Adrien Marie Legendre in the early 19th century had developed a theory of elliptic integrals and came up with the elliptic integral standard forms. These integrals Legendre actually called "elliptic functions", but it was later realized that they didn't quite deserve that title. Niels Henrik Abel, about 1823, pointed out that to consider Legendre's elliptic integrals as fundamental would be like considering:

``` x    dt
∫  ___________ = sin-1 x
0 sqrt(1 - t2)
```

as fundamental when studying the trigonometric functions, and the same difficulties result. Abel instead proposed that one study, just as in trigonometry, the upper limit of integration for the elliptic integrals, but probably due to Abel's untimely death in 1829, it was ultimately Carl Jacobi who, in his landmark book published in the same year, Fundamenta Nova Theoriae Functionum Ellipticarum (Foundations of a New Theory of Elliptic Functions, I think) that gave substance to this idea, and the functions that result are called the Jacobi Elliptic Functions in his honor. Jacobi developed his functions, just as Abel suggested, by inverting the elliptic integral of the first kind:

```     φ          dφ
u = ∫   ___________________
0  sqrt(1 - k2 sin2 φ)
```

The value φ Jacobi considered fundamental, and he called it the amplitude corresponding to the argument u, and he used the notation:

```φ = am (u, k) = am u
```

when the elliptic modulus k is understood, and he wrote:

```sqrt(1 - k2 sin2 φ) = Δ am u = Δ φ
```

The elliptic functions arise from taking the sine and cosine of the amplitude, and so we have the three Jacobi elliptic functions: sin am u, cos am u, and Δ am u. This notation however, was found to be quite cumbersome, and so Christoph Gudermann (famous as the mentor of Karl Weierstrass) later proposed to abbreviate the notation by using sn u = sin am u, cn u = cos am u, and dn u = Δ am u, by analogy with the trigonometric functions. There is also a notation for the ratios of two Jacobi elliptic functions, consisting of the first letter of the numerator function and the first letter of the denominator function, e.g. sn u/cn u = sc u (which Gudermann called tn u for obvious reasons), and cn u/dn u = cd u, and an analogous notation for the reciprocals of an elliptic function done by reversing the letters, e.g. 1/cn u = nc u. This is the modern, accepted notation that is in use today. It is immediately obvious from the above definition, that:

sn2 u + cn2 u = 1

and also:

k2sn2 u + dn2 u = 1

From the elliptic integral that was used to define sn u, it is clear that as the amplitude φ approaches 0, the argument u also approaches 0, hence, sn 0 = 0, and from the identities above we have cn 0 = dn 0 = 1. If, on the other hand, we negate the amplitude, the argument also becomes negative, hence sn -u = -sn u, i.e. sn u is an odd function. Again, using the above identities, it is equally obvious that cn u and dn u are both even functions.

Simple formulae for the derivative of these elliptic functions are also easily obtained:

```d         d                      d am φ
__ sn u = __ sin am φ = cos am φ ______ = cn u dn u
du        du                       du
```

and similarly

```d
__ cn u = - sn u dn u
du
d
__ dn u = -k2 sn u cn u
du
```

In the study of the trigonometric functions we have the transcendental number π, with the property that 2π is the period of the sine and cosine functions. A convenient definition of π for this purpose would be:

```π    1     dx
_ = ∫  ____________ = sin-1 1
2    0 sqrt(1 - x2)
```

This gives a quarter period for the sine and cosine functions.

It may be shown that for the Jacobi elliptic functions there exists a whole family of π-like transcendents as a function of the elliptic modulus k. Analogously, if we take the complete elliptic integral of the first kind:

```     1           dx                π/2         dp
K = ∫  ________________________ = ∫    ___________________ = F(k|π/2)
0 sqrt((1 - x2)(1 - k2x2))    0   sqrt(1 - k2 sin2 φ)
```

we obtain a transcendental number that we shall see is related to the periodicity of the Jacobi elliptic functions. A second transcendental number is also obtainable, using the same formula, but with the complementary modulus k' = sqrt(1 - k2), K' = F(k'|π/2). These values are called the quarter periods of the Jacobi elliptic functions.

In particular, it may readily be seen that sn u is periodic on the real axis with period 4K. Same too with cn u. It may also be shown that dn u has a real period of 2K, as does the function sc u = sn u/cn u. These special values for the elliptic functions are also seen to hold:

sn 2K = 0

cn 2K = -1

dn 2K = 1

sn 4K = 0

cn 4K = 1

dn 4K = 1

where the elliptic modulus is k, of course.

One of Jacobi's important discoveries about these functions is what happens for imaginary arguments. He was able to derive what are now called the Jacobi imaginary transformations, given by these formulae:

sn(iu, k) = i sc(u, k')

cn(iu, k) = nc(u, k')

dn(iu, k) = dc(u, k')

It is obvious from these relations that sn u is imaginary for imaginary values of its argument, and cn u and dn u are real for imaginary arguments. By using these formulae, it is easy to see that the Jacobi elliptic functions also possess a second period, the functions being doubly periodic:

sn u has periods 4K and 2iK'

cn u has periods 4K and 2K + 2iK'

dn u has periods 2K and 4iK'

If we take the parallelogram-shaped cells on the complex plane defined by these periods, it may be seen that the functions behave within any particular cell the same way they do in any other cell.

It may be shown sn u has zeroes at 2mK + 2niK', and simple poles at 2mK | (2n+1)iK', for any integers m and n. In the same way cn u has zeroes at (2m + 1)K + 2niK' and simple poles at 2mK + (2n + 1)iK', and dn u has zeroes at (2m+1)K + (2n+1)K' and poles at 2mK + (2n + 1)iK'.

The Jacobi elliptic functions also obey algebraic addition theorems, similar to trigonometric identities, so that we have:

sn (a + b) = (sn a cn b dn b + sn b cn a dn a)/(1 - k2 sn2 a sn2 b)

cn (a + b) = (cn a cn b - sn a sn b dn a dn b)/(1 - k2 sn2 a sn2 b)

dn (a + b) = (dn a dn b - k2sn a sn b cn a cn b)/(1 - k2 sn2 a sn2 b)

Together with Jacobi's imaginary transform, these formulas may be used to compute the values of arbitrary complex arguments of the Jacobi elliptic functions if one has a way of calculating them for real values of the argument.

Algorithms based on Landen's transformation or the Gaussian arithmetic-geometric mean may be used to calculate specific values for these functions for numerical applications The Jacobi theta functions, with their very rapidly converging infinite series representations are closely related to the Jacobi elliptic functions (they are to these functions what the exponential function is to the trigonometric functions), and may be used for numerical calculation as well. A table for the elliptic integral of the first kind may be used to obtain rough values for them as well.

The Jacobi elliptic functions have many applications in physics and engineering. They are the exact solutions to the equations of motion for a simple pendulum for example. They are also used in analog and digital filter design, and the elliptic filters based on them, because they have equiripple both in the passband and the stopband, are the lowest order filters for a given transition bandwidth; a typical Butterworth or Chebyshev filter with the same transition band would have nearly double the order (their phase response is, however, poor for audio applications).

References:

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions, 1964.

Arthur L. Baker, Elliptic Functions, 1890.

Roland Bulirsch, Numerical Calculation of Elliptic Integrals and Elliptic Functions, Numer. Math. 7, pp. 78-90, 1965.

Harris Hancock, Elliptic Integrals, 1917.

Harris Hancock, Lectures on the Theory of Elliptic Functions, 1910.

E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 1917.

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