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The complex plane | |
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The complex numbers and the complex plane | |
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A formal view of the complex numbers | |
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Some geometry | |
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Subsets of the plane | |
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Functions and limits | |
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The exponential, logarithm, and trigonometric functions | |
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Line integrals and Green's theorem | |
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Basic properties of analytic functions | |
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Analytic and harmonic functions; the Cauchy-Riemann equations | |
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Flows, fields, and analytic functions | |
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Power series | |
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Cauchy's theorem and Cauchy's formula | |
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The Cauchy-Goursat theorem | |
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Consequences of Cauchy's formula | |
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Isolated singularities | |
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The residue theorem and its application to the evaluation of definite integrals | |
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Analytic functions as mappings | |
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The zeros of an analytic function | |
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The stability of solutions of a system of linear differential equations | |
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Maximum modulus and mean value | |
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Linear fractional transformations | |
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Conformal mapping | |
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Conformal mapping and flows | |
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The Riemann mapping theorem and Schwarz-Christoffel transformations | |
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Analytic and harmonic functions in applications | |
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Harmonic functions | |
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Harmonic functions as solutions to physical problems | |
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Integral representations of harmonic functions | |
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Boundary-value problems | |
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Impulse functions and the Green's function of a domain | |
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Transform methods | |
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The Fourier transform: basic properties | |
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Formulas Relating u and u | |
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The Laplace transform | |
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Applications of the Laplace transform to differential equations | |
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The Z-Transform | |
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The stability of a discrete linear system | |
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The stability of a discrete linear system | |
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A Table of Conformal Mappings | |
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A Table of Laplace Transforms | |
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Solutions to Odd-Numbered Exercises | |
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Index | |