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Elliptic integrals of the first, second and third kinds. Jacobian elliptic functions. Identities, formulas, series expansions, derivatives, integrals.

Incomplete elliptic integral of the first kind. The integral

where the second integral (in v) is obtained from the integral in θ by the change of variables v = sin θ. The upper limits f and x on the integrals are related by x = sin f. The variable f is called the amplitude of u and often written am u. The constant k (0 < k < 1) is the modulus.

The integral expressed in terms of θ

is Legendre’s normal form.

Complete elliptic integral of the first kind. The integral

Incomplete elliptic integral of the second kind. The integral

Complete elliptic integral of the second kind. The integral

Incomplete elliptic integral of the third kind. The integral

where a is an arbitrary constant.

Complete elliptic integral of the third kind. The integral

Jacobi’s elliptic functions. The Jacobi elliptic functions consist of sn u, cn u, and dn u and a few more functions defined directly from them. The functions sn u, cn u, and dn u are defined from 1) above as follows:

As a consequence of these definitions the following hold:

sn2 u + cn2 u = 1

k2sn2 u + dn2 u = 1

In addition to sn u, cn u, and dn u we define the following functions:

Inverse functions. The inverse functions of sn u, cn u, and dn u are denoted by sn-1 x, cn-1 x, and dn-1 x, respectively.

Reciprocals. The reciprocals of sn u, cn u, and dn u are

Note how the names of the reciprocals are formed. They are formed by reversing the letters in the names of the functions. For example, the letters “sn” in the name of the sine function are reversed to “ns” to form the name of the reciprocal. Similarly, the letters cn of the cosine function are reversed to nc to form the name of the reciprocal. This makes the names easy to remember.

Quotients. The following quotients are defined:

Again note the system used for forming the names of the quotients. The name of the quotient is formed from the first letter in the name in the numerator plus the first letter in the name in the denominator. Thus in the quotient

the name sc was formed from the first letter s in sn and the first letter c in cn. This makes the names easy to remember.

Periods of elliptic functions. Let

and

where k is the modulus and

is the complementary modulus. Then the functions sn u, cn u, and dn u are doubly periodic functions with periods as follows:

sn u has periods 4K and 2iK'

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

dn u has periods 2K and 4iK'

Derivatives

Series expansions

Catalan’s constant

Identities involving elliptic functions

Special values

Integrals

Legendre’s relation. Legendre’s relation is

EK' + E'K - KK' = π/2

where

References.

1. James/James. Mathematics Dictionary

2. The International Dictionary of Applied Mathematics. D. Van Nostrand Co.

3. Murray R. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)