TRIGONOMETRIC SERIES - Canadian Mathematical Congress Lecture Series - 2. Trigonometric Series - A Survey by R.L. Jeffery

Sep 29, 2026

A survey lecture by R.L. Jeffery for the Canadian Mathematical Congress, presented to Section III of the Royal Society of Canada in 1953, with an emphasis on the historical development and mathematical challenges behind trigonometric series. The lecture frames the study through applied origins in differential equations and then returns to early work from d’Alembert, Euler, and Daniel Bernoulli on the vibrating string. It highlights key questions about determining Fourier-type coefficients and the extent to which trigonometric series represent functions beyond a chosen interval.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages52

Reading Time2 h 11 min

AccessFree to download

Introduction

The period of d'Alembert, Euler and Daniel Bernoulli

TRIGONOMETRICSERIES # A Survey by R.L.JEFFERY GEORGE AVERY BUNTING LIBRARYWASHINGTON COLLEGE https://archive.org/details/trigonometricser0000unse # TRIGONOMETRIC SERIES CANADIAN MATHEMATICAL CONGRESS LECTURE SERIES 1.Introduction to the Theory of Distributions.By ISRAEL HALPERIN,based on the lectures given by LAURENT SCHWARTZ 2.Trigonometric Series.A Survey by R.L.JEFFERY # TRIGONOMETRICSERIES A Survey by R.L.JEFFERY Professor of Mathematics,Queen's University Presidential AddressRoyal Society of CanadaSection III,1953 UNIVERSITY OF TORONTO PRESSTORONTO,1956 Copyright OCanada,1956by University of Toronto PressPrinted in CanadaLondon:Geoffrey CumberlegeOxford University Press QA 404 可45 h①儿 TRIGONOMETRIC SERIES Mr.Chairman and Fellows of Section III of the Royal Society of Canada: It was with apprehension that I looked forward to the time when,asPresident of this Section,I would be called upon to give an address.Toprepare a mathematical talk for an audience that is predominantly non-mathematical is a heavy undertaking.Fortunately I have a side interestthat often intrudes itself into your work.The Astronomer,the Chemist,the Engineer and the Physicist are continually meeting with trigonometricseries as solutions of the differential equations that arise in the problemsthey are called upon to solve.In fact the origin of the study of trigonometricseries was in the search for solutions of such equations,and no topic arisingout of applied and experimental work has been so great a challenge to theprofessional mathematician or so great a stimulus to the advancement ofpure mathematics.The main purpose of my address is to show the extentto which this is true.I shall go back to the beginning,describe the mainproblems that have arisen,and give some indication of the manner in whichthey have been solved.The time it has taken to settle the various stageswill be emphasized as a measure of the difficulties encountered.To bringthe subject up to the present in a general way is the purpose of the firstpart of the address.In the second part the proofs of the main-line theoremswill be given. PART I Introduction.There are at least four separate discussions of theearlier phases of the subject(1,2,3,4).Consequently I shall take fromthis period only what I need to illustrate the points I wish to make. 1.The period of d'Alembert,Euler and Daniel Bernoulli.Mostof you are familiar with the story of the vibrating string.Let an elasticstring be stretched taut on the x-axis with ends at (1,0),(一1,0).If thisstring is displaced and released it vibrates in such a way that the ordinateof a point on the string is a function of the time t and the x-coordinateof the point,y=y(t,x). As early as 1747 the French mathematician d'Alembert knew that thisfunction satisfied the differential equation (1) This in itself is remarkable when we consider that both Newton and Leib-nitz who invented the Calculus were alive in the early 1700's.Between1747 and 1753,d'Alembert,Euler and Bernoulli gave their attention to thesolution of this equation and showed that it involved representing theinitial position of the string at the time of release, y(0,x)=f(x), by a trigonometric series of the form (2) This posed two questions: I.If y(0,x)could be represented by such a series how could the co-efficients ak,b be determined? II.It was clear that there was a considerable degree of arbitrariness inthe way the string could be constrained in its initial position.For exampley(0,x)could be in part a straight line,in part an arc of a circle,in part apiece of a sine curve.Was it reasonable to expect that the single expression (2)could represent a straight line on part of the interval(一l,1),a circleon another part of this interval,and a sine curve on still another part?To the mathematicians of that day this seemed absurd.In this point ofview Euler was the most emphatic.He took his stand on the ground thatthe function(2)was periodic

FAQs

What is the lecture’s main goal in addressing trigonometric series?

To show how trigonometric series connect to applied and experimental problems while also driving advances in pure mathematics.

How does the vibrating string example motivate the theory?

An elastic string’s displacement is modeled by a function y(t, x) satisfying a differential equation, leading to representing the initial position y(0, x) using a trigonometric series.

What two central questions arise about representing the initial position by a trigonometric series?

How to determine the coefficients a_k, b_k, and whether a single trigonometric expression can represent different shapes on different parts of an interval.

Sep 29, 2026

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