# std::comp_ellint_1, std::comp_ellint_1f, std::comp_ellint_1l

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< cpp‎ | numeric‎ | special math

double comp_ellint_1( double k ); float comp_ellint_1( float k ); long double comp_ellint_1( long double k ); float comp_ellint_1f( float k ); long double comp_ellint_1l( long double k ); | (1) | (since C++17)
double comp_ellint_1( IntegralType k ); | (2) | (since C++17)

1) Computes the complete elliptic integral of the first kind of `k`.
2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

### Parameters

k | - | elliptic modulus or eccentricity (a value of a floating-point or integral type)

### Return value

If no errors occur, value of the complete elliptic integral of the first kind of `k`, that is ellint_1(k,π/2), 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 period of a pendulum of length l, given acceleration due to gravity g, and initial angle θ equals 4√l/gK(sin2
(θ/2)), where K is `std::comp_ellint_1`.

### Example

Run this code

    #include <cmath>
    #include <iostream>
    int main()
    {
        double hpi = std::acos(-1)/2;
        std::cout << "K(0) = " << std::comp_ellint_1(0) << '\n'
                  << "π/2 = " << hpi << '\n'
                  << "K(0.5) = " << std::comp_ellint_1(0.5) << '\n'
                  << "F(0.5, π/2) = " << std::ellint_1(0.5, hpi) << '\n';
        std::cout << "Period of a pendulum length 1 m at 90 degree initial angle is "
                  << 4*std::sqrt(1/9.80665)*
                     std::comp_ellint_1(std::pow(std::sin(hpi/2),2)) << " s\n";
    }

Output:

    K(0) = 1.5708
    π/2 = 1.5708
    K(0.5) = 1.68575
    F(0.5, π/2) = 1.68575
    Period of a pendulum length 1 m at 90 degree initial angle is 2.15324 s

### External links

Weisstein, Eric W. "Complete Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource.

### See also

ellint_1ellint_1fellint_1l (C++17)(C++17)(C++17) | (incomplete) elliptic integral of the first kind (function)

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