GREEN’S FUNCTIONS WITH APPLICATIONS Second Edition

1 Chapter 1 Historical DevelopmentOne of the fundamental problems of field theory1is the constructionof solutions to linear differential equations when there is a specified sourceand the differential equation must satisfy certain boundary conditions. Thepurpose of this book is to show how Green s FUNCTIONS provide a powerfulmethod for obtaining these solutions. In this chapter, we present a historicaloverview of their MR. GREEN S ESSAYIn 1828 George Green(1793 1841) published anEssay on the Applicationof Mathematical Analysis to the Theory of Electricity and Magnetism.

2 In thisseminal work of mathematical physics, Green sought to determine the electricpotential within a vacuum bounded by conductors with specified today s notation we would say that he examined the solutions of 2u= fwithin a volumeVthat satisfy certain boundary conditions along theory in which the basic quantities are fields, such as electromagnetic 2015 by Taylor & Francis Group, LLCC opyrighted Material Taylor & Francis2 Green s FUNCTIONS with ApplicationsTo solve this problem, Green first considered a problemwhere the sourceis a point charge.

3 In modern notation, he sought to solve the partial differen-tial equation: 2g(r|r0)= 4 (r r0),( )where (r r0) is the Dirac delta function. We now know that the solutionto Equation isg=1/R,whereR2=(x )2+(y )2+(z ) the singular nature ofg,heproceededasfollows:First Green proved the theorem that bears his name: V( 2 2 )dV= S ( ) ndS,( )where the outwardly pointing normal is denoted bynand and are scalarfunctions that possess bounded derivatives. Then, by introducing a small ballabout the singularity atr0(because Equation )andthen excluding it from the volumeV, he obtained Vg 2udV+ S g u ndS= Vu 2gdV+ S u g ndS 4 u(r0)( )because the surface integral over the small ball is 4 u(r0)astheradiusofthe ball tends to zero.

4 Next, Green required thatgsatisfies the homogeneousboundary conditiong= 0 along the surfaceS. Since 2u= fand 2g=0withinV(recall that the pointr0is excluded fromV), he found thatu(r)=14 S u g ndS,( )whenf=0(Laplace sequation) thevalue ofuonS. This solved the boundary-value problem oncegwas knew thatghad to exist; it physically described the electrical potentialfrom a point charge located s essay remained relatively unknown until it was published2atthe urging of Kelvin between 1850 and 1854. Later Poincar e3summarizedour knowledge of Green s FUNCTIONS near the turn of the twentieth subsequent evolution of Green s FUNCTIONS can be divided into two parts:before and after the publication in 1946 ofMethods of Theoretical PhysicsbyP.

5 M. Morse and H. this paper-back version of classnotes that2 Green, G., 1850, 1852, 1854: An essay on the application of mathematical analysis tothe theories of electricity and Reine Angew. Math.,39,73 89;44,356 374;47,161 e, H., 1894: Sur les equations de la physique math Circ. ,8,57 , P. M., and H. Feshbach, 1946:Methods of Theoretical Technol-ogy Press, 497 pp. 2015 by Taylor & Francis Group, LLCC opyrighted Material Taylor & FrancisHistorical Development3they developed since the late 1930s to teach mathematical methods to physicsgraduate students, they laid out the four properties that a Green s functionmust possess.

6 Using the sturm - liouville problem given byddx[f(x)dydx]+p(x)y= q(x),( )these four properties are: The Green s function satisfies the homogeneous differential equation whenx&= ,thesourcepoint. The Green s function satisfies homogeneous boundary conditions. The Green s function is symmetric in the variablesx, . The Green s functiong(x| )satisfiestheconditiondgdx x= + dgdx x= = 1f( ).( )Prior to the publication of Morse and Feshbach s notes, authors used var-ious tricks to find Green s FUNCTIONS that satisfied these four properties.

7 Morseand Feshbach s great contribution was to show that the Green s function isthe point source solution [to a boundary-value problem] satisfying appropriateboundary conditions. Thus the Green s function could be found by simplysolving (in the case of sturm - liouville problem)ddx[f(x)dgdx]+p(x)g= (x )( )with homogeneous boundary conditions, where (x ) was the recently in-troduced delta function by Dirac. The advantage of this formulation wasthat the powerful techniques of eigenvalue expansions and transform methodscould be used in a straightforward manner to find Green s FUNCTIONS .

8 Theywill be the primary methods used in this the 1960 s many textbooks began to champion the use of Green sfunctions. For example, in Mackie s 1965 book5he sought to give a generalaccount of how certain mathematical techniques, notably those of Green sfunctions and of integral transforms, can be used to solve important and com-monly occurring boundary value problems in ordinary and partial differentialequations. In the following sectionswe turn to the development of Green sfunctions as they evolved within each general class of differential , A.

9 G., 1965:Boundary Value ,252pp. 2015 by Taylor & Francis Group, LLCC opyrighted Material Taylor & Francis4 Green s FUNCTIONS with POTENTIAL EQUATIONS hortly after the publication of Green s monograph on the European con-tinent, the German mathematician and pedagogue Carl Gottfried Neumann(1832 1925) developed the concept of Green s function as it applies to thetwo-dimensional (in contrast to three-dimensional) potential the two-dimensional Green s function, showed that it possesses theproperty of reciprocity, and found that it behaves as ln(r)asr.

10 Usingelliptic coordinates he rederived Poisson s integral formula and developed aneigenfunction expansion for the two-dimensional Green s function. In 1875 Paul Meutzner (1849 1914) extended Neumann s particular, he ob-tained the Green s function for the region within an ellipse (Ellipsenfl ache)and a circle (Ringfl ache). Finally, in his book on the logarithmic potential,A. Harnack8(1851 1888) gave the Green s function for a circle and of these authors used a technique that would become one of the fun-damental techniques in constructing a Green s function, namely eigenfunctionexpansions.