## Fourier Series and Integral TransformsThis volume provides the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications. |

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### Contents

Notation and Terminology | 1 |

2 Calculus Notation | 2 |

3 Useful Trigonometric Formulae | 4 |

Background Inner Product Spaces | 5 |

2 The Norm | 10 |

3 Orthogonal and Orthonormal Systems | 15 |

4 Orthogonal Projections and Approximation in the Mean | 19 |

5 Infinite Orthonormal Systems | 24 |

The Fourier Transform | 93 |

2 Examples | 98 |

3 Properties and Formulae | 102 |

4 The Inverse Fourier Transform and Plancherels Identity | 108 |

6 Applications of the Residue Theorem | 119 |

7 Applications to Partial Differential Equations | 125 |

8 Applications to Signal Processing | 130 |

The Laplace Transform | 140 |

Fourier Series | 32 |

2 Evenness Oddness and Additional Examples | 40 |

3 Complex Fourier Series | 42 |

4 Pointwise Convergence and Dirichlets Theorem | 46 |

5 Uniform Convergence | 56 |

6 Parsevals Identity | 63 |

7 The Gibbs Phenomenon | 68 |

8 Sine and Cosine Series | 72 |

9 Differentiation and Integration of Fourier Series | 76 |

10 Fourier Series on Other Intervals | 81 |

11 Applications to Partial Differential Equations | 85 |

2 More Formulae and Examples | 143 |

3 Applications to Ordinary Differential Equations | 149 |

4 The Heaviside and DiracDelta Functions | 155 |

5 Convolution | 162 |

6 More Examples and Applications | 168 |

7 The Inverse Transform Formula | 173 |

8 Applications of the Inverse Transform | 175 |

The Residue Theorem and Related Results | 182 |

Leibnizs Rule and Fubinis Theorem | 186 |

Index | 188 |

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### Common terms and phrases

absolutely integrable Assume band-limited Bessel's inequality calculate the Laplace called Cauchy-Schwarz inequality complex Fourier series complex number consider constant converges uniformly convolution cosine series cosnx define the function definition denote the Fourier Determine the Fourier Dirichlet's Theorem Example exists fact finite number follows Fourier coefficients Fubini's Theorem function g heat equation infinite orthonormal system inner product space inverse Fourier transform inverse Laplace transform J-oo Leibniz's rule Let f linear combination linear space method n=l denote n=l n=l natural number number of points obtain odd function Parseval's identity partial sums piecewise continuous function Plancherel's identity points of discontinuity pointwise convergence polynomial properties prove real number real-valued Residue Theorem right-hand side satisfying the conditions scalars series converges series of g Shannon Sampling Theorem sinnx Sm(x solution solve subinterval subspace substitution Theorem 3.5 Theorem Theorem uniform convergence

### References to this book

Handbook of Mathematics for Engineers and Scientists Andrei D. Polyanin,Alexander V. Manzhirov No preview available - 2006 |

Numerische Verbrennungssimulation: Effiziente numerische Simulation ... Peter Gerlinger No preview available - 2005 |