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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 | |