Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.5 SUPERPOSITION OF EIGENFUNCTIONS: GREEN’S FUNCTIONS Now, the boundary conditions require thatB=0andsin (√ 1 4 −λ ) π=0,andso ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS 17.6 A useful generalisation Sometimes we encounter inhomogeneous equations of ...

17.7 EXERCISES We note that ifμ=λn,i.e.ifμequals one of the eigenvalues ofL,thenG(x, z) becomes infinite and this method runs in ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS interval, if necessary by changing the signs of all eigenvalues. Fora≤x 1 ≤x 2 ...

17.7 EXERCISES whereκis a constant and f(x)= { x 0 ≤x≤π/ 2 , π−xπ/ 2 <x≤π. 17.10 Consider the following two approaches to con ...

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS 17.14 Express the solution of Poisson’s equation in electrostatics, ∇^2 φ(r)=−ρ ...

18 Special functions In the previous two chapters, we introduced the most important second-order linear ODEs in physics and engi ...

SPECIAL FUNCTIONS which on collecting terms gives ∑∞ n=0 {(n+2)(n+1)an+2−[n(n+1)−(+1)]an}xn=0. The recurrence relation is ther ...

18.1 LEGENDRE FUNCTIONS P 0 P 1 P 2 P 3 − 1 − 1 − 0. 5 0.^5 1 1 x − 2 2 Figure 18.1 The first four Legendre polynomials. The fir ...

SPECIAL FUNCTIONS whereP(x) is a polynomial of order, and so converges for allx,andQ(x)is an infinite series that converges o ...

18.1 LEGENDRE FUNCTIONS Q 0 Q 1 Q 2 x − 1 − 1 − 0. 5 − 0. 5 0. 5 0. 5 1 1 Figure 18.2 The first three Legendre functions of the ...

SPECIAL FUNCTIONS which reduces to (x^2 −1)u(+2)+2xu(+1)−(+1)u()=0. Changing the sign all through, we recover Legendre’s eq ...

18.1 LEGENDRE FUNCTIONS Mutual orthogonality In section 17.4, we noted that Legendre’s equation was of Sturm–Liouville form with ...

SPECIAL FUNCTIONS
Prove the expression (18.14) for the coefficients in the Legendre polynomial expansion of a functionf(x). If ...

18.1 LEGENDRE FUNCTIONS Equation (18.16) can then be written, using (18.15), as h ∑ Pnhn=(1− 2 xh+h^2 ) ∑ Pn′hn, and equating th ...

SPECIAL FUNCTIONS randr′must be exchanged in (18.22) or the series would not converge. This result may be used, for example, to ...

18.2 ASSOCIATED LEGENDRE FUNCTIONS 18.2 Associated Legendre functions The associated Legendre equation has the form (1−x^2 )y′′− ...

SPECIAL FUNCTIONS in (18.3) and (18.4), which we now denote byu 1 (x)andu 2 (x), we may obtain two linearly-independent series s ...

18.2 ASSOCIATED LEGENDRE FUNCTIONS write (x^2 −1) = (x+1)(x−1) and use Leibnitz’ theorem to evaluate the derivative, which yield ...

SPECIAL FUNCTIONS to be zero, sinceQm(x) is singular atx=±1, with the result that the general solution is simply some multiple ...