Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Formulation of physical problems
- 2 Classification of equations with two independent variables
- 3 One-dimensional waves
- 4 Finite domains and separation of variables
- 5 Elements of Fourier series
- 6 Introduction to Green's functions
- 7 Unbounded domains and Fourier transforms
- 8 Bessel functions and circular boundaries
- 9 Complex variables
- 10 Laplace transform and initial value problems
- 11 Conformal mapping and hydrodynamics
- 12 Riemann–Hilbert problems in hydrodynamics and elasticity
- 13 Perturbation methods – the art of approximation
- 14 Computer algebra for perturbation analysis
- Appendices
- Bibliography
- Index
9 - Complex variables
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Formulation of physical problems
- 2 Classification of equations with two independent variables
- 3 One-dimensional waves
- 4 Finite domains and separation of variables
- 5 Elements of Fourier series
- 6 Introduction to Green's functions
- 7 Unbounded domains and Fourier transforms
- 8 Bessel functions and circular boundaries
- 9 Complex variables
- 10 Laplace transform and initial value problems
- 11 Conformal mapping and hydrodynamics
- 12 Riemann–Hilbert problems in hydrodynamics and elasticity
- 13 Perturbation methods – the art of approximation
- 14 Computer algebra for perturbation analysis
- Appendices
- Bibliography
- Index
Summary
Thus far we have only dealt with real variables; the use of complex representation for a sinusoidal function of time is just a matter of convenience involving only the real variable t and no new principles. To an analytical engineer the techniques of complex variables are essential because of their wide range of applications. Many two-dimensional potential theories in classical hydrodynamics, static electricity, steady diffusion, etc., can be directly solved by complex functions. The inverse Fourier and Laplace transforms are often most efficiently evaluated in a complex plane. In contrast to most methods of real variables where the mathematical details are tailored to suit the geometry of the boundaries, conformal mapping is a radically different tool whose effectiveness lies in altering the boundaries themselves.
In the following four chapters, we give a guided tour of the basic principles of complex functions, together with applications that range from the elementary to the slightly advanced. In the present chapter the basics of analytic functions and the rules of differential and integration are explained. In Chapter 10 these basics are applied to the techniques of Laplace transform. In Chapter 11 elements of conformal mapping are introduced with examples from hydrodynamics. One of the most beautiful applications of complex functions in continuum mechanics is the formulation and solution of certain mixed boundary-value problems. In Chapter 12 two examples from hydrodynamics and elasticity are examined and the Riemann–Hilbert technique is explained.
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- Mathematical Analysis in EngineeringHow to Use the Basic Tools, pp. 210 - 259Publisher: Cambridge University PressPrint publication year: 1995