Preface

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An elliptic function is a function meromorphic on the complec plane ℂ that is periodic in two directions. Elliptic integrals were first encountered by John Wallis around 1655. Historically, elliptic functions were first discovered by Niels Henrik Abel (1802--1829) as inverse functions of elliptic integrals. However, their theory was developed by the German mathematician Carl Gustav Jacob Jacobi (1804--1851), who called December 23, 1751 the birthday of elliptic functions, because on that day Euler (1017--1783) was asked his opinion of a paper by Fagnano (1715--1797) on arcs of lemniscates.

There are known two standard forms of elliptic functions: Jacobi elliptic functions and Weierstrass elliptic functions. Jacobi elliptic functions arise as solutions to differential equations of the form

and Weierstrass elliptic functions arise as solutions to differential equations of the form

 

There are known four incomplete elliptic integrals: \begin{align} F (\phi , k ) &= \int_0^{\phi} \frac{{\text d}\theta}{\sqrt{1 - k^2 \sin^2 \theta}} = \int_0^{\sin\phi} \frac{{\text d}t}{\sqrt{1 - t^2} \,\sqrt{1 - k^2 t^2}} , \label{elliptic.1} \\ E (\phi , k ) &= \int_0^{\phi} \sqrt{1 - k^2 \sin^2 \theta}\,{\text d}\theta = \int_0^{\sin\phi} \frac{\sqrt{1 - k^2 t^2}}{\sqrt{1- t^2}}\,{\text d}t , \label{elliptic.2} \\ D (\phi , k ) &= \int_0^{\phi} \frac{\sin^2 \theta \,{\text d}\theta}{\sqrt{1 - k^2 \sin^2 \theta}} = \int_0^{\sin\phi} \frac{t^2 {\text d}t}{\sqrt{1 - t^2} \,\sqrt{1 - k^2 t^2}} = \frac{1}{k^2} \left[ F(\phi , k) - E (\phi , k) \right] , \label{elliptic.3} \\ \Pi (\phi , \alpha^2 , k ) &= \int_0^{\phi} \frac{{\text d}\theta}{\sqrt{1 - k^2 \sin^2 \theta} \left( 1 - \alpha^2 \sin^2 \theta \right)} = \int_0^{\sin\phi} \frac{{\text d}t}{\sqrt{1- t^2} \,\sqrt{1 - k^2 t^2} \left( 1 - \alpha^2 t^2 \right)} . \label{elliptic.4} \end{align} The cases with ϕ = π/2 are known as the complete integrals: \begin{align} & \quad {\bf complete }\ {\bf elliptic } \ {\bf integral }\ {\bf of }\ {\bf the }\ {\bf first }\ {\bf kind:} \notag \\ K(k) &= F \left( \frac{\pi}{2} , k \right) = \int_0^1 \frac{{\text d}t}{\sqrt{1- t^2}\,\sqrt{1 - k^2 t^2}} , \label{elliptic.5} \\ & \quad {\bf complete }\ {\bf elliptic } \ {\bf integral }\ {\bf of }\ {\bf the }\ {\bf second }\ {\bf kind:} \notag \\ E(k) &= E\left( \frac{\pi}{2} , k \right) = \int_0^1 \sqrt{\frac{1 - k^2 t^2}{1-t^2}} \,{\text d}t , \label{elliptic.6} \\ D(k) &= D \left( \frac{\pi}{2} , k \right) = \frac{1}{k^2} \left[ K(k) - E(k) \right] , \label{elliptic.7} \\ & \quad {\bf complete }\ {\bf elliptic } \ {\bf integral }\ {\bf of }\ {\bf the }\ {\bf third }\ {\bf kind:} \notag \\ \Pi \left( \alpha^2 k \right) &= \Pi \left( \frac{\pi}{2} , \alpha^2 , k \right) . \label{elliptic.8} \end{align} Mathematica has dedicated commands for evaluating elliptic integrals:
EllipticK[ m ]
EllipticE[ m ]
EllipticPi[ n, m ]

		Plot[{Callout[ElliptiK[m], "EllipticK(m)", Left, Frame -> True, FrameMargins -> 10], Callout[EllipticE[m], "EllipticE(m)", Right, Frame -> True, CalloutStyle -> Yellow], Callout[(EllipticK[m] - EllipticE[m])/m^2, "EllipticD(m)", Above, Frame -> True, Background -> Pink]}, {m, -1, 1}, PlotStyle -> Thickness[0.01]]
Complete elliptic functions K(m), E(m), and D(m).		Mathematica code

		Plot3D[EllipticPi[n, m], {n, -1, 1}, {m, -1, 1}, PlotStyle -> Directive[Orange, Specularity[White, 40]]]
Elliptic integral of the third kind Π(n, m).		Mathematica code

Using the binomial series

and the integral we find the series expansions of the complete elliptic integrals: \begin{align} K(m) &= \int_0^1 \frac{{\text d}t}{\sqrt{1- t^2}\,\sqrt{1 - m^2 t^2}} = \frac{\pi}{2} \left\{ 1 + \sum_{n\ge 1} \left[ \frac{(2n-1)!!}{(2n)!!} \right]^2 m^n \right\} = _2F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; m \right) \frac{\pi}{2} , \label{elliptic.9} \\ E(m) &= \int_0^1 \sqrt{\frac{1 - m^2 t^2}{1-t^2}} \,{\text d}t = \frac{\pi}{2} \left\{ 1 + \sum_{n\ge 1} \left[ \frac{(2n-1)!!}{(2n)!!} \right]^2 \frac{m^n}{2n-1} \right\} = _2F_1 \left( -\frac{1}{2}, \frac{1}{2}; 1; m \right) \frac{\pi}{2} . \label{elliptic.10} \end{align} Some limiting values: and

Jacobi Elliptic Functions

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There are several types of elliptic functions including the Weierstrass elliptic functions as well as related theta functions but the most common elliptic functions are the Jacobian elliptic functions, based on the inverses of the three types of elliptic integrals.

There are twelve Jacobi elliptic functions denoted by pq(x,m), where p and q are any of the letters c, s, n, and d. (Functions of the form pp(x,m) are trivially set to unity for notational completeness). The variable x is the argument, and m is the parameter, both of which may be complex. The first variable might be given in terms of the amplitude φ, or more commonly, in terms of x given below. The second variable might be given in terms of the parameter m, or as the elliptic modulus k, where k² = m, or in terms of the modular angle α, where m = sin²  α. The complements of k and m are defined as m' =1-m and \( k' = \sqrt {1-k^2} = \sqrt{1-m} . \) These four terms are used below without comment to simplify various expressions.

The elliptic sine sn x is given by

sn is pronounced roughly as “ess-en”. (Try saying three times fast: “ess-en u is the sine of the amplitude of u”). A plot of ()sn uvs. u looks very similar to a sine wave as seen in Figure

The elliptic cosine cn x is given by

The delta amplitude dn x is Here, the angle φ is called the amplitude.

Some basic relations:

Limiting values of k:

The Jacobi imaginary transformations relate various functions of the imaginary variable

First order derivatives: Second order differential equations:

Airy functions are solutions to the Airy differential equation
y''[x] - x y[x] == 0
There are two linearly independent solutions, called by Mathematica as
AiryAi[x] and AiryBi[x].

Example: The arclength of the ellipse Example: The arclength of Bernoulli’s Lemniscate

 

The principal difference between the Jacobiand the Weierstrass elliptic integrals is in the number of poles in each fundamentalcell. While the Jacobi elliptic functions has two simple poles per cell and can beconsiders as a solution to the differential equation the Weierstrass elliptic functions have just one double pole and is a solution to

 

Theta functions are the elliptic analogs of the exponential function and are typically written as θ_𝑎(u, q), where 𝑎 ranges from 1 to 4 to represent the fours variations of the theta function, u is the argument of the function and q is the Nome, given as where Example: ???

 

References

 

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