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

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< cpp‎ | numeric‎ | special math

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

Run this code

    #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.

### See also

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl (C++17)(C++17)(C++17) | cylindrical Bessel functions (of the first kind) (function)

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