An Introduction to Laplace Transforms and Fourier Series (Springer Undergraduate Mathematics Series)
In this book, there is a strong emphasis on application with the necessary mathematical grounding. There are plenty of worked examples with all solutions provided. This enlarged new edition includes generalised Fourier series and a completely new chapter on wavelets.
Only knowledge of elementary trigonometry and calculus are required as prerequisites. An Introduction to Laplace Transforms and Fourier Series will be useful for second and third year undergraduate students in engineering, physics or mathematics, as well as for graduates in any discipline such as financial mathematics, econometrics and biological modelling requiring techniques for solving initial value problems.
Why Read This Book
You will learn practical, worked methods for solving initial-value problems and representing periodic signals using Laplace transforms and Fourier series, with an emphasis on applications rather than abstruse theory. The book's many worked examples (with full solutions) and a new chapter on wavelets make it a compact, hands-on bridge from calculus to the transform techniques used in engineering signal analysis.
Who Will Benefit
Second- or third-year undergraduates in engineering, physics or mathematics, and graduates in applied fields (e.g., econometrics, biology, financial mathematics) who need transform methods to solve ODEs and analyse signals.
Level: Intermediate — Prerequisites: Elementary trigonometry and single-variable calculus (differentiation and integration); basic complex numbers are helpful but not essential.
Key Takeaways
- Solve linear initial-value problems using Laplace transforms and inverse-transform techniques.
- Construct and analyse Fourier series (including generalized Fourier series) to represent periodic signals and assess convergence issues.
- Apply transform methods to compute convolution, system responses, and basic spectral characteristics of signals.
- Use transform pairs and tables to perform routine transform manipulations and partial-fraction inversions.
- Understand the basics of wavelets and how they complement Fourier methods for time–frequency analysis.
Topics Covered
- Preface and overview
- Chapter 1: Mathematical preliminaries — complex numbers, integration and trigonometry review
- Chapter 2: The Laplace transform — definition, properties and basic transforms
- Chapter 3: Inversion of the Laplace transform — partial fractions and complex inversion ideas
- Chapter 4: Applications of Laplace transforms to ODEs and initial-value problems
- Chapter 5: Convolution, transfer functions and system response
- Chapter 6: Fourier series — periodic functions and orthogonality
- Chapter 7: Convergence, Gibbs phenomenon and generalized Fourier series
- Chapter 8: Fourier methods in boundary-value problems and signal representation
- Chapter 9: Worked examples and solutions
- Chapter 10: Wavelets — introduction and comparison with Fourier methods (new in this edition)
- Appendices: Transform tables, useful integrals and solution keys
Languages, Platforms & Tools
How It Compares
More example-driven and accessible than standard comprehensive texts like Oppenheim & Willsky's Signals and Systems, and more focused on Laplace/Fourier fundamentals than Bracewell's The Fourier Transform and Its Applications.
