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1: 18.4 Graphics
See accompanying text
Figure 18.4.5: Laguerre polynomials L n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.6: Laguerre polynomials L 3 ( α ) ( x ) , α = 0 , 1 , 2 , 3 , 4 . Magnify
See accompanying text
Figure 18.4.7: Monic Hermite polynomials h n ( x ) = 2 n H n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.8: Laguerre polynomials L 3 ( α ) ( x ) , 0 α 3 , 0 x 10 . Magnify 3D Help
See accompanying text
Figure 18.4.9: Laguerre polynomials L 4 ( α ) ( x ) , 0 α 3 , 0 x 10 . Magnify 3D Help
2: 23.14 Integrals
23.14.2 2 ( z ) d z = 1 6 ( z ) + 1 12 g 2 z ,
3: 18.41 Tables
Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ( x ) , U n ( x ) , L n ( x ) , and H n ( x ) for n = 0 ( 1 ) 12 . The ranges of x are 0.2 ( .2 ) 1 for T n ( x ) and U n ( x ) , and 0.5 , 1 , 3 , 5 , 10 for L n ( x ) and H n ( x ) . The precision is 10D, except for H n ( x ) which is 6-11S. … For P n ( x ) , L n ( x ) , and H n ( x ) see §3.5(v). …
4: 11.14 Tables
  • Abramowitz and Stegun (1964, Chapter 12) tabulates 𝐇 n ( x ) , 𝐇 n ( x ) Y n ( x ) , and I n ( x ) 𝐋 n ( x ) for n = 0 , 1 and x = 0 ( .1 ) 5 , x 1 = 0 ( .01 ) 0.2 to 6D or 7D.

  • Agrest et al. (1982) tabulates 𝐇 n ( x ) and e x 𝐋 n ( x ) for n = 0 , 1 and x = 0 ( .001 ) 5 ( .005 ) 15 ( .01 ) 100 to 11D.

  • Zhang and Jin (1996) tabulates 𝐇 n ( x ) and 𝐋 n ( x ) for n = 4 ( 1 ) 3 and x = 0 ( 1 ) 20 to 8D or 7S.

  • Abramowitz and Stegun (1964, Chapter 12) tabulates 0 x ( I 0 ( t ) 𝐋 0 ( t ) ) d t and ( 2 / π ) x t 1 𝐇 0 ( t ) d t for x = 0 ( .1 ) 5 to 5D or 7D; 0 x ( 𝐇 0 ( t ) Y 0 ( t ) ) d t ( 2 / π ) ln x , 0 x ( I 0 ( t ) 𝐋 0 ( t ) ) d t ( 2 / π ) ln x , and x t 1 ( 𝐇 0 ( t ) Y 0 ( t ) ) d t for x 1 = 0 ( .01 ) 0.2 to 6D.

  • Agrest et al. (1982) tabulates 0 x 𝐇 0 ( t ) d t and e x 0 x 𝐋 0 ( t ) d t for x = 0 ( .001 ) 5 ( .005 ) 15 ( .01 ) 100 to 11D.

  • 5: 11.15 Approximations
  • Luke (1975, pp. 416–421) gives Chebyshev-series expansions for 𝐇 n ( x ) , 𝐋 n ( x ) , 0 | x | 8 , and 𝐇 n ( x ) Y n ( x ) , x 8 , for n = 0 , 1 ; 0 x t m 𝐇 0 ( t ) d t , 0 x t m 𝐋 0 ( t ) d t , 0 | x | 8 , m = 0 , 1 and 0 x ( 𝐇 0 ( t ) Y 0 ( t ) ) d t , x t 1 ( 𝐇 0 ( t ) Y 0 ( t ) ) d t , x 8 ; the coefficients are to 20D.

  • MacLeod (1993) gives Chebyshev-series expansions for 𝐋 0 ( x ) , 𝐋 1 ( x ) , 0 x 16 , and I 0 ( x ) 𝐋 0 ( x ) , I 1 ( x ) 𝐋 1 ( x ) , x 16 ; the coefficients are to 20D.

  • Newman (1984) gives polynomial approximations for 𝐇 n ( x ) for n = 0 , 1 , 0 x 3 , and rational-fraction approximations for 𝐇 n ( x ) Y n ( x ) for n = 0 , 1 , x 3 . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

  • 6: 18.18 Sums
    18.18.10 L n ( α 1 + + α r + r 1 ) ( x 1 + + x r ) = m 1 + + m r = n L m 1 ( α 1 ) ( x 1 ) L m r ( α r ) ( x r ) .
    18.18.12 L n ( α ) ( λ x ) L n ( α ) ( 0 ) = = 0 n ( n ) λ ( 1 λ ) n L ( α ) ( x ) L ( α ) ( 0 ) .
    18.18.37 = 0 n L ( α ) ( x ) = L n ( α + 1 ) ( x ) ,
    18.18.38 = 0 n L ( α ) ( x ) L n ( β ) ( y ) = L n ( α + β + 1 ) ( x + y ) .
    18.18.40 = 0 n ( n ) H 2 ( x ) H 2 n 2 ( y ) = ( 1 ) n 2 2 n n ! L n ( x 2 + y 2 ) .
    7: 19.33 Triaxial Ellipsoids
    Let a homogeneous magnetic ellipsoid with semiaxes a , b , c , volume V = 4 π a b c / 3 , and susceptibility χ be placed in a previously uniform magnetic field H parallel to the principal axis with semiaxis c . The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength H / ( 1 + L c χ ) , where L c is the demagnetizing factor, given in cgs units by
    19.33.7 L c = 2 π a b c 0 d λ ( a 2 + λ ) ( b 2 + λ ) ( c 2 + λ ) 3 = V R D ( a 2 , b 2 , c 2 ) .
    19.33.8 L a + L b + L c = 4 π ,
    where L a and L b are obtained from L c by permutation of a , b , and c . …
    8: 23.9 Laurent and Other Power Series
    23.9.1 c n = ( 2 n 1 ) w 𝕃 { 0 } w 2 n , n = 2 , 3 , 4 , .
    23.9.2 ( z ) = 1 z 2 + n = 2 c n z 2 n 2 , 0 < | z | < | z 0 | ,
    23.9.3 ζ ( z ) = 1 z n = 2 c n 2 n 1 z 2 n 1 , 0 < | z | < | z 0 | .
    23.9.6 ( ω j + t ) = e j + ( 3 e j 2 5 c 2 ) t 2 + ( 10 c 2 e j + 21 c 3 ) t 4 + ( 7 c 2 e j 2 + 21 c 3 e j + 5 c 2 2 ) t 6 + O ( t 8 ) ,
    23.9.7 σ ( z ) = m , n = 0 a m , n ( 10 c 2 ) m ( 56 c 3 ) n z 4 m + 6 n + 1 ( 4 m + 6 n + 1 ) ! ,
    9: 11.4 Basic Properties
    §11.4(v) Recurrence Relations and Derivatives
    where ν ( z ) denotes either 𝐇 ν ( z ) or 𝐋 ν ( z ) . …
    §11.4(vi) Derivatives with Respect to Order
    For properties of zeros of 𝐇 ν ( x ) see Steinig (1970). For asymptotic expansions of zeros of 𝐇 0 ( x ) see MacLeod (2002a).
    10: 18.5 Explicit Representations
    18.5.6 L n ( α ) ( 1 x ) = ( 1 ) n n ! x n + α + 1 e 1 / x d n d x n ( x α 1 e 1 / x ) .
    Similarly in the cases of the ultraspherical polynomials C n ( λ ) ( x ) and the Laguerre polynomials L n ( α ) ( x ) we assume that λ > 1 2 , λ 0 , and α > 1 , unless stated otherwise. …
    L 0 ( x ) = 1 ,
    L 1 ( x ) = x + 1 ,
    L 6 ( x ) = 1 720 x 6 1 20 x 5 + 5 8 x 4 10 3 x 3 + 15 2 x 2 6 x + 1 .