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11: 19.19 Taylor and Related Series
For N = 0 , 1 , 2 , define the homogeneous hypergeometric polynomial … If n = 2 , then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … and define the n -tuple 1 2 = ( 1 2 , , 1 2 ) . … The number of terms in T N can be greatly reduced by using variables Z = 1 - ( z / A ) with A chosen to make E 1 ( Z ) = 0 . …
12: 10.44 Sums
10.44.1 𝒵 ν ( λ z ) = λ ± ν k = 0 ( λ 2 - 1 ) k ( 1 2 z ) k k ! 𝒵 ν ± k ( z ) , | λ 2 - 1 | < 1 .
If 𝒵 = I and the upper signs are taken, then the restriction on λ is unnecessary. …
10.44.3 𝒵 ν ( u ± v ) = k = - ( ± 1 ) k 𝒵 ν + k ( u ) I k ( v ) , | v | < | u | .
The restriction | v | < | u | is unnecessary when 𝒵 = I and ν is an integer. …
10.44.6 K n ( z ) = n ! ( 1 2 z ) - n 2 k = 0 n - 1 ( - 1 ) k ( 1 2 z ) k I k ( z ) k ! ( n - k ) + ( - 1 ) n - 1 ( ln ( 1 2 z ) - ψ ( n + 1 ) ) I n ( z ) + ( - 1 ) n k = 1 ( n + 2 k ) I n + 2 k ( z ) k ( n + k ) ,
13: 22.16 Related Functions
With q as in (22.2.1) and ζ = π x / ( 2 K ) , … In Equations (22.16.24)–(22.16.26), - 2 K < x < 2 K . … where ξ = x / θ 3 2 ( 0 , q ) . … (Sometimes in the literature Z ( x | k ) is denoted by Z ( am ( x , k ) , k 2 ) .) … Z ( x | k ) satisfies the same quasi-addition formula as the function ( x , k ) , given by (22.16.27). …
14: 10.43 Integrals
Let 𝒵 ν ( z ) be defined as in §10.25(ii). …
z ν + 1 𝒵 ν ( z ) d z = z ν + 1 𝒵 ν + 1 ( z ) ,
10.43.2 z ν 𝒵 ν ( z ) d z = π 1 2 2 ν - 1 Γ ( ν + 1 2 ) z ( 𝒵 ν ( z ) L ν - 1 ( z ) - 𝒵 ν - 1 ( z ) L ν ( z ) ) , ν - 1 2 .
e ± z z ν 𝒵 ν ( z ) d z = e ± z z ν + 1 2 ν + 1 ( 𝒵 ν ( z ) 𝒵 ν + 1 ( z ) ) , ν - 1 2 ,
e ± z z - ν 𝒵 ν ( z ) d z = e ± z z - ν + 1 1 - 2 ν ( 𝒵 ν ( z ) 𝒵 ν - 1 ( z ) ) , ν 1 2 .
15: 10.25 Definitions
10.25.1 z 2 d 2 w d z 2 + z d w d z - ( z 2 + ν 2 ) w = 0 .
In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . … as z in | ph z | 3 2 π - δ ( < 3 2 π ) . …
Symbol 𝒵 ν ( z )
Corresponding to the symbol 𝒞 ν introduced in §10.2(ii), we sometimes use 𝒵 ν ( z ) to denote I ν ( z ) , e ν π i K ν ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν . …
16: 22.21 Tables
Curtis (1964b) tabulates sn ( m K / n , k ) , cn ( m K / n , k ) , dn ( m K / n , k ) for n = 2 ( 1 ) 15 , m = 1 ( 1 ) n - 1 , and q (not k ) = 0 ( .005 ) 0.35 to 20D. Lawden (1989, pp. 280–284 and 293–297) tabulates sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , ( x , k ) , Z ( x | k ) to 5D for k = 0.1 ( .1 ) 0.9 , x = 0 ( .1 ) X , where X ranges from 1. 5 to 2. 2. … Zhang and Jin (1996, p. 678) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) for k = 1 4 , 1 2 and x = 0 ( .1 ) 4 to 7D. …
17: 35.1 Special Notation
a , b

complex variables.

Z

complex symmetric matrix.

Z κ ( T )

zonal polynomials.

The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( T ) and (of the second kind) B ν ( T ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; T ) or F 1 1 ( a b ; T ) and (of the second kind) Ψ ( a ; b ; T ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; T ) or F 1 2 ( a 1 , a 2 b ; T ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; T ) or F q p ( a 1 , , a p b 1 , , b q ; T ) . An alternative notation for the multivariate gamma function is Π m ( a ) = Γ m ( a + 1 2 ( m + 1 ) ) (Herz (1955, p. 480)). Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( T ) = A ν ( T ) / A ν ( 0 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | S , T ) = | T | ν B ν ( S T ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( T ) = | T | ν B ν ( S T ) (Faraut and Korányi (1994, pp. 357–358)).
18: 10.66 Expansions in Series of Bessel Functions
10.66.1 ber ν x + i bei ν x = k = 0 e ( 3 ν + k ) π i / 4 x k J ν + k ( x ) 2 k / 2 k ! = k = 0 e ( 3 ν + 3 k ) π i / 4 x k I ν + k ( x ) 2 k / 2 k ! .
ber n ( x 2 ) = k = - ( - 1 ) n + k J n + 2 k ( x ) I 2 k ( x ) ,
bei n ( x 2 ) = k = - ( - 1 ) n + k J n + 2 k + 1 ( x ) I 2 k + 1 ( x ) .
19: 19.36 Methods of Computation
where, in the notation of (19.19.7) with a = - 1 2 and n = 3 , … where n = 0 , 1 , 2 , , and … This method loses significant figures in ρ if α 2 and k 2 are nearly equal unless they are given exact values—as they can be for tables. If α 2 = k 2 , then the method fails, but the function can be expressed by (19.6.13) in terms of E ( ϕ , k ) , for which Neuman (1969b) uses ascending Landen transformations. … The cases k c 2 / 2 p < and - < p < k c 2 / 2 require different treatment for numerical purposes, and again precautions are needed to avoid cancellations. …
20: 22.1 Special Notation
x , y

real variables.

k

complementary modulus, k 2 + k 2 = 1 . If k [ 0 , 1 ] , then k [ 0 , 1 ] .

The functions treated in this chapter are the three principal Jacobian elliptic functions sn ( z , k ) , cn ( z , k ) , dn ( z , k ) ; the nine subsidiary Jacobian elliptic functions cd ( z , k ) , sd ( z , k ) , nd ( z , k ) , dc ( z , k ) , nc ( z , k ) , sc ( z , k ) , ns ( z , k ) , ds ( z , k ) , cs ( z , k ) ; the amplitude function am ( x , k ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . … Other notations for sn ( z , k ) are sn ( z | m ) and sn ( z , m ) with m = k 2 ; see Abramowitz and Stegun (1964) and Walker (1996). …