This particular textual content is an advent to harmonic research at the easiest symmetric areas, specifically Euclidean area, the sector, and the Poincaré top part plane. This publication is meant for starting graduate scholars in arithmetic or researchers in physics or engineering. Written with a casual variety, the booklet locations an emphasis on motivation, concrete examples, background, and, notably, purposes in arithmetic, records, physics, and engineering.

Many corrections and updates were integrated during this re-creation. Updates contain discussions of P. Sarnak and others' paintings on quantum chaos, the paintings of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's query "Can you pay attention the form of a drum?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, ultimately, the author's comparisons of constant idea with the finite analogues.

Topics featured in the course of the textual content comprise inversion formulation for Fourier transforms, critical restrict theorems, Poisson's summation formulation and functions in crystallography and quantity conception, functions of round harmonic research to the hydrogen atom, the Radon rework, non-Euclidean geometry at the Poincaré top part aircraft H or unit disc and functions to microwave engineering, basic domain names in H for discrete teams Γ, tessellations of H from such discrete workforce activities, automorphic kinds, and the Selberg hint formulation and its functions in spectral thought in addition to quantity theory.