Editing Elliptic integral (section)

==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 as they give the same elliptic integral. Most texts adhere to a canonical naming scheme, using the following naming conventions.

For expressing one argument:
* {{math|''α''}}, the ''[[modular angle]]''
* {{math|1=''k'' = sin ''α''}}, the ''elliptic modulus'' or ''[[eccentricity (mathematics)|eccentricity]]''
* {{math|1=''m'' = ''k''<sup>2</sup> = sin<sup>2</sup> ''α''}}, 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 {{math|''φ''}}, the ''amplitude'', or as {{math|''x''}} or {{math|''u''}}, where {{math|1=''x'' = sin ''φ'' = sn ''u''}} and {{math|sn}} is one of the [[Jacobian elliptic functions]].

Specifying the value of any one of these quantities determines the others. Note that {{math|''u''}} also depends on {{math|''m''}}. Some additional relationships involving {{math|''u''}} include
<math display="block">\cos \varphi = \operatorname{cn} u, \quad \textrm{and} \quad \sqrt{1 - m \sin^2 \varphi} = \operatorname{dn} u.</math>

The latter is sometimes called the ''delta amplitude'' and written as {{math|1=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 period]]s.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude:
<math display="block"> F(\varphi, \sin \alpha) = F\left(\varphi \mid \sin^2 \alpha\right) = F(\varphi \setminus \alpha) = F(\sin \varphi ; \sin \alpha).</math>
This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by [[Abramowitz and Stegun]] and that used in the integral tables by [[Gradshteyn and Ryzhik]].

There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, {{math|''F''(''k'', ''φ'')}}, is often encountered; and similarly {{math|''E''(''k'', ''φ'')}} for the integral of the second kind. [[Abramowitz and Stegun]] substitute the integral of the first kind, {{math|''F''(''φ'', ''k'')}}, for the argument {{mvar|φ}} in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. {{math|''E''(''F''(''φ'', ''k'') {{!}} ''k''<sup>2</sup>)}} for {{math|''E''(''φ'' {{!}} ''k''<sup>2</sup>)}}. Moreover, their complete integrals employ the ''parameter'' {{math|''k''<sup>2</sup>}} as argument in place of the modulus {{math|''k''}}, i.e. {{math|''K''(''k''<sup>2</sup>)}} rather than {{math|''K''(''k'')}}. And the integral of the third kind defined by [[Gradshteyn and Ryzhik]], {{math|Π(''φ'', ''n'', ''k'')}}, puts the amplitude {{mvar|φ}} first and not the "characteristic" {{mvar|n}}.

Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, [[Wolfram Research|Wolfram]]'s [[Mathematica]] software and [[Wolfram Alpha]] define the complete elliptic integral of the first kind in terms of the parameter {{math|''m''}}, instead of the elliptic modulus {{math|''k''}}.