Arctanw

Arctanw denotes the inverse tangent of a complex variable w. It is commonly written as arctan(w) and is a multivalued analytic function with branch points at w = i and w = -i.

One convenient representation is arctanw = (1/(2i)) [ln(1 + i w) − ln(1 − i w)], where ln is the complex logarithm. The principal value assigns a single value by choosing principal branches of the logarithm and typically placing branch cuts along the imaginary axis from i to i∞ and from −i to −i∞.

On the real line, arctanw reduces to the ordinary arctan with range (−π/2, π/2). For complex arguments, the function is multi-valued: if z is a value, then z + kπ for any integer k is also a value, due to the periodicity of the tangent function.

The derivative formula holds away from branch points: d/dw arctanw = 1/(1 + w^2).

Computationally, many programming environments implement the principal value of arctan for complex inputs as atan or related functions, using the same logarithmic representation.

Applications include solving equations involving tangents in the complex plane and evaluating integrals in complex analysis.