About the Project
NIST

Jacobian elliptic-function form

AdvancedHelp

(0.009 seconds)

1—10 of 306 matching pages

1: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
§22.15(ii) Representations as Elliptic Integrals
The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. …
2: 22.2 Definitions
§22.2 Definitions
22.2.9 sc ( z , k ) = θ 3 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 2 ( ζ , q ) = 1 cs ( z , k ) .
As a function of z , with fixed k , each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … The Jacobian functions are related in the following way. …
3: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
§22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the intervals ( - , 0 ) and ( 1 , ) , or with purely imaginary moduli are related to functions with moduli in the interval [ 0 , 1 ] by the following formulas. …
§22.17(ii) Complex Moduli
For proofs of these results and further information see Walker (2003).
4: 22.5 Special Values
§22.5 Special Values
Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its z -derivative (or at a pole, the residue), for values of z that are integer multiples of K , i K . …
§22.5(ii) Limiting Values of k
Table 22.5.3: Limiting forms of Jacobian elliptic functions as k 0 .
sn ( z , k ) sin z cd ( z , k ) cos z dc ( z , k ) sec z ns ( z , k ) csc z
5: 22.16 Related Functions
For - K < x < K , …See Figure 22.16.2. …
22.16.18 ( x , k ) = - k 2 0 x cd 2 ( t , k ) d t + x + k 2 sn ( x , k ) cd ( x , k ) ,
22.16.21 ( x , k ) = - 0 x dc 2 ( t , k ) d t + x + sn ( x , k ) dc ( x , k ) ,
22.16.26 ( x , k ) = - 0 x ( cs 2 ( t , k ) - t - 2 ) d t + x - 1 - cn ( x , k ) ds ( x , k ) .
6: 22.18 Mathematical Applications
§22.18 Mathematical Applications
§22.18(i) Lengths and Parametrization of Plane Curves
The special case y 2 = ( 1 - x 2 ) ( 1 - k 2 x 2 ) is in Jacobian normal form. …
7: 22.14 Integrals
§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions
§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions
The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. …
§22.14(iv) Definite Integrals
8: 22.4 Periods, Poles, and Zeros
§22.4(i) Distribution
For the distribution of the k -zeros of the Jacobian elliptic functions see Walker (2009).
§22.4(ii) Graphical Interpretation via Glaisher’s Notation
9: 22.8 Addition Theorems
§22.8 Addition Theorems
22.8.1 sn ( u + v ) = sn u cn v dn v + sn v cn u dn u 1 - k 2 sn 2 u sn 2 v ,
§22.8(ii) Alternative Forms for Sum of Two Arguments
§22.8(iii) Special Relations Between Arguments
10: 22.3 Graphics
§22.3(i) Real Variables: Line Graphs
§22.3(iii) Complex z ; Real k
§22.3(iv) Complex k
See accompanying text
Figure 22.3.24: sn ( x + i y , k ) for - 4 x 4 , 0 y 8 , k = 1 + 1 2 i . … Magnify 3D Help
See accompanying text
Figure 22.3.25: sn ( 5 , k ) as a function of complex k 2 , - 1 ( k 2 ) 3.5 , - 1 ( k 2 ) 1 . … Magnify 3D Help