pow

pow(x1: array, x2: array, /) → array

Calculates an implementation-dependent approximation of exponentiation by raising each element x1_i (the base) of the input array x1 to the power of x2_i (the exponent), where x2_i is the corresponding element of the input array x2.

Note

If both x1 and x2 have integer data types, the result of pow when x2_i is negative (i.e., less than zero) is unspecified and thus implementation-dependent.

If x1 has an integer data type and x2 has a floating-point data type, behavior is implementation-dependent (type promotion between data type “kinds” (integer versus floating-point) is unspecified).

Note

By convention, the branch cut of the natural logarithm is the negative real axis \((-\infty, 0)\).

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component. As special cases involving complex floating-point operands should be handled according to exp(x2*log(x1)), exponentiation has the same branch cut for x1 as the natural logarithm (see log()).

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

Parameters:

  • x1 (array) – first input array whose elements correspond to the exponentiation base. Should have a numeric data type.

  • x2 (array) – second input array whose elements correspond to the exponentiation exponent. Must be compatible with x1 (see Broadcasting). Should have a numeric data type.

Returns:

out (array) – an array containing the element-wise results. The returned array must have a data type determined by Type Promotion Rules.

Notes

Special cases

For real-valued floating-point operands,

  • If x1_i is not equal to 1 and x2_i is NaN, the result is NaN.

  • If x2_i is +0, the result is 1, even if x1_i is NaN.

  • If x2_i is -0, the result is 1, even if x1_i is NaN.

  • If x1_i is NaN and x2_i is not equal to 0, the result is NaN.

  • If abs(x1_i) is greater than 1 and x2_i is +infinity, the result is +infinity.

  • If abs(x1_i) is greater than 1 and x2_i is -infinity, the result is +0.

  • If abs(x1_i) is 1 and x2_i is +infinity, the result is 1.

  • If abs(x1_i) is 1 and x2_i is -infinity, the result is 1.

  • If x1_i is 1 and x2_i is not NaN, the result is 1.

  • If abs(x1_i) is less than 1 and x2_i is +infinity, the result is +0.

  • If abs(x1_i) is less than 1 and x2_i is -infinity, the result is +infinity.

  • If x1_i is +infinity and x2_i is greater than 0, the result is +infinity.

  • If x1_i is +infinity and x2_i is less than 0, the result is +0.

  • If x1_i is -infinity, x2_i is greater than 0, and x2_i is an odd integer value, the result is -infinity.

  • If x1_i is -infinity, x2_i is greater than 0, and x2_i is not an odd integer value, the result is +infinity.

  • If x1_i is -infinity, x2_i is less than 0, and x2_i is an odd integer value, the result is -0.

  • If x1_i is -infinity, x2_i is less than 0, and x2_i is not an odd integer value, the result is +0.

  • If x1_i is +0 and x2_i is greater than 0, the result is +0.

  • If x1_i is +0 and x2_i is less than 0, the result is +infinity.

  • If x1_i is -0, x2_i is greater than 0, and x2_i is an odd integer value, the result is -0.

  • If x1_i is -0, x2_i is greater than 0, and x2_i is not an odd integer value, the result is +0.

  • If x1_i is -0, x2_i is less than 0, and x2_i is an odd integer value, the result is -infinity.

  • If x1_i is -0, x2_i is less than 0, and x2_i is not an odd integer value, the result is +infinity.

  • If x1_i is less than 0, x1_i is a finite number, x2_i is a finite number, and x2_i is not an integer value, the result is NaN.

For complex floating-point operands, special cases should be handled as if the operation is implemented as exp(x2*log(x1)).

Note

Conforming implementations are allowed to treat special cases involving complex floating-point operands more carefully than as described in this specification.

Changed in version 2022.12: Added complex data type support.