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### 4.8.3 Hyperbolic functions

`cl_R sinh (const cl_R& x)`

Returns `sinh(x)`.

`cl_N sinh (const cl_N& z)`

Returns `sinh(z)`. The range of the result is the entire complex plane.

`cl_R cosh (const cl_R& x)`

Returns `cosh(x)`. The range of the result is the interval `cosh(x) >= 1`.

`cl_N cosh (const cl_N& z)`

Returns `cosh(z)`. The range of the result is the entire complex plane.

`struct cosh_sinh_t { cl_R cosh; cl_R sinh; };`
`cosh_sinh_t cosh_sinh (const cl_R& x)`

Returns both `sinh(x)` and `cosh(x)`. This is more efficient than computing them separately. The relation `cosh^2 - sinh^2 = 1` will hold only approximately.

`cl_R tanh (const cl_R& x)`
`cl_N tanh (const cl_N& x)`

Returns `tanh(x) = sinh(x)/cosh(x)`.

`cl_N asinh (const cl_N& z)`

Returns `arsinh(z)`. This is defined as `arsinh(z) = log(z+sqrt(1+z^2))` and satisfies `arsinh(-z) = -arsinh(z)`. The range of the result is the strip in the complex domain `-pi/2 <= imagpart(arsinh(z)) <= pi/2`, excluding the numbers with `imagpart = -pi/2` and `realpart > 0` and the numbers with `imagpart = pi/2` and `realpart < 0`.

`cl_N acosh (const cl_N& z)`

Returns `arcosh(z)`. This is defined as `arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))`. The range of the result is the half-strip in the complex domain `-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0`, excluding the numbers with `realpart = 0` and `-pi < imagpart < 0`.

`cl_N atanh (const cl_N& z)`

Returns `artanh(z)`. This is defined as `artanh(z) = (log(1+z)-log(1-z)) / 2` and satisfies `artanh(-z) = -artanh(z)`. The range of the result is the strip in the complex domain `-pi/2 <= imagpart(artanh(z)) <= pi/2`, excluding the numbers with `imagpart = -pi/2` and `realpart <= 0` and the numbers with `imagpart = pi/2` and `realpart >= 0`.

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