Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2016 | OriginalPaper | Chapter

5. Functions of a Complex Variable

Author : Jon H. Davis

Published in: Methods of Applied Mathematics with a Software Overview

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In earlier chapters, complex-valued functions appeared in connection with Fourier series expansions. In this context, while the function assumes complex values, the argument of the function is real-valued. There is a highly developed theory of (complex-valued) functions of a complex-valued argument. This theory contains some remarkably powerful results which are applicable to a variety of problems.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Sturm–Liouville Theory and Boundary Value Problems
next chapter Laplace Transforms
Footnotes
1
The term “domain” has a technical meaning. See page 66.
 
2
The consequences of assuming that a function is analytic are surprisingly large. The real variable case gives little clue how “well behaved” analytic functions are.
 
3
\(\overline{z}\) is the complex conjugate of z = x + iy, defined by \(\overline{z} = x - iy\).
 
4
Recall that a domain is defined as an open set, in which any two points may be joined by a polygonal line lying in the set. A closed domain is a domain together with its boundary.
 
5
The requirement is that the domain be simply connected: every simple closed curve in the domain must have its interior in the region.
 
6
To verify, simply differentiate termwise, and evaluate at the point z = a.
 
7
The poles may not arise due to cancellation from a zero of appropriate order in f.
 
8
A consequence of the definition of (u, v) as the velocity field is that the x-acceleration of a particle at (x, y) is \(\frac{\partial u} {\partial t} + u\frac{\partial u} {\partial x} + v\frac{\partial u} {\partial y}\) rather than \(\frac{\partial u} {\partial t}\).
 
9
Here the flow field is assumed to have continuous partial derivatives.
 
10
The introduction of “solid bodies” defined by streamlines into the flow can also result in flow regions which are not simply connected. In this case, it may occur that the line integrals defining (locally) the velocity potential and stream function may fail to give a globally defined single-valued potential. This, however, causes no problem since the potentials are ambiguous only to the extent of a line integral around the obstacle boundary, and the physically relevant quantities are given by the gradient of the potentials.
 
11
This potential is singular at the origin. Again, the flow region does not include the singular point, so no problem arises.
 
12
A property of analytic functions is that it is impossible to have an infinite number of zeroes within a bounded domain, without having the function vanish identically. It is therefore not necessary to explicitly assume a finite number of zeroes. This is a consequence of the other assumptions imposed.
 
13
A Jordan curve divides the plane into an inside and an outside.
 
14
Completely parallel applications can be made in electrostatics and steady state heat transfer. What matters is that the governing partial differential equation is the two-dimensional Laplace equation.
 
15
The function \(\sqrt{z^{2 } - c^{2}}\) can be defined with a branch cut from z = −c to z = +c, or using a two-sheeted Riemann surface by pasting two such cut planes together. The + sign corresponds to the “upper” sheet.
 
Literature
1.
L.M. Milne-Thompson, Theoretical Hydrodynamics, 4th edn. (MacMillan, London, 1962)
L.V. Alfohrs, Complex Analysis, 2nd edn. (McGraw-Hill, New York, 1966)
R.V. Churchill, Complex Variables and Applications, 2nd edn. (McGraw-Hill, New York, 1960)
E.T. Copson, Theory of Functions of a Complex Variable (Oxford University Press, Oxford, 1962)
W. Fulks, Advanced Calculus (Wiley, New York, 1962)
J.F. Marsden, Basic Complex Analysis (W.H. Freeman and Company, San Francisco, 1975)
W. Miller, Lie Theory and Special Functions (Academic, New York, 1968)
Z. Nehari, Introduction to Complex Analysis (Allyn and Bacon, Boston, 1961)
E.C. Titchmarsh, Eigenfunction Expansions Associated With Second Order Differential Equations (Oxford University Press, London, 1946)
C.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1944)
E.T. Whittaker, C.N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University Press, Cambridge, 1927)
Metadata
Title
Functions of a Complex Variable
Author
Jon H. Davis
Copyright Year
2016
Book
Methods of Applied Mathematics with a Software Overview

Print ISBN: 978-3-319-43369-1

Electronic ISBN: 978-3-319-43370-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-43370-7_5

Premium Partner

    Image Credits