Mathematical Methods for Physics and Engineering : A Comprehensive Guide

16.6 EXERCISES (c) Determine the radius of convergenceRof theσ= 3 series and relate it to the positions of the singularities of ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS (b) Find one series solution in powers ofz. Give a formal expression for a s ...

16.7 HINTS AND ANSWERS (c) Show that the corresponding non-terminating series solutionsSm(z) have as their first few terms S 0 ( ...

17 Eigenfunction methods for differential equations In the previous three chapters we dealt with the solution of differential eq ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS set consists of those functionsyifor which Lyi(x)=λiyi(x), (17.2) whereλiis a c ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS class of operators calledHermitian operators(the operator in the simple harmoni ...

17.1 SETS OF FUNCTIONS where thednare a different set of coefficients. In each case, provided the basis functions are linearly i ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS
Starting from the linearly independent functionsyn(x)=xn,n=0, 1 ,..., construc ...

17.2 ADJOINT, SELF-ADJOINT AND HERMITIAN OPERATORS 17.1.1 Some useful inequalities Since for a Hilbert space〈f|f〉≥0, the inequal ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS or by the operator itself, such that the boundary terms in (17.15) vanish, then ...

17.3 PROPERTIES OF HERMITIAN OPERATORS 17.3 Properties of Hermitian operators We now provide proofs of some of the useful proper ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS which is a statement of the orthogonality ofyiandyj. If one (or more) of the ei ...

17.3 PROPERTIES OF HERMITIAN OPERATORS set and we can write ∫b a yˆi∗yˆjρdx=δij, (17.27) which is valid for all pairs of valuesi ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS are both real, is a non-zero eigenfunction corresponding to that eigenvalue. It ...

17.4 STURM–LIOUVILLE EQUATIONS certain boundary conditions are met, namely that any two eigenfunctionsyiand yjof (17.33) must sa ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS Equation p(x) q(x) λρ(x) Hypergeometric xc(1−x)a+b−c+1 0 −ab xc−^1 (1−x)a+b−c L ...

17.4 STURM–LIOUVILLE EQUATIONS (ii) From (17.39), the required integrating factor is F(x)=exp (∫x − 1 du ) =exp(−x). Thus, the L ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS integrating factor. This is given, as in equation (17.39), by F(x)=exp [∫x c−(a ...

17.5 SUPERPOSITION OF EIGENFUNCTIONS: GREEN’S FUNCTIONS 17.5 Superposition of eigenfunctions: Green’s functions We have already ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS and we assume that we may interchange the order of summation and integration, t ...