JacobiSN

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed z

For fixed m

General characteristics

Domain and analyticity

sn(z|m) is a meromorphic function of z which is defined over (ℂ^2).

Symmetries and periodicities

Parity

sn(z|m) is an odd function with respect to z.

Mirror symmetry

Periodicity

sn(z|m) is a doubly periodic function with respect to z with periods 2 i K(1-m) and 4 K(m).

Poles and essential singularities

With respect to z

For fixed m, the function sn(z|m) has an infinite set of singular points:
a) z=2 r K(m)+(2 s+1) i K(1-m),{r,s}∈ℤ, are the simple poles with residues ((((-1))^r)/(Sqrt[m]));
b) z== (OverTilde[∞]) is an essential singular point.

Branch points

With respect to m

For fixed z, the function sn(z|m) is a meromorphic function in m that has no branch points.

P. Walker

With respect to z

For fixed m, the function sn(z|m) does not have branch points.

Branch cuts

With respect to m

For fixed z, the function sn(z|m) is a meromorphic function in m that has no branch cuts.

P. Walker

With respect to z

For fixed m, the function sn(z|m) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

Expansions at z==2 r EllipticK[m]+(2 s+1) i EllipticK[1-m]

Expansions at m==0

Expansions at m==1

q-series

Other series representations

Product representations

Differential equations

Ordinary nonlinear differential equations

With respect to m

With respect to z

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

n th degree transformations:

Argument involving half-periods

Argument involving inverse Jacobi functions

Addition formulas

Half-angle formulas

Multiple arguments

Double angle formulas

Multiple angle formulas

Products of a single Jacobi function

Sums over products of two Jacobi functions

Sums over products of three Jacobi functions

Sums over products of four Jacobi functions

Sums over products of five Jacobi functions

Sums over products of six Jacobi functions

Sums over products of seven Jacobi functions

Sums over products of arbitrarily many Jacobi functions

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

With respect to z

With respect to m

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving functions of the direct function

Involving direct function and elliptic functions

Definite integration

Involving functions of the direct function

Involving direct function and elliptic functions

Representations through equivalent functions

With inverse function

With related functions

Involving JacobiAmplitude

Involving one other Jacobi elliptic function

Involving two other Jacobi elliptic functions

Involving three other Jacobi elliptic functions

Involving four other Jacobi elliptic functions

Involving five other Jacobi elliptic functions

Involving Weierstrass functions

Involving theta functions

Inequalities

Zeros

Theorems

Mapping of the upper half plane to a rectangle

The function w(z)==sn(z|m) maps the upper half w-plane to a rectangle in the z-plane with vertices {-K(m)+i K(1-m),-K(m),K(m),K(m)+i K(1-m)}.

The solutions of the Euler equations for the motion of a free rigid body

The solutions of the Euler equations for the motion of a free rigid body:

A p'(t)-(B-C) q(t) r(t)==B q'(t)-(C-A) r(t) p(t)==C r'(t)-(A-B) p(t) q(t)==0;p(t)==(Subscript[c, p])cn(α (t-(Subscript[t, 0]))|m)∧q(t)==(Subscript[c, q])sn(α (t-(Subscript[t, 0]))|m)∧r(t)==(Subscript[c, r])  dn(α (t-(Subscript[t, 0]))|m),

where n and m follow algebraically from A,  B,  C and the initial conditions.

Parametrization of any biquadratic curve

Any biquadratic curve a (x^2)(y^2)+b((x^2)y+x (y^2))+c((x^2)+(y^2))+2d x y+e(x+y)+f==0 can be parametrized in the form x(t)==(α sn(t|m)+β)/(γ sn(t|m)+δ), y(t)==(α sn(t+η|m)+β)/(γ sn(t+η|m)+δ).

The curvature and the torsion

The curvature and the torsion of the knotted and unknotted centerlines of an elastic rod can be expressed in elliptic functions.

History

– K. F. Gauss (1799)
– C. G. J. Jacobi (1827)
– N. H. Abel (1827)
– C. Gudermann (1838) introduced the notation sn.

Global References

A. Khare, U. Sukhatme: Cyclic Identities Involving Jacobi Elliptic Functions  math-ph/0201004 (2002)
http://arXiv.org/abs/math-ph/0201004

A. Khare, U. Sukhatme: Cyclic Identities Involving Jacobi Elliptic Functions Journal of Mathematical Physics 43, 3798-3806 (2002)

A. Khare, A. Lakshminarayan, U. Sukhatme: Cyclic Identities Involving Jacobi Elliptic Functions. II  math-ph/0207019 (2002)
http://arXiv.org/abs/math-ph/0207019

A. Khare, A. Lakshminarayan, U. Sukhatme: Cyclic Identities Involving Jacobi Elliptic Functions Journal of Mathematical Physics 44, 1822-1841 (2003)

A. Khare, A. Lakshminarayan, U. Sukhatme: Local Identities Involving Jacobi Elliptic Functions  math-ph/0306028 (2003)
http://arXiv.org/abs/math-ph/0306028

Copyright