Corollary and Application of Argument Principle

Ni Ruan

《International Core Journal of Engineering》 8卷8期 (2022/08) Pp. 861-866

https://doi.org/10.6919/ICJE.202208_8(8).0117

摘要

From the point of view of the properties of argument principle, this paper uses the residue theorem and residue operation to get some important inferences of argument principle. First, C is a circumferential line, and function f(z)、φ(z)satisfies the condition that f(z) is meromorphic inside C. φ(z)resolves on the closed field I(C) and f(z) resolves on C without zeros. So, f(z) has different zeros and poles inside the perimeter of C that satisfy an integral equation. Second, let C be a circumferential line, ∀α∊R, and f(z)-α satisfy that it is meromorphic inside C. It resolves on C and has no zeros. So, there's an expression for a logarithmic residue of f(z)-α. By analyzing typical problems, this paper discusses the argument principle and corollary in the complex field, including the number of zeros and distribution of the polynomial (or rational function) in a given region.

關鍵字

Argument Principle ； Rouché Theorem ； Logarithmic Residues ； Meromorphic Functions

參考文獻

Y.Q. Zhong: Theory of Complex Functions (third edition). Beijing: Higher Education Press, 2003.

Z.J. Yan: Complex Function. Hefei: University of Science and Technology of China Press, 2002.

Z.C. Ou yang: The Argument Principle and its Application. Western science and technology of China, Vol. 10(2011) No. 27, p. 35-36.

L.L. Lu: Analysis on the application of the Argument Principle. Journal of Guangdong Institute of Education (Natural Science Edition), (1994), p. 28-32.