tmp/tmpcebv17uo/{from.md → to.md}
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#### Irregular modified cylindrical Bessel functions <a id="sf.cmath.cyl_bessel_k">[[sf.cmath.cyl_bessel_k]]</a>
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``` cpp
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double cyl_bessel_k(double nu, double x);
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float cyl_bessel_kf(float nu, float x);
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long double cyl_bessel_kl(long double nu, long double x);
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```
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*Effects:* These functions compute the irregular modified cylindrical
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Bessel functions of their respective arguments `nu` and `x`.
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*Returns:* $$%
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\mathsf{K}_\nu(x) =
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(\pi/2)i^{\nu+1} ( \mathsf{J}_\nu(ix)
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+ i \mathsf{N}_\nu(ix)
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)
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=
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\left\{
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\begin{array}{cl}
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\displaystyle
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\frac{\pi}{2}
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\frac{\mathsf{I}_{-\nu}(x) - \mathsf{I}_{\nu}(x)}
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{\sin \nu\pi },
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& \mbox{for $x \ge 0$ and non-integral $\nu$}
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\\
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\\
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\displaystyle
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\frac{\pi}{2}
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\lim_{\mu \rightarrow \nu} \frac{\mathsf{I}_{-\mu}(x) - \mathsf{I}_{\mu}(x)}
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{\sin \mu\pi },
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& \mbox{for $x \ge 0$ and integral $\nu$}
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\end{array}
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\right.$$ where $\nu$ is `nu` and x is `x`.
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*Remarks:* The effect of calling each of these functions is
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*implementation-defined* if `nu >= 128`.
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See also [[sf.cmath.cyl_bessel_i]], [[sf.cmath.cyl_bessel_j]],
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[[sf.cmath.cyl_neumann]].
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