Complex Analysis and Differential Equations

✍ Scribed by Barreira, Luis;Valls, Claudia

• Publisher: Springer London
• Year: 2012
• Tongue: English
• Leaves: 417
• Series: Springer Undergraduate Mathematics Series
• Category: Library

✦ Synopsis

This rigorous yet accessible introduction to complex analysis and differential equations covers complex numbers, holomorphic functions, analytic functions, ordinary differential equations, Fourier series and applications to partial differential equations.

✦ Table of Contents

Complex Analysis and Differential Equations......Page 3
Preface......Page 5
Contents......Page 6
Part I: Complex Analysis......Page 8
1.1 Complex Numbers......Page 9
1.2 Polar Form......Page 14
1.3 Conjugate......Page 17
1.4 Complex Functions......Page 20
1.5 Solved Problems and Exercises......Page 27
2.1 Limits and Continuity......Page 42
2.2 Differentiability......Page 44
2.3 Differentiability Condition......Page 53
2.4 Paths and Integrals......Page 57
2.5 Primitives......Page 67
2.6 Index of a Closed Path......Page 74
2.7 Cauchy's Integral Formula......Page 78
2.8 Integrals and Homotopy of Paths......Page 79
2.9 Harmonic Conjugate Functions......Page 83
2.10 Solved Problems and Exercises......Page 87
3.1 Sequences......Page 114
3.2 Series of Complex Numbers......Page 116
3.3 Series of Real Numbers......Page 120
3.4 Uniform Convergence......Page 127
3.5 Solved Problems and Exercises......Page 133
4.1 Power Series......Page 154
4.2 Zeros......Page 167
4.3 Laurent Series and Singularities......Page 169
4.4 Residues......Page 179
4.5 Meromorphic Functions......Page 182
4.6 Solved Problems and Exercises......Page 188
Part II: Differential Equations......Page 226
5.1 Basic Notions......Page 227
5.2 Existence and Uniqueness of Solutions......Page 230
5.3 Linear Equations: Scalar Case......Page 237
5.4 Linear Equations: General Case......Page 240
5.5 Computing Exponentials of Matrices......Page 248
5.6 Solved Problems and Exercises......Page 254
6.1 Exact Equations......Page 284
6.2 Equations Reducible to Exact......Page 288
6.3 Scalar Equations of Order Greater than 1......Page 290
6.4 Laplace Transform......Page 299
6.5 Solved Problems and Exercises......Page 314
7.1 An Example......Page 336
7.2 Fourier Series......Page 341
7.3 Uniqueness and Orthogonality......Page 350
7.4 Even and Odd Functions......Page 356
7.5 Series of Cosines and Series of Sines......Page 358
7.6 Integration and Differentiation Term by Term......Page 362
7.7 Solved Problems and Exercises......Page 365
8.1 Heat Equation and Its Modifications......Page 376
8.2 Laplace Equation......Page 386
8.3 Wave Equation......Page 391
8.4 Solved Problems and Exercises......Page 395
Index......Page 415