2 \pi i ) ^ {n} }
  
Formula (2) leads to a generalization of the concept of a logarithmic residue to regular functions of several complex variables in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064048.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064050.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064051.png" /> be a [[Holomorphic mapping|holomorphic mapping]] such that the [[Jacobian|Jacobian]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064052.png" /> and the set of zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064053.png" /> is isolated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064054.png" />. Then for any domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064055.png" /> bounded by a simple closed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064056.png" /> not passing through the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064057.png" /> one has a formula for the index of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064058.png" /> with respect to the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064059.png" />:
+
\int\limits _ {\Gamma _  \epsilon  }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{d f _ {1} \wedge \dots \wedge d f _ {n} }{f _ {1} \dots f _ {n} }
 +
  = N ,
 +
$$
  
where the integration is carried out with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064061.png" />-dimensional frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064062.png" /> with sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064063.png" />. The integral in (3) also expresses the sum of the multiplicities of the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064064.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064065.png" /> (see [[#References|[2]]]).
+
where the integration is carried out with respect to the $  n $-
 +
dimensional frame $  \Gamma _  \epsilon  = \{ {z \in G } : {| f _  \nu  ( z) | = \epsilon,  \nu = 1 \dots n } \} $
 +
with sufficiently small $  \epsilon > 0 $.  
 +
The integral in (3) also expresses the sum of the multiplicities of the zeros of $  f $
 +
in $  G $(
 +
see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The index of the origin with respect to a curve in the complex plane (also called the winding number of the curve, cf. [[Winding number|Winding number]]) is the number of times that the curve encircles the origin. More precisely, it is the change in the argument of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064066.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064067.png" /> traverses the curve (cf. [[#References|[a1]]], [[#References|[a3]]]). In higher dimensions, the index of a point with respect to a closed surface may be defined as the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064068.png" /> such that the surface is homologous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060640/l06064069.png" /> times the boundary of a ball centred at the point (cf. [[#References|[a2]]], [[#References|[a4]]]).
+
The index of the origin with respect to a curve in the complex plane (also called the winding number of the curve, cf. [[Winding number|Winding number]]) is the number of times that the curve encircles the origin. More precisely, it is the change in the argument of $  \mathop{\rm ln}  w $
 +
as $  w $
 +
traverses the curve (cf. [[#References|[a1]]], [[#References|[a3]]]). In higher dimensions, the index of a point with respect to a closed surface may be defined as the number $  N $
 +
such that the surface is homologous to $  N $
 +
times the boundary of a ball centred at the point (cf. [[#References|[a2]]], [[#References|[a4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Burchel,  "An introduction to classical complex analysis" , '''1''' , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.A. Aizenberg,  A.P. Yuzhakov,  "Integral representations and residues in multidimensional complex analysis" , ''Transl. Math. Monogr.'' , '''58''' , Amer. Math. Soc.  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1979)  pp. 241</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.W. Milnor,  "Toplogy from the differentiable viewpoint" , Univ. Virginia Press  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Burchel,  "An introduction to classical complex analysis" , '''1''' , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.A. Aizenberg,  A.P. Yuzhakov,  "Integral representations and residues in multidimensional complex analysis" , ''Transl. Math. Monogr.'' , '''58''' , Amer. Math. Soc.  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1979)  pp. 241</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.W. Milnor,  "Toplogy from the differentiable viewpoint" , Univ. Virginia Press  (1969)</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


of a meromorphic function $ w = f ( z) $ at a point $ a $ of the extended complex $ z $- plane

The residue

$$ \mathop{\rm res} _ {a} \ \frac{f ^ { \prime } ( z) }{f ( z) } $$

of the logarithmic derivative $ f ^ { \prime } ( z) / f ( z) $ at the point $ a $. Representing the function $ \mathop{\rm ln} f ( z) $ in a neighbourhood $ V ( a) $ of a point $ a \neq \infty $ in the form $ \mathop{\rm ln} f ( z) = A \mathop{\rm ln} ( z - a ) + \phi ( z) $, where $ \phi ( z) $ is a regular function in $ V ( a) $, one obtains

$$ \mathop{\rm res} _ {a} \ \frac{f ^ { \prime } ( z) }{f ( z) } = A . $$

The corresponding formulas for the case $ a = \infty $ have the form

$$ \mathop{\rm ln} f ( z) = A \mathop{\rm ln} \left ( \frac{1}{z} \right ) + \phi ( z) , $$

$$ \mathop{\rm res} _ \infty \frac{f ^ { \prime } ( z) }{f ( z) } = A . $$

If $ a $ is a zero or a pole of $ f ( z) $ of multiplicity $ m $, then the logarithmic residue of $ f ( z) $ at $ a $ is equal to $ m $ or $ - m $, respectively; at all other points the logarithmic residue is zero.

If $ f ( z) $ is a meromorphic function in a domain $ D $ and $ \Gamma $ is a rectifiable Jordan curve situated in $ D $ and not passing through the zeros or poles of $ f ( z) $, then the logarithmic residue of $ f ( z) $ with respect to the contour $ \Gamma $ is the integral

$$ \tag{1 } \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{f ^ { \prime } ( z) }{f ( z) } \ d z = N - P , $$

where $ N $ is the number of zeros and $ P $ is the number of poles of $ f ( z) $ inside $ \Gamma $( taking account of multiplicity). The geometrical meaning of (1) is that as $ \Gamma $ is traversed in the positive sense, the vector $ w = f ( z) $ performs $ N - P $ rotations about the origin $ w = 0 $ of the $ w $- plane (see Argument, principle of the). In particular, if $ f ( z) $ is regular in $ D $, that is, $ P = 0 $, then from (1) one obtains a formula for the calculation of the index of the point $ w = 0 $ with respect to the image $ \Gamma ^ {*} = f ( \Gamma ) $ of $ \Gamma $ by means of the logarithmic residue:

$$ \tag{2 } \mathop{\rm ind} _ {0} \Gamma ^ {*} = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{f ^ { \prime } ( z) }{f ( z) } \ d z . $$

Formula (2) leads to a generalization of the concept of a logarithmic residue to regular functions of several complex variables in a domain $ D $ of the complex space $ \mathbf C ^ {n} $, $ n \geq 1 $. Let $ w = f ( z) = ( f _ {1} \dots f _ {n} ) : D \rightarrow \mathbf C ^ {n} $ be a holomorphic mapping such that the Jacobian $ J _ {f} ( z) \not\equiv 0 $ and the set of zeros $ E = f ^ { - 1 } ( 0) $ is isolated in $ D $. Then for any domain $ G \subset \overline{G}\; \subset D $ bounded by a simple closed surface $ \Gamma $ not passing through the zeros of $ f $ one has a formula for the index of the point $ w = 0 $ with respect to the image $ \Gamma ^ {*} = f ( \Gamma ) $:

$$ \tag{3 } \mathop{\rm ind} _ {0} \Gamma ^ {*} = \ \frac{1}{( 2 \pi i ) ^ {n} } \int\limits _ {\Gamma _ \epsilon } \frac{d f _ {1} \wedge \dots \wedge d f _ {n} }{f _ {1} \dots f _ {n} } = N , $$

where the integration is carried out with respect to the $ n $- dimensional frame $ \Gamma _ \epsilon = \{ {z \in G } : {| f _ \nu ( z) | = \epsilon, \nu = 1 \dots n } \} $ with sufficiently small $ \epsilon > 0 $. The integral in (3) also expresses the sum of the multiplicities of the zeros of $ f $ in $ G $( see [2]).

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)

Comments

The index of the origin with respect to a curve in the complex plane (also called the winding number of the curve, cf. Winding number) is the number of times that the curve encircles the origin. More precisely, it is the change in the argument of $ \mathop{\rm ln} w $ as $ w $ traverses the curve (cf. [a1], [a3]). In higher dimensions, the index of a point with respect to a closed surface may be defined as the number $ N $ such that the surface is homologous to $ N $ times the boundary of a ball centred at the point (cf. [a2], [a4]).

References

[a1] R.B. Burchel, "An introduction to classical complex analysis" , 1 , Acad. Press (1979)
[a2] L.A. Aizenberg, A.P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis" , Transl. Math. Monogr. , 58 , Amer. Math. Soc. (1983) (Translated from Russian)
[a3] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
[a4] J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1969)
How to Cite This Entry:
Logarithmic residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_residue&oldid=47703
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article