Complex variable theory is attractive for engineers as it offers elegant approaches for certain types of differential equations in engineering including heat transfer, solid mechanics, and fluid mechanics. However, a gap exists between books written by mathematicians and books written by engineers in their specific fields. Naturally, mathematicians tend to emphasize rigorousness and consistency while less emphasizing applications. On the other hand, books written by engineers often jump directly to the specific topics assuming that the readers already have sufficient background of complex variables and the pathway from theory to the application is not clearly elucidated. This book closes the gap in the literature. providing a smooth transition from basic theory to the application is accomplished. Although it is not possible to cover all the topics in engineering exhaustively, the readers can at least find the logic of how and why complex variables are effective for some of the engineering problems. Another motivation for writing this book is to demonstrate that the readers can take advantage of a computer algebra system, Mathematica, to facilitate tedious algebra and visualize complex functions so that they can focus on principles instead of spending endless hours on algebra by hand. Unlike numerical tools such as MATLAB and FORTRAN, Mathematica can expand, differentiate, and integrate complex-valued functions symbolically. Mathematica can be used as a stand-alone symbolic calculator or a programming tool using the Wolfram Language. If Mathematica is not available locally, Wolfram Cloud Basic can be used online as a free service to execute Mathematica statements.
Author(s): Seiichi Nomura
Series: Synthesis Lectures on Mechanical Engineering
Publisher: Springer
Year: 2022
Language: English
Pages: 176
City: Cham
Preface
Contents
1 Functions of Complex Variables
1.1 Complex Numbers
1.2 Complex Plane
1.3 Complex-Valued Functions
1.3.1 Exponential Function, ez
1.3.2 Trigonometric Functions
1.3.3 Logarithmic Function, log z
1.3.4 Branch Cut and Branch Points
1.4 Numerics
1.5 Problems
2 Calculus of Functions of Complex Variables
2.1 Differentiability (Cauchy-Riemann Equations)
2.1.1 Cauchy-Riemann Equations
2.1.2 Alternative Form of Cauchy-Riemann Equations
2.1.3 Harmonic Functions
2.1.4 Uniqueness of Analytic Functions
2.2 Problems
3 Integrations of Functions of Complex Variables
3.1 Integral Calculus
3.2 Cauchy's Theorem
3.2.1 Morera's Theorem
3.3 Cauchy's Integral Formula
3.3.1 Contour Integral of zn
3.3.2 Cauchy's Integral Formula
3.3.3 Generalized Cauchy's Integral Formula
3.3.4 Liouville's Theorem
3.3.5 Fundamental Theorem of Algebra
3.4 Problems
4 Series of Complex Variable Functions
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4.1 Taylor Series
4.1.1 Taylor Series of f(z) About z=a
4.1.2 Analytic Continuation
4.1.3 Can We Prove 1 + 2 + 3 + 4 + @汥瑀瑯步渠= -112 ?
4.2 Laurent Series
4.3 Mathematica Code
4.4 Problems
5 Residues
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5.1 Types of Singularities
5.2 Residues
5.2.1 Definition of Residues
5.2.2 Calculation of Residues
5.3 Residue Theorem
5.3.1 Residue at Infinity
5.4 Application of Residue Theorem to Certain Integrals
5.4.1 First Type
5.4.2 Second Type
5.4.3 Third Type
5.4.4 Mathematica Code
5.5 Problems
6 Applications to Engineering Problems
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6.1 Conformal Mapping
6.1.1 Solving Laplace Equation by Conformal Mapping
6.1.2 Bilinear (Möbius) Transformation
6.2 General Solution to Laplace Equation (Δφ(x, y) = 0)
6.3 General Solution to Bi-harmonic Equation (ΔΔφ(x, y) = 0)
6.4 Heat Conduction
6.5 Solid Mechanics
6.6 Fluid Mechanics
6.7 Problems
A Introduction to Mathematica
A.1 Essential Commands/Statements
A.2 Complex Numbers
A.3 Equations
A.4 Differentiation/Integration
A.5 Matrices/Vectors/Tensors
A.6 Functions
A.7 Graphics
A.8 Other Useful Functions
A.8.1 Control Statements
Appendix References
Index
