Art of Special Functions

ART OF SPECIAL FUNCTIONS

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A complete, proof-driven handbook for mastering the functions that shape advanced calculus, analysis, and applied mathematics.

Art of Special Functions is a rigorous, problem-centered guide to the special functions that appear throughout mathematics, physics, and engineering.
The book focuses not only on formulas, but on structure, derivation, and problem-solving control—showing where these functions come from, why they work, and how to use them effectively.

Rather than presenting special functions as isolated objects, the text builds them systematically from first principles, highlighting unifying ideas such as integral representations, differential equations, transforms, and asymptotic behavior. Every major result is carefully derived, and techniques are reinforced through extensive worked examples and exercises.

📘 What this book contains

• A complete treatment of classical and advanced special functions
• 350+ fully solved problems, explained step by step
• 500+ practice exercises, ranging from routine to challenging
• Rigorous proofs and derivations of fundamental identities and theorems
• Clear connections between theory, computation, and applications
• A structured, consistent approach designed for both study and reference

The book spans 1150 pages and is designed to be self-contained.

✍️ Topics you’ll master

• Gamma, Beta, Digamma, and Polygamma functions
• Special integrals and classical transcendental functions
• Riemann zeta, Hurwitz zeta, polylogarithms, and related sums
• Hypergeometric functions and orthogonal polynomials
• Bessel functions, Legendre functions, and spherical harmonics
• Airy functions, turning-point theory, and asymptotics
• Elliptic integrals, elliptic functions, and theta functions
• Integral transforms: Fourier, Laplace, Mellin, Hankel, Hilbert, Abel
• Applications to differential equations, wave phenomena, probability, and signal processing

🎯 Who this book is for

• Students who struggle with advanced integrals and want deeper understanding
• Undergraduates and graduate students in mathematics, physics, or engineering
• Contest solvers and self-learners aiming to strengthen analytical intuition
• Anyone who wants a serious, unified reference on special functions

This book is written to build understanding, not memorization.

📎 Format

• PDF (high-resolution, LaTeX typeset)
• Instant download
• Lifetime access

🔎 Why this book is different

Most books on special functions list formulas.

This book explains structure.

You learn how special functions arise, how they connect to one another, and how to actually use them in calculations, proofs, and applications. If you’ve ever felt that special functions were a collection of disconnected facts, this book is designed to change that.