atanh

atanh(x: array, /) → array

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

Note

The principal value of the inverse hyperbolic tangent of a complex number \(z\) is

\[\operatorname{atanh}(z) = \frac{\ln(1+z)-\ln(z-1)}{2}\]

For any \(z\),

\[\operatorname{atanh}(z) = \frac{\operatorname{atan}(zj)}{j}\]

Note

For complex floating-point operands, atanh(conj(x)) must equal conj(atanh(x)) and atanh(-x) must equal -atanh(x).

Note

The inverse hyperbolic tangent is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments \((-\infty, 1]\) and \([1, \infty)\) of the real axis.

Accordingly, for complex arguments, the function returns the inverse hyperbolic tangent in the range of a half-strip unbounded along the real axis and in the interval \([-\pi j/2, +\pi j/2]\) 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 tangent 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 greater than 1, the result is NaN.

  • If x_i is -1, the result is -infinity.

  • If x_i is +1, the result is +infinity.

  • If x_i is +0, the result is +0.

  • If x_i is -0, the result is -0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

  • If a is +0 and b is +0, the result is +0 + 0j.

  • If a is +0 and b is NaN, the result is +0 + NaN j.

  • If a is 1 and b is +0, the result is +infinity + 0j.

  • If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is +0 + πj/2.

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

  • If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 + πj/2.

  • If a is +infinity and b is +infinity, the result is +0 + πj/2.

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

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

Changed in version 2022.12: Added complex data type support.