Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.2 ASSOCIATED LEGENDRE FUNCTIONS Sinced^2 (1−x^2 )/dx^2 =(−1)(2)!, and noting that (−1)^2 +2m= 1, we have Im= 1 22 (! ...

SPECIAL FUNCTIONS Generating function The generating function for associated Legendre functions can be easily derived by combini ...

18.3 SPHERICAL HARMONICS be derived in a number of ways, such as using the generating function (18.40) or by differentiation of ...

SPECIAL FUNCTIONS orthonormal set, i.e. ∫ 1 − 1 ∫ 2 π 0 [ Ym(θ, φ) ]∗ Ym ′ ′(θ, φ)dφ d(cosθ)=δ′δmm′. (18.46) In addition, th ...

18.4 CHEBYSHEV FUNCTIONS Sinceδ(Ω−Ω′) can depend only on the angleγbetween the two directions Ω and Ω′, we may also expand it in ...

SPECIAL FUNCTIONS and has three regular singular points, atx=− 1 , 1 ,∞. By comparing it with (18.1), we see that the Chebyshev ...

18.4 CHEBYSHEV FUNCTIONS T 0 T 1 T 2 T 3 − 1 − 1 − 0. 5 − 0. 5 0. 5 0. 5 1 1 Figure 18.3 The first four Chebyshev polynomials of ...

SPECIAL FUNCTIONS U 0 U 1 U 2 U 3 − (^1) − 0. 5 0. 5 1 − 2 2 − 4 4 Figure 18.4 The first four Chebyshev polynomials of the secon ...

18.4 CHEBYSHEV FUNCTIONS Evaluating the first and second derivatives ofVn+1,weobtain Vn′+1=(1−x^2 )^1 /^2 Un′−x(1−x^2 )−^1 /^2 U ...

SPECIAL FUNCTIONS The normalisation, whenm=n, is easily found by making the substitution x=cosθand using (18.55). We immediately ...

18.4 CHEBYSHEV FUNCTIONS in which the coefficientsanare given by an= 2 π ∫ 1 − 1 f(x)Un(x)(1−x^2 )^1 /^2 dx. Generating function ...

SPECIAL FUNCTIONS Using (18.65) and settingx=cosθimmediately gives a rearrangement of the required result (18.69). Similarly, ad ...

18.5 BESSEL FUNCTIONS generality. The equation arises from physical situations similar to those involving Legendre’s equation bu ...

SPECIAL FUNCTIONS the form of a Frobenius series corresponding to the larger root,σ 1 =ν=m/2, as described above. However, for t ...

18.5 BESSEL FUNCTIONS We note that Bessel functions of half-integer order are expressible in closed form in terms of trigonometr ...

SPECIAL FUNCTIONS J 0 J 1 J 2 246810 x − 0. 5 0. 5 1 1. 5 Figure 18.5 The first three integer-order Bessel functions of the firs ...

18.5 BESSEL FUNCTIONS and hence thatJν(x)andJ−ν(x) are linearly dependent. So, in this case, we cannot write the general solutio ...

SPECIAL FUNCTIONS Y 0 Y (^1) Y 2 246810 x − 1 − 0. 5 0. 5 1 Figure 18.6 The first three integer-order Bessel functions of the se ...

18.5 BESSEL FUNCTIONS To determine the required boundary conditions for this result to hold, let us consider the functionsf(x)=J ...

SPECIAL FUNCTIONS evaluated using l’Hˆopital’s rule, or alternatively we may calculate the relevant integral directly.
Evaluate ...