JacobiDN

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

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

Symmetries and periodicities

Parity

dn(z|m) is an even function with respect to z.

Mirror symmetry

Periodicity

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

Poles and essential singularities

With respect to z

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

Branch points

With respect to m

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

P. Walker

With respect to z

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

Branch cuts

With respect to m

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

P. Walker

With respect to z

For fixed m, the function dn(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

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

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 involving the group of functions

Cyclic Identities of rank 2

p==3

General case: (p-1)/2∈Integers

Cyclic Identities of rank 3

p==2

p==3

p==4

General case: p∈Integers

General case: (p-1)/2∈Integers

Cyclic Identities of rank 4 and above

r==4, p==2

r==4, p==3

r==5, p==3

Local identities of rank 2

Local identities of rank 3

Rank 3 identities with 3 distinct arguments

Rank 3 identities with 2 distinct arguments

Local identities of rank 4

Rank 4 identities with 4 distinct arguments

Rank 4 identities with 3 distinct arguments

Rank 4 identities with 2 distinct arguments

Higher order local identities

Examples of local identities of rank 5

Local identities of arbitrary rank

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

Zeros

Theorems

The zero modes of the periodic supersymmetric partner potentials

The wave functions (Subscript[ψ, 0]^((±)))(z)=((dn(z|m))^(∓1)) are the zero modes of the periodic supersymmetric partner potentials (SubPlus[V])(z)=2-m+2(m-1)/((dn(z|m))^2) and (SubMinus[V])(z)=2-m-2((dn(z|m))^2).

A periodic solution of the nonlinear Schrödinger equation

The function ψ(x,t)==r exp(i (p x-((p^2)-(2-(k^2)) (r^2)) t)) dn(r x-2 p r t|(k^2)) is a periodic solution of the non-linear Schrödinger equation i ((∂ψ(x,t))/(∂t))+ (((∂^2)ψ(x,t))/(∂(x^2)))+2((ψ(x,t))^2)(OverBar[ψ(x,t)])==0.

The time dependence of the x- and y-coordinates of a mathematical pendulum  in a gravitational field

The time dependence of the x - and y-coordinates of a mathematical pendulum of length l in a gravitational field with acceleration g and maximal elongation angle α is given by

x(t)==2l sin((α/2)) sn((Sqrt[(g/l)])t|(sin^2)((α/2))) dn((Sqrt[(g/l)])t|(sin^2)((α/2))), y(t)==l (2(dn^2)((Sqrt[(g/l)])t|(sin^2)((α/2)))-1).

History

– C. G. J. Jacobi (1827)
– N. H. Abel (1827)
– C. Gudermann (1838) introduced the notations dn

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

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