Argument, principle of the

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)