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1: 22.16 Related Functions
§22.16(iii) Jacobi’s Zeta Function
Definition
Properties
See accompanying text
Figure 22.16.3: Jacobi’s zeta function Z ( x | k ) for 0 x 10 π and k = 0.4 , 0.7 , 0.99 , 0.999999 . Magnify
2: 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 ) .
3: 22.21 Tables
4: 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 ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . …
5: 22.20 Methods of Computation
Jacobi’s zeta function can then be found by use of (22.16.32). …
6: 27.1 Special Notation
(For other notation see Notation for the Special Functions.)
d , k , m , n

positive integers (unless otherwise indicated).

ζ ( s )

Riemann zeta function; see §25.2(i).

( n | P )

Jacobi symbol; see §27.9.

7: 20.10 Integrals
20.10.1 0 x s - 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 - 2 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.2 0 x s - 1 ( θ 3 ( 0 | i x 2 ) - 1 ) d x = π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.3 0 x s - 1 ( 1 - θ 4 ( 0 | i x 2 ) ) d x = ( 1 - 2 1 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
8: 31.7 Relations to Other Functions
§31.7 Relations to Other Functions
§31.7(i) Reductions to the Gauss Hypergeometric Function
§31.7(ii) Relations to Lamé Functions
With z = sn 2 ( ζ , k ) and …equation (31.2.1) becomes Lamé’s equation with independent variable ζ ; compare (29.2.1) and (31.2.8). …
9: 22.2 Definitions
22.2.4 sn ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 1 ( ζ , q ) θ 4 ( ζ , q ) = 1 ns ( z , k ) ,
22.2.5 cn ( z , k ) = θ 4 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 4 ( ζ , q ) = 1 nc ( z , k ) ,
22.2.6 dn ( z , k ) = θ 4 ( 0 , q ) θ 3 ( 0 , q ) θ 3 ( ζ , q ) θ 4 ( ζ , q ) = 1 nd ( z , k ) ,
22.2.7 sd ( z , k ) = θ 3 2 ( 0 , q ) θ 2 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 3 ( ζ , q ) = 1 ds ( z , k ) ,
22.2.8 cd ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 3 ( ζ , q ) = 1 dc ( z , k ) ,
10: 22.11 Fourier and Hyperbolic Series
22.11.1 sn ( z , k ) = 2 π K k n = 0 q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
22.11.7 ns ( z , k ) - π 2 K csc ζ = 2 π K n = 0 q 2 n + 1 sin ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
22.11.8 ds ( z , k ) - π 2 K csc ζ = - 2 π K n = 0 q 2 n + 1 sin ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.9 cs ( z , k ) - π 2 K cot ζ = - 2 π K n = 1 q 2 n sin ( 2 n ζ ) 1 + q 2 n ,
22.11.10 dc ( z , k ) - π 2 K sec ζ = 2 π K n = 0 ( - 1 ) n q 2 n + 1 cos ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,