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Difference between revisions of "cpp/numeric/special functions/ellint 1"

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{{cpp/title|ellint_1|ellint_1f|ellint_1l}}
 
{{cpp/title|ellint_1|ellint_1f|ellint_1l}}
−
{{cpp/numeric/special_math/navbar}}
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{{cpp/numeric/special_functions/navbar}}
 
{{dcl begin}}
 
{{dcl begin}}
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{{dcl |num=1|since=c++17|
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{{dcl header|cmath}}
−
double      ellint_1( double k, double φ );
+
{{dcl rev multi|num=1|since1=c++17|dcl1=
−
float      ellint_1f( float k, float φ  );
+
float      ellint_1 ( float k, float phi );
−
long double ellint_1l( long double k, long double φ );
+
double      ellint_1 ( double k, double phi );
 +
long double ellint_1 ( long double k, long double phi );
 +
|since2=c++23|dcl2=
 +
/* floating-point-type */ ellint_1( /* floating-point-type */ k,
 +
                                    /* floating-point-type */ phi );
 
}}
 
}}
−
{{dcl |num=2|since=c++17|
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{{dcl|num=2|since=c++17|
−
Promoted    ellint_1( Arithmetic k, Arithmetic φ );
+
float      ellint_1f( float k, float phi );
 +
}}
 +
{{dcl|num=3|since=c++17|
 +
long double ellint_1l( long double k, long double phi );
 +
}}
 +
{{dcl h|[[#Notes|Additional overloads]]}}
 +
{{dcl header|cmath}}
 +
{{dcl|num=A|since=c++17|
 +
template< class Arithmetic1, class Arithmetic2 >
 +
/* common-floating-point-type */
 +
    ellint_1( Arithmetic1 k, Arithmetic2 phi );
 
}}
 
}}
 
{{dcl end}}
 
{{dcl end}}
  
−
@1@ Computes the [[enwiki:Elliptic_integral#Elliptic_integral_of_the_first_kind|incomplete elliptic integral of the first kind]] of {{tt|k}} and {{tt|φ}}.
+
@1-3@ Computes the {{enwiki|Elliptic integral#Elliptic integral of the first kind|incomplete elliptic integral of the first kind}} of {{c|k}} and {{c|phi}}.{{rev inl|since=c++23| The library provides overloads of {{tt|std::ellint_1}} for all cv-unqualified floating-point types as the type of the parameters {{c|k}} and {{c|phi}}.}}
−
@2@ A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by {{v|1}}. If any argument has [[cpp/types/is_integral|integral type]], it is cast to {{c|double}}. If any argument is {{c|long double}}, then the return type {{tt|Promoted}} is also {{c|long double}}, otherwise the return type is always {{c|double}}.
+
@A@ Additional overloads are provided for all other combinations of arithmetic types.
  
 
===Parameters===
 
===Parameters===
 
{{par begin}}
 
{{par begin}}
−
{{par | k | elliptic modulus or eccentricity (a value of a floating-point or integral type)}}
+
{{par|k|elliptic modulus or eccentricity (a floating-point or integer value)}}
−
{{par | φ | Jacobi amplitude (a value of floating-point or integral type, measured in radians)}}
+
{{par|phi|Jacobi amplitude (a floating-point or integer value, measured in radians)}}
 
{{par end}}
 
{{par end}}
  
 
===Return value===
 
===Return value===
−
If no errors occur, value of the incomplete elliptic integral of the first kind of {{tt|k}} and {{tt|φ}}, that is {{math|{{minteg|0|φ|{{mfrac|dθ|{{mrad|1-k{{su|p=2}}sin{{su|p=2}}θ}}}}}}}}, is returned.
+
If no errors occur, value of the incomplete elliptic integral of the first kind of {{c|k}} and {{c|phi}}, that is {{math|{{minteg|0|phi|{{mfrac|dθ|{{mrad|1-k{{su|p=2}}sin{{su|p=2}}θ}}}}}}}}, is returned.
  
 
===Error handling===
 
===Error handling===
−
Errors may be reported as specified in [[cpp/numeric/math/math_errhandling|math_errhandling]]
+
Errors may be reported as specified in {{lc|math_errhandling}}:
−
 
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* If the argument is NaN, NaN is returned and domain error is not reported.
−
* If the argument is NaN, NaN is returned and domain error is not reported
+
* If {{math|{{!}}k{{!}}>1}}, a domain error may occur.
−
* If {{math|{{!}}k{{!}}>1}}, a domain error may occur
+
  
 
===Notes===
 
===Notes===
−
{{cpp/numeric/special_math/older_impl_note}}
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{{cpp/numeric/special functions/older impl note}}
  
−
An implementation of this function is also [http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/math_toolkit/ellint/ellint_1.html available in boost.math]
+
An implementation of this function is also available in [https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/ellint/ellint_1.html boost.math].
 +
 
 +
{{cpp/numeric/special functions/additional overload note|ellint_1}}
  
 
===Example===
 
===Example===
−
{{example|code=
+
{{example
 +
|code=
 
#include <cmath>
 
#include <cmath>
 
#include <iostream>
 
#include <iostream>
 +
#include <numbers>
 +
 
int main()
 
int main()
 
{
 
{
−
     double hpi = std::acos(-1)/2;
+
     const double hpi = std::numbers::pi / 2.0;
−
     std::cout << "F(0,π/2) = " << std::ellint_1(0, hpi) << '\n'
+
   
 +
     std::cout << "F(0,π/2) = " << std::ellint_1(0, hpi) << '\n'
 
               << "F(0,-π/2) = " << std::ellint_1(0, -hpi) << '\n'
 
               << "F(0,-π/2) = " << std::ellint_1(0, -hpi) << '\n'
−
               << "π/2 = " << hpi << '\n'
+
               << "π/2       = " << hpi << '\n'
−
               << "F(0.7,0) = " << std::ellint_1(0.7, 0) << '\n';
+
               << "F(0.7,0) = " << std::ellint_1(0.7, 0) << '\n';
 
}
 
}
 
|output=
 
|output=
−
F(0,π/2) = 1.5708
+
F(0,π/2) = 1.5708
 
F(0,-π/2) = -1.5708
 
F(0,-π/2) = -1.5708
−
π/2 = 1.5708
+
π/2       = 1.5708
−
F(0.7,0) = 0
+
F(0.7,0) = 0
 
}}
 
}}
−
 
−
===External links===
 
−
[http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html Weisstein, Eric W. "Elliptic Integral of the First Kind."] From MathWorld--A Wolfram Web Resource.
 
  
 
===See also===
 
===See also===
 
{{dsc begin}}
 
{{dsc begin}}
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{{dsc inc | cpp/numeric/special_math/dsc comp_ellint_1}}
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{{dsc inc|cpp/numeric/special_functions/dsc comp_ellint_1}}
 
{{dsc end}}
 
{{dsc end}}
  
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[[de:cpp/numeric/special_math/ellint_1]]
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===External links===
−
[[es:cpp/numeric/special_math/ellint_1]]
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{{eli|[https://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html Weisstein, Eric W. "Elliptic Integral of the First Kind."] From MathWorld — A Wolfram Web Resource.}}
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[[fr:cpp/numeric/special_math/ellint_1]]
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[[it:cpp/numeric/special_math/ellint_1]]
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{{langlinks|de|es|fr|it|ja|pt|ru|zh}}
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[[ja:cpp/numeric/special_math/ellint_1]]
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[[pt:cpp/numeric/special_math/ellint_1]]
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[[ru:cpp/numeric/special_math/ellint_1]]
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[[zh:cpp/numeric/special_math/ellint_1]]
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Latest revision as of 15:11, 22 March 2023

 
C++
 
Numerics library
Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Floating-point environment (C++11)
Complex numbers
Numeric array (valarray)
Pseudo-random number generation
Factor operations
(C++17)
(C++17)
Interpolations
(C++20)
(C++20)
Saturation arithmetic
(C++26)
(C++26)
(C++26)
(C++26)
(C++26)

Generic numeric operations
Bit operations
(C++20)    
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++23)
(C++20)
 
Mathematical special functions
 
Defined in header <cmath>
(1)
float       ellint_1 ( float k, float phi );

double      ellint_1 ( double k, double phi );

long double ellint_1 ( long double k, long double phi );
 (since C++17)
(until C++23)
/* floating-point-type */ ellint_1( /* floating-point-type */ k,
                                    /* floating-point-type */ phi );
 (since C++23)
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)
Defined in header <cmath>
template< class Arithmetic1, class Arithmetic2 >

/* common-floating-point-type */

    ellint_1( Arithmetic1 k, Arithmetic2 phi );
 (A) (since C++17)
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.

Contents

[edit] Parameters

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

[edit] Return value

If no errors occur, value of the incomplete elliptic integral of the first kind of k and phi, that is ∫phi0
dθ
√1-k2sin2θ
, is returned.

[edit] 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.

[edit] 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 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))    .
(until C++23)

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, 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.

(since C++23)

[edit] 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

[edit] See also

(C++17)(C++17)(C++17)
 (complete) elliptic integral of the first kind
(function) [edit]

[edit] External links

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