Argument Notation
Incomplete elliptic integrals are functions of two arguments; complete elliptic integrals are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral). Most texts adhere to a canonical naming scheme, using the following naming conventions.
For expressing one argument:
- α, the modular angle;
- k = sin α, the elliptic modulus or eccentricity;
- m = k2 = sin2α, the parameter.
Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably.
The other argument can likewise be expressed as φ, the amplitude, or as x or u, where x = sin φ = sn u and sn is one of the Jacobian elliptic functions.
Specifying the value of any one of these quantities determines the others. Note that u also depends on m. Some additional relationships involving u include
The latter is sometimes called the delta amplitude and written as Δ(φ) = dn u. Sometimes the literature also refers to the complementary parameter, the complementary modulus, or the complementary modular angle. These are further defined in the article on quarter periods.
Read more about this topic: Elliptic Integral
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