Content-Length: 441376 | pFad | https://dlmf.nist.gov/./../././././././././././././bib/../././././././bib/../././22.16#Px11.p1
DLMF: §22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.16 Related Functions
Contents
§22.16(i) Jacobi’s Amplitude (am ) Function
§22.16(ii) Jacobi’s Epsilon Function
§22.16(iii) Jacobi’s Zeta Function
§22.16(iv) Graphs
§22.16(i) Jacobi’s Amplitude (am ) Function
Definition
22.16.1
am ( x , k ) = Arcsin ( sn ( x , k ) ) ,
x ∈ ℝ ,
where the inverse sine has its principal value when − K ≤ x ≤ K and is
defined by continuity elsewhere. See Figure 22.16.1 .
am ( x , k ) is an infinitely differentiable function of x .
Quasi-Periodicity
22.16.2
am ( x + 2 K , k ) = am ( x , k ) + π .
Integral Representation
22.16.3
am ( x , k ) = ∫ 0 x dn ( t , k ) d t .
Special Values
22.16.4
am ( x , 0 )
= x ,
22.16.5
am ( x , 1 )
= gd ( x ) .
For the Gudermannian function gd ( x ) see §4.23(viii) .
Approximation for Small x
22.16.6
am ( x , k ) = x − k 2 x 3 3 ! + k 2 ( 4 + k 2 ) x 5 5 ! + O ( x 7 ) .
Approximations for Small k , k ′
22.16.7
am ( x , k ) = x − 1 4 k 2 ( x − sin x cos x ) + O ( k 4 ) ,
22.16.8
am ( x , k ) = gd x − 1 4 k ′ 2 ( x − sinh x cosh x ) sech x + O ( k ′ 4 ) .
Fourier Series
With q as in (22.2.1 ) and ζ = π x / ( 2 K ) ,
22.16.9
am ( x , k ) = π 2 K x + 2 ∑ n = 1 ∞ q n sin ( 2 n ζ ) n ( 1 + q 2 n ) .
Relation to Elliptic Integrals
If − K ≤ x ≤ K , then the following four equations
are equivalent:
22.16.12
sn ( x , k ) = sin ϕ = sin ( am ( x , k ) ) ,
22.16.13
cn ( x , k ) = cos ϕ = cos ( am ( x , k ) ) .
For F ( ϕ , k ) see §19.2(ii) .
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
For − K < x < K ,
22.16.14
ℰ ( x , k ) = ∫ 0 sn ( x , k ) 1 − k 2 t 2 1 − t 2 d t ;
compare (19.2.5 ). See Figure 22.16.2 .
22.16.15
ℰ ( x , k )
= − k 2 ∫ 0 x sn 2 ( t , k ) d t + x ,
22.16.16
ℰ ( x , k )
= k 2 ∫ 0 x cn 2 ( t , k ) d t + k ′ 2 x ,
22.16.17
ℰ ( x , k )
= ∫ 0 x dn 2 ( t , k ) d t .
22.16.18
ℰ ( x , k )
= − k 2 ∫ 0 x cd 2 ( t , k ) d t + x + k 2 sn ( x , k ) cd ( x , k ) ,
22.16.19
ℰ ( x , k )
= k 2 k ′ 2 ∫ 0 x sd 2 ( t , k ) d t + k ′ 2 x + k 2 sn ( x , k ) cd ( x , k ) ,
22.16.20
ℰ ( x , k )
= k ′ 2 ∫ 0 x nd 2 ( t , k ) d t + k 2 sn ( x , k ) cd ( x , k ) .
In Equations (22.16.21 )–(22.16.23 ),
− K < x < K .
22.16.21
ℰ ( x , k )
= − ∫ 0 x dc 2 ( t , k ) d t + x + sn ( x , k ) dc ( x , k ) ,
22.16.22
ℰ ( x , k )
= − k ′ 2 ∫ 0 x nc 2 ( t , k ) d t + k ′ 2 x + sn ( x , k ) dc ( x , k ) ,
22.16.23
ℰ ( x , k )
= − k ′ 2 ∫ 0 x sc 2 ( t , k ) d t + sn ( x , k ) dc ( x , k ) .
In Equations (22.16.24 )–(22.16.26 ),
− 2 K < x < 2 K .
22.16.24
ℰ ( x , k )
= − ∫ 0 x ( ns 2 ( t , k ) − t − 2 ) d t + x − 1 + x − cn ( x , k ) ds ( x , k ) ,
22.16.25
ℰ ( x , k )
= − ∫ 0 x ( ds 2 ( t , k ) − t − 2 ) d t + x − 1 + k ′ 2 x − cn ( x , k ) ds ( x , k ) ,
22.16.26
ℰ ( x , k )
= − ∫ 0 x ( cs 2 ( t , k ) − t − 2 ) d t + x − 1 − cn ( x , k ) ds ( x , k ) .
Quasi-Addition and Quasi-Periodic Formulas
22.16.27
ℰ ( x 1 + x 2 , k ) = ℰ ( x 1 , k ) + ℰ ( x 2 , k ) − k 2 sn ( x 1 , k ) sn ( x 2 , k ) sn ( x 1 + x 2 , k ) ,
22.16.28
ℰ ( x + K , k ) = ℰ ( x , k ) + E ( k ) − k 2 sn ( x , k ) cd ( x , k ) ,
22.16.29
ℰ ( x + 2 K , k ) = ℰ ( x , k ) + 2 E ( k ) .
For E ( k ) see §19.2(ii) .
Relation to Theta Functions
22.16.30
ℰ ( x , k ) = 1 θ 3 2 ( 0 , q ) θ 4 ( ξ , q ) d d ξ θ 4 ( ξ , q ) + E ( k ) K ( k ) x ,
where ξ = x / θ 3 2 ( 0 , q ) . For θ j see §20.2(i) . For E ( k ) see §19.2(ii) .
Relation to the Elliptic Integral E ( ϕ , k )
22.16.31
E ( am ( x , k ) , k ) = ℰ ( x , k ) ,
− K ≤ x ≤ K .
For E ( ϕ , k ) see §19.2(ii) . See also (22.16.14 ).
§22.16(iii) Jacobi’s Zeta Function
Definition
With E ( k ) and K ( k ) as in §19.2(ii) and
x ∈ ℝ ,
22.16.32
Z ( x | k ) = ℰ ( x , k ) − ( E ( k ) / K ( k ) ) x .
See Figure 22.16.3 . (Sometimes in the literature
Z ( x | k ) is denoted by Z ( am ( x , k ) , k 2 ) .)
Properties
Z ( x | k ) satisfies the same quasi-addition formula as the function
ℰ ( x , k ) , given by (22.16.27 ). Also,
22.16.33
Z ( x + K | k ) = Z ( x | k ) − k 2 sn ( x , k ) cd ( x , k ) ,
22.16.34
Z ( x + 2 K | k ) = Z ( x | k ) .
§22.16(iv) Graphs
Figure 22.16.1: Jacobi’s amplitude function am ( x , k )
for 0 ≤ x ≤ 10 π
and k = 0.4 , 0.7 , 0.99 , 0.999999 .
Values of k greater than 1 are illustrated in
Figure 22.19.1 .
Magnify
Figure 22.16.2: Jacobi’s epsilon function ℰ ( x , k )
for 0 ≤ x ≤ 10 π
and k = 0.4 , 0.7 , 0.99 , 0.999999 .
(These graphs are similar to those in Figure 22.16.1 ;
compare (22.16.3 ), (22.16.17 ), and the graphs of
dn ( x , k ) in §22.3(i) .)
Magnify
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