1 #ifndef STAN_MATH_REV_FUN_ASINH_HPP
2 #define STAN_MATH_REV_FUN_ASINH_HPP
3 
4 #include <stan/math/prim/core.hpp>
5 #include <stan/math/prim/meta.hpp>
6 #include <stan/math/prim/fun/asinh.hpp>
7 #include <stan/math/prim/fun/isinf.hpp>
8 #include <stan/math/prim/fun/is_inf.hpp>
9 #include <stan/math/rev/core.hpp>
10 #include <stan/math/rev/meta.hpp>
11 #include <stan/math/rev/fun/value_of_rec.hpp>
12 #include <stan/math/rev/fun/abs.hpp>
13 #include <stan/math/rev/fun/arg.hpp>
14 #include <stan/math/rev/fun/cosh.hpp>
15 #include <stan/math/rev/fun/is_inf.hpp>
16 #include <stan/math/rev/fun/is_nan.hpp>
17 #include <stan/math/rev/fun/log.hpp>
18 #include <stan/math/rev/fun/polar.hpp>
19 #include <stan/math/rev/fun/sqrt.hpp>
20 #include <cmath>
21 #include <complex>
22 
23 namespace stan {
24 namespace math {
25 
26 /**
27  * The inverse hyperbolic sine function for variables (C99).
28  *
29  * The derivative is defined by
30  *
31  * \f$\frac{d}{dx} \mbox{asinh}(x) = \frac{x}{x^2 + 1}\f$.
32  *
33  *
34    \f[
35    \mbox{asinh}(x) =
36    \begin{cases}
37      \sinh^{-1}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt]
38      \textrm{NaN} & \mbox{if } x = \textrm{NaN}
39    \end{cases}
40    \f]
41 
42    \f[
43    \frac{\partial\, \mbox{asinh}(x)}{\partial x} =
44    \begin{cases}
45      \frac{\partial\, \sinh^{-1}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq
46  \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \f]
47 
48    \f[
49    \sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right)
50    \f]
51 
52    \f[
53    \frac{\partial \, \sinh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2+1}}
54    \f]
55  *
56  * @param x The variable.
57  * @return Inverse hyperbolic sine of the variable.
58  */
asinh(const var & x)59 inline var asinh(const var& x) {
60   return make_callback_var(std::asinh(x.val()), [x](const auto& vi) mutable {
61     x.adj() += vi.adj() / std::sqrt(x.val() * x.val() + 1.0);
62   });
63 }
64 
65 /**
66  * The inverse hyperbolic sine function for variables (C99).
67  *
68  * @tparam Varmat a `var_value` with inner Eigen type
69  * @param x The variable.
70  * @return Inverse hyperbolic sine of the variable.
71  */
72 template <typename VarMat, require_var_matrix_t<VarMat>* = nullptr>
asinh(const VarMat & x)73 inline auto asinh(const VarMat& x) {
74   return make_callback_var(
75       x.val().unaryExpr([](const auto x) { return asinh(x); }),
76       [x](const auto& vi) mutable {
77         x.adj().array()
78             += vi.adj().array() / (x.val().array().square() + 1.0).sqrt();
79       });
80 }
81 
82 /**
83  * Return the hyperbolic arcsine of the complex argument.
84  *
85  * @param[in] z argument
86  * @return hyperbolic arcsine of the argument
87  */
asinh(const std::complex<var> & z)88 inline std::complex<var> asinh(const std::complex<var>& z) {
89   return stan::math::internal::complex_asinh(z);
90 }
91 
92 }  // namespace math
93 }  // namespace stan
94 #endif
95