A LaTeX reference document providing physics and mathematics applications for every section of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products (7th edition, 2007).
Gradshteyn-Ryzhik (G&R) is the standard reference table of integrals used across physics, engineering, and mathematics. It tells you what an integral evaluates to. This companion tells you why you might need it---where each integral, identity, or special function arises in practice.
For each topic group in G&R, the companion provides:
- Physics applications (2--4 per group): concrete problems from quantum mechanics, electrodynamics, statistical mechanics, fluid dynamics, optics, general relativity, and more.
- Mathematics applications (1--3 per group): connections to analysis, algebra, number theory, topology, and combinatorics.
- Index entries: over 5,200 hierarchical entries linking concepts to the G&R sections where the relevant formulas appear.
- Bibliography: 33 references to foundational papers and textbooks.
| G&R Section | Topic | Companion File |
|---|---|---|
| 0 | Introduction | sections/sec00_introduction.tex |
| 1 | Elementary Functions and Finite Sums | sections/sec01_elementary.tex |
| 2 | Indefinite Integrals of Elementary Functions | sections/sec02_indefinite_elementary.tex |
| 3--4 | Definite Integrals of Elementary Functions | sections/sec0304_definite_elementary.tex |
| 5 | Indefinite Integrals of Special Functions | sections/sec05_indefinite_special.tex |
| 6--7 | Definite Integrals of Special Functions | sections/sec0607_definite_special.tex |
| 8--9 | Special Functions | sections/sec0809_special_functions.tex |
| 10 | Vector Field Theory | sections/sec10_vector_field.tex |
| 11 | Algebraic Inequalities | sections/sec11_algebraic_ineq.tex |
| 12 | Integral Inequalities | sections/sec12_integral_ineq.tex |
| 13 | Matrices and Quadratic Forms | sections/sec13_matrices.tex |
| 14 | Determinants | sections/sec14_determinants.tex |
| 15 | Norms | sections/sec15_norms.tex |
| 16 | Ordinary Differential Equations | sections/sec16_odes.tex |
| 17 | Integral Transforms | sections/sec17_transforms.tex |
| 18 | Z-Transform | sections/sec18_ztransform.tex |
Requires a TeX Live (or equivalent) installation with pdflatex, biber, and makeindex.
pdflatex integrals_and_their_use.tex
biber integrals_and_their_use
makeindex integrals_and_their_use
pdflatex integrals_and_their_use.tex
pdflatex integrals_and_their_use.texThe final PDF is approximately 370 pages. Three passes of pdflatex are needed for cross-references, bibliography back-references, and index page numbers to settle.
A pre-built PDF (integrals_and_their_use.pdf) is included in the repository for convenience.
As a G&R supplement. When you look up an integral in G&R, check the corresponding section here to see where that integral appears in physics and mathematics. The index is the fastest entry point---search for a concept (e.g., "hydrogen atom", "Fourier transform", "Selberg integral") and follow the page references.
As a survey of applications. Each section can be read independently as a tour of how a particular class of integrals connects to research problems. For example, the section on definite integrals of exponentials (G&R 3.3--3.4) covers everything from Gaussian path integrals to the Sommerfeld expansion for degenerate electron gases.
As a teaching resource. The applications provide "why does this matter?" context for integral calculus, special functions, and mathematical methods courses.
integrals_and_their_use.tex Main document (inputs all sections)
references.bib Bibliography (33 entries)
sections/ Per-section LaTeX files (16 files)
This work is provided for educational and research purposes.
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., edited by A. Jeffrey and D. Zwillinger. Academic Press, 2007.
- NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/