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. 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.

2 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. 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.

3 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. 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.

4 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. 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.

5 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 . 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. 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.

6 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. The investigator would first find an eigenfunction expansion thatsatisfied both the homogeneous differential equation and boundary geometry of the problem would determine the coordinate system that wasused. Then the Fourier coefficients would be chosen so that the Green s func-tion exhibited the proper behavior (such as 1/r) ,9 John Dougall (1867 1960) derived three-dimensional Green s functionsin cylindrical and spherical 1879 Alfred George Greenhill10(1847 1927) applied the method ofimages to construct the Green s function for a rectangular his results are expressed as an infinite summation of theta FUNCTIONS ,it was not very useful and has essentially been Munro Macdonald11(1865 1935) took a slightly different ap-proach in the 1890s.

7 As before, he began with the eigenfunction expansion6 Neumann, C., 1861: Ueber die Integration der partiellen Differentialgleichung: 2 x2+ 2 y2= Reine Angew. Math.,59,335 , P., 1875: Untersuchungen im Gebiete des logarithmischen Ann.,8,319 338. Foranalternativederivation,seeSections15 and17inNeu-mann, C., 1906: Uber das logarithmische Verh. K. Sachs. Ges. , Klasse,58,482 Chapter 2 in Harnack, A., 1887:Die Grundlagen der Theorie des logarithmischenPotentiales und der eindeutigen Potentialfunktion in der , B. G. Teubner,170 , J., 1900: The determination of Green s function by means of cylindrical orspherical Edinburgh Math. Soc., Ser. 1,18,33 , A. G.,1879: On Green s function for a rectangular Philos. Soc.,3,289 , H. M., 1895: The electrical distribution on a conductor bounded by twospherical surfaces cutting at any London Math. Soc., Ser. 1,26,156 172; 2015 by Taylor & Francis Group, LLCC opyrighted Material Taylor & FrancisHistorical Development5 Figure :CarlGottfriedNeumann(1832 1925)wasaleadingGermanmathematicianand teacher.

8 Today he is best known for his work on the Dirichlet principle and inte-gral equations, and his co-founding with Alfred Clebsch ofMathematische Universit atarchiv Leipzig; right photographc Photo Deutsches satisfied the boundary conditions. But now, the Fourier coefficients werechosen so that the expansion satisfied the general Poisson equation. Thenhe considered the special case of a point source. We illustrate his method inExample Because you must solve the general Poisson equation first, histechnique never became the late 1890 s Arnold Sommerfeld12(1868 1951) developed a tech-nique using integration on the complex plane to extend the method of imagesto several other useful geometries in three dimensions. Ernst William Hob-son (1856 1933) then used this method13to find the Green s function for acircular disk. Later, Ludwig Waldmann (1913 1980), a young assistant toSommerfeld, applied this technique in electrostatic calculations of an this technique is very complicated and we will presentMacdonald, H.

9 M., 1900: Demonstration of Green s formula for electric density near thevertex of a right Cambridge Philos. Soc.,18,292 , A., 1897: Uber verzweigte Potentiale im London , Ser. 1,28,395 , E. W., 1900: On Green s function for a circular disc, with APPLICATIONS toelectrostatic Cambridge Philos. Soc.,18,277 , L., 1937: Zwei Anwendungen der Sommerfeld schen Methode der ver-zweigten Z.,38,654 663. 2015 by Taylor & Francis Group, LLCC opyrighted Material Taylor & Francis6 Green s FUNCTIONS with Applicationsan improved version in Example the beginning of twentieth century the method ofbilinear expansionswas developed:g(x, y, z| , , )= n=1 n(x, y, z) n( , , ) n,( )where nand n(x, y, z)arethenth eigenvalue and eigenfunction, respec-tively. Adolf Kneser15(1862 1930) showed that the Green s function was thesymmetric kernel of the integral equation n( , , )= n g(x, y, z| , , ) n(x, y, z)dx dy dz.

10 ( )Assuming that the Green s function can be expressed as an eigenfunctionexpansion, Equation follows. As examples, Kneser found the bilinearexpansion for rectangular and circular areas and for the surface of a summary then, by 1950 there were essentially three methods16forfinding Green FUNCTIONS . The first method simply used a Green s functiondeveloped for Helmholtz s equation 2u+k20u=0andtookthelimitask0 0. The Second method wrote the Green s function as a sum of eigenfunctionsthat satisfied the boundary conditions. The coefficients were then chosen sothat the correct singular behavior occurred at the source point. Finally, thethird method wrote the Green s function as the sum of the free-space solutionplus a harmonic harmonic solution was chosen so that theGreen s function satisfied the boundary on, Kelvin s classic inversion18that maps the interior of a circleor sphere to the exterior andvice versawas developed to find the Green sfunction for Poisson s equation.