1 #ifndef STAN_MATH_PRIM_FUN_BESSEL_SECOND_KIND_HPP
2 #define STAN_MATH_PRIM_FUN_BESSEL_SECOND_KIND_HPP
3 
4 #include <stan/math/prim/meta.hpp>
5 #include <stan/math/prim/functor/apply_scalar_binary.hpp>
6 #include <boost/math/special_functions/bessel.hpp>
7 
8 namespace stan {
9 namespace math {
10 
11 /**
12  *
13    \f[
14    \mbox{bessel\_second\_kind}(v, x) =
15    \begin{cases}
16      \textrm{error} & \mbox{if } x \leq 0 \\
17      Y_v(x) & \mbox{if } x > 0 \\[6pt]
18      \textrm{NaN} & \mbox{if } x = \textrm{NaN}
19    \end{cases}
20    \f]
21 
22    \f[
23    \frac{\partial\, \mbox{bessel\_second\_kind}(v, x)}{\partial x} =
24    \begin{cases}
25      \textrm{error} & \mbox{if } x \leq 0 \\
26      \frac{\partial\, Y_v(x)}{\partial x} & \mbox{if } x > 0 \\[6pt]
27      \textrm{NaN} & \mbox{if } x = \textrm{NaN}
28    \end{cases}
29    \f]
30 
31    \f[
32    Y_v(x)=\frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)}
33    \f]
34 
35    \f[
36    \frac{\partial \, Y_v(x)}{\partial x} = \frac{v}{x}Y_v(x)-Y_{v+1}(x)
37    \f]
38  *
39  */
40 template <typename T2, require_arithmetic_t<T2>* = nullptr>
bessel_second_kind(int v,const T2 z)41 inline T2 bessel_second_kind(int v, const T2 z) {
42   return boost::math::cyl_neumann(v, z);
43 }
44 
45 /**
46  * Enables the vectorised application of the bessel second kind function, when
47  * the first and/or second arguments are containers.
48  *
49  * @tparam T1 type of first input
50  * @tparam T2 type of second input
51  * @param a First input
52  * @param b Second input
53  * @return Bessel second kind function applied to the two inputs.
54  */
55 template <typename T1, typename T2, require_any_container_t<T1, T2>* = nullptr>
bessel_second_kind(const T1 & a,const T2 & b)56 inline auto bessel_second_kind(const T1& a, const T2& b) {
57   return apply_scalar_binary(a, b, [&](const auto& c, const auto& d) {
58     return bessel_second_kind(c, d);
59   });
60 }
61 
62 }  // namespace math
63 }  // namespace stan
64 
65 #endif
66