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9 - Complex variables

Published online by Cambridge University Press:  05 June 2012

Chiang C. Mei
Affiliation:
Massachusetts Institute of Technology
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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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Chapter
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Mathematical Analysis in Engineering
How to Use the Basic Tools
, pp. 210 - 259
Publisher: Cambridge University Press
Print publication year: 1995

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  • Complex variables
  • Chiang C. Mei, Massachusetts Institute of Technology
  • Book: Mathematical Analysis in Engineering
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511810749.010
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  • Complex variables
  • Chiang C. Mei, Massachusetts Institute of Technology
  • Book: Mathematical Analysis in Engineering
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511810749.010
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Complex variables
  • Chiang C. Mei, Massachusetts Institute of Technology
  • Book: Mathematical Analysis in Engineering
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511810749.010
Available formats
×