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''argument principle''
 
''argument principle''
  
A geometric principle in the theory of functions of a complex variable. It is formulated as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132501.png" /> be a bounded domain in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132502.png" />, and let, moreover, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132503.png" /> be a continuous curve, oriented so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132504.png" /> lies on the left. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132505.png" /> is meromorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132506.png" /> and has no zeros or poles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132507.png" />, then the difference between the number of its zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132508.png" /> and the number of its poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a0132509.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325010.png" /> (counted according to their multiplicity) is equal to the increase of the argument of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325011.png" /> when travelling once around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325012.png" />, divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325013.png" />, i.e.
+
A geometric principle in the theory of functions of a complex variable. It is formulated as follows: Let $  D $
 +
be a bounded domain in the complex plane $  \mathbf C $,  
 +
and let, moreover, the boundary $  \partial  D $
 +
be a continuous curve, oriented so that $  D $
 +
lies on the left. If a function $  w = f (z) $
 +
is meromorphic in a neighbourhood of $  \overline{D}\; $
 +
and has no zeros or poles on $  \partial  D $,  
 +
then the difference between the number of its zeros $  N $
 +
and the number of its poles $  P $
 +
inside $  D $(
 +
counted according to their multiplicity) is equal to the increase of the argument of $  f $
 +
when travelling once around $  \partial  D $,  
 +
divided by $  2 \pi $,  
 +
i.e.
  
<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/a/a013/a013250/a01325014.png" /></td> </tr></table>
+
$$
 +
N - =
 +
\frac{1}{2 \pi }
 +
\Delta _ {\partial  D }  \mathop{\rm arg}  f ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325015.png" /> denotes any continuous branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325016.png" /> on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325017.png" />. The expression on the right-hand side equals the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325018.png" /> of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325019.png" /> with respect to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325020.png" />.
+
where $  \mathop{\rm arg}  f $
 +
denotes any continuous branch of $  \mathop{\rm Arg}  f $
 +
on the curve $  \partial  D $.  
 +
The expression on the right-hand side equals the index $  \mathop{\rm ind} _ {0}  f ( \partial  D ) $
 +
of the curve $  f ( \partial  D ) $
 +
with respect to the point $  w = 0 $.
  
 
The principle of the argument is used in the proofs of various statements on the zeros of holomorphic functions (such as the fundamental theorem of algebra on polynomials, the theorem of Hurwitz on zeros, etc.). From the principle of the argument follow many other important geometric principles of function theory, e.g. the principle of invariance of domain (cf. [[Invariance, principle of|Invariance, principle of]]), the [[Maximum-modulus principle|maximum-modulus principle]] and the theorem on the local inverse of a holomorphic function. In many questions the principle of the argument is used implicitly, in the form of its corollary: the [[Rouché theorem|Rouché theorem]].
 
The principle of the argument is used in the proofs of various statements on the zeros of holomorphic functions (such as the fundamental theorem of algebra on polynomials, the theorem of Hurwitz on zeros, etc.). From the principle of the argument follow many other important geometric principles of function theory, e.g. the principle of invariance of domain (cf. [[Invariance, principle of|Invariance, principle of]]), the [[Maximum-modulus principle|maximum-modulus principle]] and the theorem on the local inverse of a holomorphic function. In many questions the principle of the argument is used implicitly, in the form of its corollary: the [[Rouché theorem|Rouché theorem]].
  
There are generalizations of the principle of the argument. The condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325021.png" /> be meromorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325022.png" /> may be replaced by the following: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325023.png" /> has only a finite number of poles and zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325024.png" /> and extends continuously to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325025.png" />. Instead of the complex plane, an arbitrary Riemann surface may be considered: the boundedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325026.png" /> is then replaced by the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325027.png" /> be compact. From the principle of the argument for a compact Riemann surface it follows that the number of zeros of an arbitrary meromorphic function, not identically equal to zero, is equal to the number of poles. The principle of the argument for domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325028.png" /> is equivalent to the theorem on the sum of the logarithmic residues (cf. [[Logarithmic residue|Logarithmic residue]]). For this reason, the following statement is sometimes called the generalized principle of the argument. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325029.png" /> is meromorphic in a neighbourhood of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325030.png" /> which is bounded by a finite number of continuous curves and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325031.png" /> has no zeros or poles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325032.png" />, then for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325033.png" /> which is holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325034.png" /> the equality
+
There are generalizations of the principle of the argument. The condition that $  f $
 +
be meromorphic in a neighbourhood of $  \overline{D}\; $
 +
may be replaced by the following: $  f $
 +
has only a finite number of poles and zeros in $  D $
 +
and extends continuously to $  \partial  D $.  
 +
Instead of the complex plane, an arbitrary Riemann surface may be considered: the boundedness of $  D $
 +
is then replaced by the condition that $  \overline{D}\; $
 +
be compact. From the principle of the argument for a compact Riemann surface it follows that the number of zeros of an arbitrary meromorphic function, not identically equal to zero, is equal to the number of poles. The principle of the argument for domains in $  \mathbf C $
 +
is equivalent to the theorem on the sum of the logarithmic residues (cf. [[Logarithmic residue|Logarithmic residue]]). For this reason, the following statement is sometimes called the generalized principle of the argument. If $  f $
 +
is meromorphic in a neighbourhood of a domain $  \overline{D}\; $
 +
which is bounded by a finite number of continuous curves and if $  f $
 +
has no zeros or poles on $  \partial  D $,  
 +
then for any function $  \phi $
 +
which is holomorphic in a neighbourhood of $  \overline{D}\; $
 +
the equality
 +
 
 +
$$
 +
 
 +
\frac{1}{2 \pi i }
 +
\int\limits _ {\partial  D } \phi (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/a/a013/a013250/a01325035.png" /></td> </tr></table>
+
\frac{f ^ { \prime } (z) }{f (z) }
 +
  dz  = \sum _ { k=1 } ^ { N }  \phi ( a _ {k} ) - \sum _ { k=1 } ^ { P }  \phi ( b _ {k} )
 +
$$
  
holds, where the first sum extends over all zeros and the second sum extends over all poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325037.png" />. There is also a topological generalized principle of the argument: The principle of the argument is valid for any open mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325038.png" /> that is locally finite-to-one and extends continuously to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325039.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325040.png" />.
+
holds, where the first sum extends over all zeros and the second sum extends over all poles of $  f $
 +
in $  D $.  
 +
There is also a topological generalized principle of the argument: The principle of the argument is valid for any open mapping $  f: D \rightarrow \overline{\mathbf C}\; $
 +
that is locally finite-to-one and extends continuously to $  \partial  D $,  
 +
while 0 , \infty \notin f ( \partial  D ) $.
  
An analogue of the principle of the argument for functions of several complex variables is, for example, the following theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325041.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325042.png" /> with Jordan boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325043.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325044.png" /> be a holomorphic mapping of a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325046.png" />; then the number of pre-images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325047.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325048.png" /> (counted according to multiplicity) is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013250/a01325049.png" />.
+
An analogue of the principle of the argument for functions of several complex variables is, for example, the following theorem: Let $  D $
 +
be a bounded domain in $  \mathbf C  ^ {n} $
 +
with Jordan boundary $  \partial  D $
 +
and let $  f: \overline{D}\; \rightarrow \mathbf C  ^ {n} $
 +
be a holomorphic mapping of a neighbourhood of $  \overline{D}\; $
 +
such that 0 \notin f ( \partial  D ) $;  
 +
then the number of pre-images of 0 $
 +
in $  D $(
 +
counted according to multiplicity) is equal to $  \mathop{\rm ind} _ {0}  f ( \partial  D ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. [M.A. Lavrent'ev] Lawrentjew,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. [M.A. Lavrent'ev] Lawrentjew,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>

Latest revision as of 18:48, 5 April 2020


argument principle

A geometric principle in the theory of functions of a complex variable. It is formulated as follows: Let $ D $ be a bounded domain in the complex plane $ \mathbf C $, and let, moreover, the boundary $ \partial D $ be a continuous curve, oriented so that $ D $ lies on the left. If a function $ w = f (z) $ is meromorphic in a neighbourhood of $ \overline{D}\; $ and has no zeros or poles on $ \partial D $, then the difference between the number of its zeros $ N $ and the number of its poles $ P $ inside $ D $( counted according to their multiplicity) is equal to the increase of the argument of $ f $ when travelling once around $ \partial D $, divided by $ 2 \pi $, i.e.

$$ N - P = \frac{1}{2 \pi } \Delta _ {\partial D } \mathop{\rm arg} f , $$

where $ \mathop{\rm arg} f $ denotes any continuous branch of $ \mathop{\rm Arg} f $ on the curve $ \partial D $. The expression on the right-hand side equals the index $ \mathop{\rm ind} _ {0} f ( \partial D ) $ of the curve $ f ( \partial D ) $ with respect to the point $ w = 0 $.

The principle of the argument is used in the proofs of various statements on the zeros of holomorphic functions (such as the fundamental theorem of algebra on polynomials, the theorem of Hurwitz on zeros, etc.). From the principle of the argument follow many other important geometric principles of function theory, e.g. the principle of invariance of domain (cf. Invariance, principle of), the maximum-modulus principle and the theorem on the local inverse of a holomorphic function. In many questions the principle of the argument is used implicitly, in the form of its corollary: the Rouché theorem.

There are generalizations of the principle of the argument. The condition that $ f $ be meromorphic in a neighbourhood of $ \overline{D}\; $ may be replaced by the following: $ f $ has only a finite number of poles and zeros in $ D $ and extends continuously to $ \partial D $. Instead of the complex plane, an arbitrary Riemann surface may be considered: the boundedness of $ D $ is then replaced by the condition that $ \overline{D}\; $ be compact. From the principle of the argument for a compact Riemann surface it follows that the number of zeros of an arbitrary meromorphic function, not identically equal to zero, is equal to the number of poles. The principle of the argument for domains in $ \mathbf C $ is equivalent to the theorem on the sum of the logarithmic residues (cf. Logarithmic residue). For this reason, the following statement is sometimes called the generalized principle of the argument. If $ f $ is meromorphic in a neighbourhood of a domain $ \overline{D}\; $ which is bounded by a finite number of continuous curves and if $ f $ has no zeros or poles on $ \partial D $, then for any function $ \phi $ which is holomorphic in a neighbourhood of $ \overline{D}\; $ the equality

$$ \frac{1}{2 \pi i } \int\limits _ {\partial D } \phi (z) \frac{f ^ { \prime } (z) }{f (z) } dz = \sum _ { k=1 } ^ { N } \phi ( a _ {k} ) - \sum _ { k=1 } ^ { P } \phi ( b _ {k} ) $$

holds, where the first sum extends over all zeros and the second sum extends over all poles of $ f $ in $ D $. There is also a topological generalized principle of the argument: The principle of the argument is valid for any open mapping $ f: D \rightarrow \overline{\mathbf C}\; $ that is locally finite-to-one and extends continuously to $ \partial D $, while $ 0 , \infty \notin f ( \partial D ) $.

An analogue of the principle of the argument for functions of several complex variables is, for example, the following theorem: Let $ D $ be a bounded domain in $ \mathbf C ^ {n} $ with Jordan boundary $ \partial D $ and let $ f: \overline{D}\; \rightarrow \mathbf C ^ {n} $ be a holomorphic mapping of a neighbourhood of $ \overline{D}\; $ such that $ 0 \notin f ( \partial D ) $; then the number of pre-images of $ 0 $ in $ D $( counted according to multiplicity) is equal to $ \mathop{\rm ind} _ {0} f ( \partial D ) $.

References

[1] M.A. [M.A. Lavrent'ev] Lawrentjew, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
How to Cite This Entry:
Argument, principle of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Argument,_principle_of_the&oldid=15915
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article