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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


ole.gif


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


             ole1.gif


is Legendre’s normal form.



Complete elliptic integral of the first kind. The integral


ole2.gif


                         ole3.gif



Incomplete elliptic integral of the second kind. The integral


ole4.gif




Complete elliptic integral of the second kind. The integral


ole5.gif


                         ole6.gif



 

Incomplete elliptic integral of the third kind. The integral


ole7.gif


where a is an arbitrary constant.




Complete elliptic integral of the third kind. The integral


ole8.gif


 



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:


             ole9.gif

             ole10.gif

             ole11.gif


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


             ole12.gif


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:


             ole13.gif


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


             ole14.gif


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


ole15.gif


and


ole16.gif



where k is the modulus and


             ole17.gif


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'




Addition formulas


ole18.gif




Derivatives 


ole19.gif  




Series expansions


ole20.gif




Catalan’s constant


ole21.gif




Identities involving elliptic functions


ole22.gif





Special values


             ole23.gif



Integrals



ole24.gif




Legendre’s relation. Legendre’s relation is



            EK' + E'K - KK' = π/2


where


             ole25.gif




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)



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