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1: 22.16 Related Functions
§22.16(ii) Jacobis Epsilon Function
Integral Representations
§22.16(iii) Jacobis Zeta Function
See accompanying text
Figure 22.16.2: Jacobis epsilon function ( x , k ) for 0 x 10 π and k = 0.4 , 0.7 , 0.99 , 0.999999 . … Magnify
See accompanying text
Figure 22.16.3: Jacobis zeta function Z ( x | k ) for 0 x 10 π and k = 0.4 , 0.7 , 0.99 , 0.999999 . Magnify
2: 22.21 Tables
§22.21 Tables
3: 20.4 Values at z = 0
Jacobis Identity
4: 20.1 Special Notation
Jacobis original notation: Θ ( z | τ ) , Θ 1 ( z | τ ) , H ( z | τ ) , H 1 ( z | τ ) , respectively, for θ 4 ( u | τ ) , θ 3 ( u | τ ) , θ 1 ( u | τ ) , θ 2 ( u | τ ) , where u = z / θ 3 2 ( 0 | τ ) . … Neville’s notation: θ s ( z | τ ) , θ c ( z | τ ) , θ d ( z | τ ) , θ n ( z | τ ) , respectively, for θ 3 2 ( 0 | τ ) θ 1 ( u | τ ) / θ 1 ( 0 | τ ) , θ 2 ( u | τ ) / θ 2 ( 0 | τ ) , θ 3 ( u | τ ) / θ 3 ( 0 | τ ) , θ 4 ( u | τ ) / θ 4 ( 0 | τ ) , where again u = z / θ 3 2 ( 0 | τ ) . … McKean and Moll’s notation: ϑ j ( z | τ ) = θ j ( π z | τ ) , j = 1 , 2 , 3 , 4 . …
5: 27.13 Functions
27.13.5 ( ϑ ( x ) ) 2 = 1 + n = 1 r 2 ( n ) x n .
One of Jacobis identities implies that … Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobis identities. …
6: 22.6 Elementary Identities
22.6.22 p q 2 ( 1 2 z , k ) = p s ( z , k ) + r s ( z , k ) q s ( z , k ) + r s ( z , k ) = p q ( z , k ) + r q ( z , k ) 1 + r q ( z , k ) = p r ( z , k ) + 1 q r ( z , k ) + 1 .
§22.6(iv) Rotation of Argument (Jacobis Imaginary Transformation)
Table 22.6.1: Jacobis imaginary transformation of Jacobian elliptic functions.
sn ( i z , k ) = i sc ( z , k ) dc ( i z , k ) = dn ( z , k )
7: 25.1 Special Notation
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
8: 22.1 Special Notation
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 ) ; Jacobis 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). …
9: 20.9 Relations to Other Functions
The relations (20.9.1) and (20.9.2) between k and τ (or q ) are solutions of Jacobis inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …
10: 19.5 Maclaurin and Related Expansions
For Jacobis nome q :
19.5.5 q = exp ( - π K ( k ) / K ( k ) ) = r + 8 r 2 + 84 r 3 + 992 r 4 + , r = 1 16 k 2 , 0 k 1 .