Bateman errata
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A short collection of comments and unpublished corrections compiled by Tom Koornwinder, last modified July 9, 2008, for the Bateman Manuscript Project volumes edited by Erdélyi. It lists corrections to specific formulas in Higher Transcendental Functions Vols. 1 and 2 (Legendre and Gegenbauer functions, other special functions) and Tables of Integral Transforms Vol. 1, mostly involving Jacobi polynomials.
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Errata and Comments to Higher Transcendental Functions and
Tables of Integral Transforms
collected by Tom Koornwinder, [email protected]
last modified: July 9, 2008
These are comments and possibly not yet published errata to the volumes
A. Erd´ elyi, Higher transcendental functions, Vols. 1,2,3 McGraw-Hill, 1953, 1953, 1955,
and Tables of integral transforms, Vols. 1,2 , McGraw-Hill, 1954, 1954.
See also the lists of errata which are included in the volumes, and the errata collected by
H. van Haeringen and L. P. Kok in Math. Comp. 41 (1983), 778–780, see http://www.
jstor.org/stable/2007718 .
Higher transcendental functions, Vol. 1
3.4(8) : On the right, after the equality sign, replace iπby−iπ
(observed by E. Diekema; see Ch. IV, (99) in L. Robin, Fonctions sph´ eriques de Legendre
et fonctions sph´ eroidales, Tome II , Gauthier-Villars, 1958).
3.15(4) : This formula is valid for z∈C\(−∞,1]. For z∈(−1,1) the formula remains
valid if we replace ( z2−1)1
4−1
2νby (1−z2)1
4−1
2νandP1
2−ν
n+ν−1
2by P1
2−ν
n+ν−1
2:
Cν
n(x) = 2ν−1
2Γ(n+ 2ν) Γ(ν+1
2)
Γ(2ν) Γ(n+ 1)(1−x2)1
4−1
2νP1
2−ν
n+ν−1
2(x) ( x∈(−1,1)).
Higher transcendental functions, Vol. 2
10.9(8) : In the formula for Kninsert a factor n! on the right.
10.10(5) : In the formula for Cndelete the minus sign on the right.
10.12(2) : The formula for rnshould read: rn=−n(n+α).
Tables of integral transforms, Vol. 1
1.10(5) : On the left replace the expression for f(x) (0< x < 1) by
(1−x)ν(1 + x)µP(ν,µ)
2n(x) + (1 + x)ν(1−x)µP(µ,ν)
2n(x).
1.10(6) : On the left replace the expression for f(x) (0< x < 1) by
(1−x)ν(1 + x)µP(ν,µ)
2n+1(x)−(1 + x)ν(1−x)µP(µ,ν)
2n+1(x).
On the right replace ( −1)n+1by (−1)n.
2.10(6) : On the left replace the expression for f(x) (0< x < 1) by
(1−x)ν(1 + x)µP(ν,µ)
2n(x)−(1 + x)ν(1−x)µP(µ,ν)
2n(x).
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2.10(7) : On the left replace the expression for f(x) (0< x < 1) by
(1−x)ν(1 + x)µP(ν,µ)
2n+1(x) + (1 + x)ν(1−x)µP(µ,ν)
2n+1(x).
On the right replace ( −1)n+1by (−1)n.
3.3(4) : On the left replace P(ν,ν)
n byP(ν,µ)
n.
This formula implies 1.10(5), 1.10(6), 2.10(6) and 2.10(7).
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