Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.5 BESSEL FUNCTIONS
Prove the expression (18.91). If we multiply (18.90) byxJν(αmx) and integrate fromx=0tox=bthen we obtain ...

SPECIAL FUNCTIONS Finally, subtracting (18.95) from (18.94) and dividing byxgives Jν− 1 (x)+Jν+1(x)= 2 ν x Jν(x). (18.97)
Given ...

18.5 BESSEL FUNCTIONS in subsection 18.1.2. The generating function for Bessel functions of integer order is given by G(x, h)=ex ...

SPECIAL FUNCTIONS Using de Moivre’s theorem, expiθ=cosθ+isinθ, we then obtain exp(ixsinθ)=cos(xsinθ)+isin(xsinθ)= ∑∞ m=−∞ Jm(x)( ...

18.6 SPHERICAL BESSEL FUNCTIONS whereis an integer. This equation looks very much like Bessel’s equation and can in fact be red ...

SPECIAL FUNCTIONS
Show that theth spherical Bessel function is given by f(x)=(−1)x ( 1 x d dx ) f 0 (x), (18.106) wheref( ...

18.7 LAGUERRE FUNCTIONS it has a regular singularity atx= 0 and an essential singularity atx=∞.The parameterνis a given real num ...

SPECIAL FUNCTIONS L 0 L 1 L 2 L 3 123 4 5 567 − 5 10 − 10 x Figure 18.7 The first four Laguerre polynomials. 18.7.1 Properties o ...

18.7 LAGUERRE FUNCTIONS
Prove that the expression (18.112) yields thenth Laguerre polynomial. Evaluating thenth derivative in ( ...

SPECIAL FUNCTIONS The above orthogonality and normalisation conditions allow us to expand any (reasonable) function in the inter ...

18.8 ASSOCIATED LAGUERRE FUNCTIONS which trivially rearranges to give the recurrence relation (18.115). To obtain the recurrence ...

SPECIAL FUNCTIONS In particular, we note thatL^0 n(x)=Ln(x). As discussed in the previous section, Ln(x) is a polynomial of orde ...

18.8 ASSOCIATED LAGUERRE FUNCTIONS
Show that I≡ ∫∞ 0 Lmn(x)Lmn(x)xme−xdx= (n+m)! n! . (18.122) Using the Rodrigues’ formula (18 ...

SPECIAL FUNCTIONS where, in the second equality, we have expanded the RHS using the binomial theorem. On equating coefficients o ...

18.9 HERMITE FUNCTIONS H 0 H 1 H 2 H 3 − 1. 5 − 1 − 0. 5 0.^511.^5 5 10 − 5 − 10 x Figure 18.8 The first four Hermite polynomial ...

SPECIAL FUNCTIONS 18.9.1 Properties of Hermite polynomials The Hermite polynomials and functions derived from them are important ...

18.9 HERMITE FUNCTIONS
Show that I≡ ∫∞ −∞ Hn(x)Hn(x)e−x 2 dx=2nn! √ π. (18.132) Using the Rodrigues’ formula (18.130), we may w ...

SPECIAL FUNCTIONS Differentiating this formktimes with respect tohgives ∑∞ n=k Hn (n−k)! hn−k= ∂kG ∂hk =ex 2 ∂k ∂hk e−(x−h) 2 =( ...

18.10 HYPERGEOMETRIC FUNCTIONS by making appropriate changes of the independent and dependent variables, any second-order differ ...

SPECIAL FUNCTIONS gamma function.§It is straightforward to show that the hypergeometric series converges in the range|x|<1. I ...