acosh¶
- acosh(x: array, /) array ¶
Calculates an implementation-dependent approximation to the inverse hyperbolic cosine for each element
x_i
of the input arrayx
.Note
The principal value of the inverse hyperbolic cosine of a complex number \(z\) is
\[\operatorname{acosh}(z) = \ln(z + \sqrt{z+1}\sqrt{z-1})\]For any \(z\),
\[\operatorname{acosh}(z) = \frac{\sqrt{z-1}}{\sqrt{1-z}}\operatorname{acos}(z)\]or simply
\[\operatorname{acosh}(z) = j\ \operatorname{acos}(z)\]in the upper half of the complex plane.
Note
For complex floating-point operands,
acosh(conj(x))
must equalconj(acosh(x))
.Note
The inverse hyperbolic cosine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segment \((-\infty, 1)\) of the real axis.
Accordingly, for complex arguments, the function returns the inverse hyperbolic cosine in the interval \([0, \infty)\) along the real axis and in the interval \([-\pi j, +\pi j]\) along the imaginary axis.
Note: branch cuts follow C99 and have provisional status (see Branch Cuts).
- Parameters:
x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
- Returns:
out (array) – an array containing the inverse hyperbolic cosine of each element in
x
. The returned array must have a floating-point data type determined by Type Promotion Rules.
Notes
Special cases
For real-valued floating-point operands,
If
x_i
isNaN
, the result isNaN
.If
x_i
is less than1
, the result isNaN
.If
x_i
is1
, the result is+0
.If
x_i
is+infinity
, the result is+infinity
.
For complex floating-point operands, let
a = real(x_i)
,b = imag(x_i)
, andIf
a
is either+0
or-0
andb
is+0
, the result is+0 + πj/2
.If
a
is a finite number andb
is+infinity
, the result is+infinity + πj/2
.If
a
is a nonzero finite number andb
isNaN
, the result isNaN + NaN j
.If
a
is+0
andb
isNaN
, the result isNaN ± πj/2
(sign of imaginary component is unspecified).If
a
is-infinity
andb
is a positive (i.e., greater than0
) finite number, the result is+infinity + πj
.If
a
is+infinity
andb
is a positive (i.e., greater than0
) finite number, the result is+infinity + 0j
.If
a
is-infinity
andb
is+infinity
, the result is+infinity + 3πj/4
.If
a
is+infinity
andb
is+infinity
, the result is+infinity + πj/4
.If
a
is either+infinity
or-infinity
andb
isNaN
, the result is+infinity + NaN j
.If
a
isNaN
andb
is a finite number, the result isNaN + NaN j
.If
a
isNaN
andb
is+infinity
, the result is+infinity + NaN j
.If
a
isNaN
andb
isNaN
, the result isNaN + NaN j
.
Changed in version 2022.12: Added complex data type support.