Introduction to Special Functions: A Computational Approach

Special functions first emerged as classical tools for solving difficult equations in mathematics and physics. With the rise of modern computation, their role has expanded far beyond closed-form solutions, revealing rich structure, deep interconnections and remarkable practical power. This book traces that evolution by pairing theory with the computational capabilities of Wolfram Language, allowing readers to move seamlessly from definition to analysis, visualization and numerical exploration.

The text begins with the historical development of special functions and the foundational methods used to study them, including symbolic evaluation, analytic continuation and high-precision numerical computation. Carefully designed exercises encourage readers to experiment, compute and verify identities on their own, strengthening both intuition and theoretical understanding. As the chapters progress, the book explores the major families of special functions, including gamma and zeta functions, special integrals, Bessel and Airy functions, orthogonal polynomials and hypergeometric functions. Advanced topics follow, covering elliptic and Heun-related functions, multivariate generalizations such as Appell functions, and unifying superfunctions like the Meijer G- and Fox H-functions. While these topics are central to many areas of mathematics, physics and engineering, they are often underrepresented in traditional university curricula. This book is designed to fill that gap by integrating theory with practical computation and visualization.

This textbook concludes with both a summary "nutshell" chapter to be used as a study guide and a sample exam. Readers can also download the free, interactive ebook edition to engage with live Wolfram Language code. Additional materials, including videos and quizzes, can be found online in the companion Wolfram U course.

Information and Media Inquiries

Spring 2026 Publication
Publicity and Interviews: publishing@wolfram.com
Non-Fiction

Contents

• What Is a Special Function?
• Review of Mathematical Operations in Calculus
• Differential Equations as Data Structures
• The Euler Gamma Function
• The Beta Function
• The Polygamma Function
• The Error Function
• The Exponential Integral
• Fresnel Functions
• Trigonometric Integrals
• Legendre Polynomials
• Hermite Polynomials
• Laguerre Polynomials
• Chebyshev Polynomials
• Bessel Functions
• Kelvin, Hankel and Struve Functions
• Airy Functions
• The Generalized Hypergeometric Function
• Gauss Hypergeometric Function
• The Confluent Hypergeometric Function
• Jacobian Elliptic Functions
• Elliptic Integrals
• Weierstrass Elliptic Functions
• The Riemann Zeta Function
• Mathieu, Spheroidal and Lamé Functions
• Heun Functions
• Multivariate Hypergeometric Functions: Appell Functions
• Meijer's G-Function
• The Fox H-Function
• Special Functions in a Nutshell

About the Author

Tigran Ishkhanyan is an applied mathematician specializing in special functions and algorithm development, with recent work on AI-driven methods for symbolic mathematical computation. He earned his BSc and MSc in applied mathematics and physics from the Moscow Institute of Physics and Technology, a PhD in laser physics from the Institute for Physical Research and a doctorate in physics from Université Bourgogne Franche-Comté. He has spent eight years at Wolfram Research as a Kernel Developer and Special Functions Specialist in the Mathematical Computations group. He also serves as the head coach and lead trainer of Armenia's national team for the International Olympiad in Artificial Intelligence.