Laplace Transforms in Theory and Practice A Computational Approach

Laplace transforms can seem difficult to understand, but this book makes them approachable. Hrachya Khachatryan introduces the subject in clear language alongside rigorous mathematics, supported by examples, illustrations and Wolfram Language visualizations. Readers move from the forward and inverse transforms of basic functions through advanced topics such as the complex inversion formula and Cauchy's residue theorem. Applications highlight how Laplace transforms simplify ordinary differential equations, partial differential equations, fractional systems and control problems—linking theory to practice across mathematics and engineering.

Readers can check their understanding with end-of-chapter exercises, detailed solutions and a sample final exam. Created with students and instructors in mind, the book also connects directly to the free online course from Wolfram U, Introduction to Laplace Transforms, which features videos that reinforce selected topics. Together, the text and course provide a thorough learning path that blends foundational concepts with worked examples and computational tools, making one of applied mathematics' central techniques both accessible and practical.

Information and Media Inquiries

Fall 2025 Publication
Publicity and Interviews: publishing@wolfram.com
Non-Fiction

Contents

• Introduction
• What Is a Laplace Transform?
• Formal Properties: Part 1
• Formal Properties: Part 2
• Special Functions
• Elementary Functions
• Piecewise and Generalized Functions
• Periodic Functions
• Numerical Evaluation
• Definition of the Inverse Laplace Transform
• Formal Properties: Part 1
• Rational Functions
• Formal Properties: Part 2
• Other Functions
• Complex Inversion
• Series and Asymptotic Expansions
• Multivariate Transforms
• Numerical Inverse Laplace Transform
• ODEs with Constant Coefficients
• ODEs: Applications in Physics and Engineering
• ODEs with Polynomial Coefficients
• Partial Differential Equations
• Integral Equations
• Fractional Differential Equations
• Evaluation of Integrals and Sums
• Control Systems
• Laplace Transforms in a Nutshell
• Sample Exam
• Table of Laplace Transforms
• References

About the Author

Hrachya Khachatryan is an Analog Circuit Design Engineer at Ferroelectric Memory GmbH. He previously worked as a Kernel Developer for Laplace and related integral transforms at Wolfram. He holds a BSc in physics from Yerevan State University, an MSc in physics from École normale supérieure and a PhD in theoretical particle physics from the International School for Advanced Studies (SISSA).