[sf.cmath.cyl.neumann]

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floating-point-type cyl_neumann(floating-point-type nu, floating-point-type x);
float        cyl_neumannf(float nu, float x);
long double  cyl_neumannl(long double nu, long double x);

Effects: These functions compute the cylindrical Neumann functions, also known as the cylindrical Bessel functions of the second kind, of their respective arguments nu and x.

Returns:

{N ν (x), where {N ν is given by Formula 29.31, ν is nu, and x is x .}_{ν is given by Formula 29.31, ν is nu, and x is x .}}_{ν (x), where {N ν is given by Formula 29.31, ν is nu, and x is x .}_{ν is given by Formula 29.31, ν is nu, and x is x .}}

\begin{matrix} {N ν (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} {J ν (x) cos ν π - {J - ν (x) sin ν π, for x \geq 0 and non-integral ν}_{- ν (x) sin ν π, for x \geq 0 and non-integral ν lim}}_{ν (x) cos ν π - {J - ν (x) sin ν π, for x \geq 0 and non-integral ν lim}_{- ν (x) sin ν π, for x \geq 0 and non-integral ν lim}} \end{matrix}}_{ν (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} {J ν (x) cos ν π - {J - ν (x) sin ν π, for x \geq 0 and non-integral ν lim}_{- ν (x) sin ν π, for x \geq 0 and non-integral ν lim}}_{ν (x) cos ν π - {J - ν (x) sin ν π, for x \geq 0 and non-integral ν lim}_{- ν (x) sin ν π, for x \geq 0 and non-integral ν lim {μ \to ν}_{μ \to ν}}} \end{matrix}} \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

29 Numerics library [numerics]

29.7 Mathematical functions for floating-point types [c.math]

29.7.6 Mathematical special functions [sf.cmath]

29.7.6.11 Cylindrical Neumann functions [sf.cmath.cyl.neumann]