std::cyl_bessel_j,std::cyl_bessel_jf,std::cyl_bessel_jl(3)

NAME
std::cyl_bessel_j,std::cyl_bessel_jf,std::cyl_bessel_jl - std::cyl_bessel_j,std::cyl_bessel_jf,std::cyl_bessel_jl

Synopsis
   Defined in header <cmath>
   float cyl_bessel_j ( float nu, float x );
                        (since C++17)
   double cyl_bessel_j ( double nu, double x ); (until C++23)
   long double cyl_bessel_j ( long double nu, long double x );
   /* floating-point-type */ cy_bessel_j( /* floating-point-type
   */ nu, (since C++23)
                        /* floating-point-type
   */ x ); (1)
   float cyl_bessel_jf( float nu, float x ); (2) (since C++17)
   long double cyl_bessel_jl( long double nu, long double x ); (3) (since C++17)
   Additional overloads
   Defined in header <cmath>
   template< class Arithmetic1, class Arithmetic2 >
   /* common-floating-point-type */ (A) (since C++17)
       cyl_bessel_j( Arithmetic1 nu, Arithmetic2 x );
   1) Computes the cylindrical Bessel function of the first kind of nu and x.
   The library provides overloads of std::cyl_bessel_j for all cv-unqualified
   floating-point types as the type of the parameters nu and x.
   (since C++23)
   A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters
   nu - the order of the function
   x - the argument of the function

Return value
   If no errors occur, value of the cylindrical Bessel function of the first kind of nu
   and x, that is J
   nu(x) = Σ∞
   k=0
   (-1)k
   (x/2)nu+2k
   k!Γ(nu+k+1)
   (for x≥0), 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 nu≥128, the behavior is implementation-defined.

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::cyl_bessel_j(num1,
       num2) has the same effect as std::cyl_bessel_j(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::cyl_bessel_j(num1, num2) has the same effect as (until C++23)
       std::cyl_bessel_j(static_cast<double>(num1),
                        static_cast<double>(num2)).
     * Otherwise, if num1 or num2 has type float, then
       std::cyl_bessel_j(num1, num2) has the same effect as
       std::cyl_bessel_j(static_cast<float>(num1),
                        static_cast<float>(num2)).
   If num1 and num2 have arithmetic types, then std::cyl_bessel_j(num1,
   num2) has the same effect as std::cyl_bessel_j(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 (since C++23)
   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.

Example
// Run this code
 #include <cmath>
 #include <iostream>
 int main()
 {
     // spot check for nu == 0
     const double x = 1.2345;
     std::cout << "J_0(" << x << ") = " << std::cyl_bessel_j(0, x) << '\n';
     // series expansion for J_0
     double fct = 1;
     double sum = 0;
     for (int k = 0; k < 6; fct *= ++k)
     {
         sum += std::pow(-1, k) * std::pow(x / 2, 2 * k) / std::pow(fct, 2);
         std::cout << "sum = " << sum << '\n';
     }
 }

Output:
 J_0(1.2345) = 0.653792
 sum = 1
 sum = 0.619002
 sum = 0.655292
 sum = 0.653756
 sum = 0.653793
 sum = 0.653792

See also
   cyl_bessel_i
   cyl_bessel_if
   cyl_bessel_il regular modified cylindrical Bessel functions
   (C++17) (function)
   (C++17)
   (C++17)

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