std::ellint_1,std::ellint_1f,std::ellint_1l(3)

NAME
std::ellint_1,std::ellint_1f,std::ellint_1l - std::ellint_1,std::ellint_1f,std::ellint_1l

Synopsis
   Defined in header <cmath>
   float ellint_1 ( float k, float phi );
                        (since C++17)
   double ellint_1 ( double k, double phi ); (until C++23)
   long double ellint_1 ( long double k, long double phi );
   /* floating-point-type */ ellint_1( /* floating-point-type */
   k, (since C++23)
                        /* floating-point-type */
   phi ); (1)
   float ellint_1f( float k, float phi ); (2) (since C++17)
   long double ellint_1l( long double k, long double phi ); (3) (since C++17)
   Additional overloads
   Defined in header <cmath>
   template< class Arithmetic1, class Arithmetic2 >
   /* common-floating-point-type */ (A) (since C++17)
       ellint_1( Arithmetic1 k, Arithmetic2 phi );
   1-3) Computes the incomplete elliptic integral of the first kind of k and phi.
   The library provides overloads of std::ellint_1 for all cv-unqualified
   floating-point types as the type of the parameters k and phi.
   (since C++23)
   A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters
   k - elliptic modulus or eccentricity (a floating-point or integer value)
   phi - Jacobi amplitude (a floating-point or integer value, measured in radians)

Return value
   If no errors occur, value of the incomplete elliptic integral of the first kind of k
   and phi, that is ∫phi
   0
   dθ
   √
   1-k2
   sin2
   θ
   , is returned.

Error handling
   Errors may be reported as specified in math_errhandling:
     * If the argument is NaN, NaN is returned and domain error is not reported.
     * If |k|>1, a domain error may occur.

Notes
   Implementations that do not support C++17, but support ISO 29124:2010, provide this
   function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value
   at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before
   including any standard library headers.
   Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1),
   provide this function in the header tr1/cmath and namespace std::tr1.
   An implementation of this function is also available in boost.math.
   The additional overloads are not required to be provided exactly as (A). They only
   need to be sufficient to ensure that for their first argument num1 and second
   argument num2:
     * If num1 or num2 has type long double, then std::ellint_1(num1,
       num2) has the same effect as std::ellint_1(static_cast<long
       double>(num1),
                     static_cast<long double>(num2)).
     * Otherwise, if num1 and/or num2 has type double or an integer type,
       then std::ellint_1(num1, num2) has the same effect as (until C++23)
       std::ellint_1(static_cast<double>(num1),
                     static_cast<double>(num2)).
     * Otherwise, if num1 or num2 has type float, then
       std::ellint_1(num1, num2) has the same effect as
       std::ellint_1(static_cast<float>(num1),
                     static_cast<float>(num2)).
   If num1 and num2 have arithmetic types, then std::ellint_1(num1, num2)
   has the same effect as std::ellint_1(static_cast</*
   common-floating-point-type */>(num1),
                 static_cast</* common-floating-point-type */>(num2)),
   where /* common-floating-point-type */ is the floating-point type with
   the greatest floating-point conversion rank and greatest
   floating-point conversion subrank between the types of num1 and num2, (since C++23)
   arguments of integer type are considered to have the same
   floating-point conversion rank as double.
   If no such floating-point type with the greatest rank and subrank
   exists, then overload resolution does not result in a usable candidate
   from the overloads provided.

Example
// Run this code
 #include <cmath>
 #include <iostream>
 #include <numbers>
 int main()
 {
     const double hpi = std::numbers::pi / 2.0;
     std::cout << "F(0,π/2) = " << std::ellint_1(0, hpi) << '\n'
               << "F(0,-π/2) = " << std::ellint_1(0, -hpi) << '\n'
               << "π/2 = " << hpi << '\n'
               << "F(0.7,0) = " << std::ellint_1(0.7, 0) << '\n';
 }

Output:
 F(0,π/2) = 1.5708
 F(0,-π/2) = -1.5708
 π/2 = 1.5708
 F(0.7,0) = 0

See also
   comp_ellint_1
   comp_ellint_1f
   comp_ellint_1l (complete) elliptic integral of the first kind
   (C++17) (function)
   (C++17)
   (C++17)

External links
   Weisstein, Eric W. "Elliptic Integral of the First Kind." From MathWorld — A Wolfram
   Web Resource.