Publication: Numerical Algorithms Based on the Theory of Complex Variable
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Numerical Algorithms Based on the Theory of Complex Variable

- Part of a collection -
 

Author(s)
J. N. Lyness

Published in
Proceedings of the 1967 22nd National Conference

Year
1967

Publisher
Association for Computing Machinery

Abstract
Since its introduction in the early part of the nineteenth century, the theory of complex variables has played a steadily increasing role in mathematics, and in scientific research. In some fields complex algebra is used to simplify the description of a physical system. The use of a complex impedance Z in network theory is an example of this. In other fields complex algebra seems to be a basic ingredient of the physical laws. In Wave Mechanics for example a probability density P(x,t) is related to the square modulus of a wave function ψ(x,t) which is itself complex, being obtained from a wave equation whose coefficients may be complex.In mathematical research itself, it is rare to find a topic which is naturally restricted to real variables, and in many topics the extension to complex variables results in a simpler theory. For example a polynomial of degree n has exactly n zeros in the field of complex numbers.

AD Theory and Techniques
Complex Step Differentiation

BibTeX
@INPROCEEDINGS{
         Lyness1967NAB,
       author = "Lyness, J. N.",
       title = "Numerical Algorithms Based on the Theory of Complex Variable",
       year = "1967",
       isbn = "9781450374941",
       publisher = "Association for Computing Machinery",
       address = "New York, NY, USA",
       url = "https://doi.org/10.1145/800196.805983",
       doi = "10.1145/800196.805983",
       abstract = "Since its introduction in the early part of the nineteenth century, the theory of
         complex variables has played a steadily increasing role in mathematics, and in scientific research.
         In some fields complex algebra is used to simplify the description of a physical system. The use of
         a complex impedance Z in network theory is an example of this. In other fields complex algebra seems
         to be a basic ingredient of the physical laws. In Wave Mechanics for example a probability density
         P(x,t) is related to the square modulus of a wave function ψ(x,t) which is itself complex,
         being obtained from a wave equation whose coefficients may be complex.In mathematical research
         itself, it is rare to find a topic which is naturally restricted to real variables, and in many
         topics the extension to complex variables results in a simpler theory. For example a polynomial of
         degree n has exactly n zeros in the field of complex numbers.",
       booktitle = "Proceedings of the 1967 22nd National Conference",
       pages = "125--133",
       numpages = "9",
       location = "Washington, D.C., USA",
       series = "ACM '67",
       ad_theotech = "Complex Step Differentiation"
}


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