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std::cyl_bessel_i, std::cyl_bessel_if, std::cyl_bessel_il

double cyl_bessel_i( double ν, double x ); float cyl_bessel_if( float ν, float x ); long double cyl_bessel_il( long double ν, long double x );	(1)	(since C++17)
Promoted cyl_bessel_i( Arithmetic ν, Arithmetic x );	(2)	(since C++17)

1) Computes the regular modified cylindrical Bessel function of ν and x.

2) A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by (1). If any argument has integral type, it is cast to double. If any argument is long double, then the return type Promoted is also long double, otherwise the return type is always double.

Parameters

ν	-	the order of the function
x	-	the argument of the function)

Return value

If no errors occur, value of the regular modified cylindrical Bessel function of ν and x, that is I
ν(x) = Σ∞
k=0

(x/2)ν+2k

k!Γ(ν+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 ν>=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.

Example

#include <cmath>
#include <iostream>
int main()
{
    // spot check for ν == 0
    double x = 1.2345;
    std::cout << "I_0(" << x << ") = " << std::cyl_bessel_i(0, x) << '\n';
 
    // series expansion for I_0
    double fct = 1;
    double sum = 0;
    for(int k = 0; k < 5; fct*=++k) {
        sum += std::pow((x/2),2*k) / std::pow(fct,2);
        std::cout << "sum = " << sum << '\n';
    }        
}

Output:

I_0(1.2345) = 1.41886
sum = 1
sum = 1.381
sum = 1.41729
sum = 1.41882
sum = 1.41886

External links

Weisstein, Eric W. "Modified Bessel Function of the First Kind." From MathWorld--A Wolfram Web Resource.