acosh

acosh(x: array, /) → array

Calculates an implementation-dependent approximation to the inverse hyperbolic cosine for each element x_i of the input array x.

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 equal conj(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 is NaN, the result is NaN.

  • If x_i is less than 1, the result is NaN.

  • If x_i is 1, 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), and

  • If a is either +0 or -0 and b is +0, the result is +0 + πj/2.

  • If a is a finite number and b is +infinity, the result is +infinity + πj/2.

  • If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.

  • If a is +0 and b is NaN, the result is NaN ± πj/2 (sign of imaginary component is unspecified).

  • If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.

  • If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.

  • If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.

  • If a is +infinity and b is +infinity, the result is +infinity + πj/4.

  • If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.

  • If a is NaN and b is a finite number, the result is NaN + NaN j.

  • If a is NaN and b is +infinity, the result is +infinity + NaN j.

  • If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.