atan

atan(x: array, /) → array

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

Each element-wise result is expressed in radians.

Note

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

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

Note

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

Note

The inverse tangent (or arc tangent) is a multi-valued function and requires a branch on the complex plane. By convention, a branch cut is placed at the line segments \((-\infty j, -j)\) and \((+j, \infty j)\) of the imaginary axis.

Accordingly, for complex arguments, the function returns the inverse tangent in the range of a strip unbounded along the imaginary axis and in the interval \([-\pi/2, +\pi/2]\) along the real axis.

Note: branch cuts follow C99 and have provisional status (see Branch Cuts).

Parameters:

x (array) – input array. Should have a floating-point data type.

Returns:

out (array) – an array containing the inverse 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 +0, the result is +0.

  • If x_i is -0, the result is -0.

  • If x_i is +infinity, the result is an implementation-dependent approximation to +π/2.

  • If x_i is -infinity, the result is an implementation-dependent approximation to -π/2.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * atanh(x*1j).

Changed in version 2022.12: Added complex data type support.