complexvalued

Complex-valued describes objects that take complex numbers as values. In mathematics and related fields, a complex-valued function f is one whose output lies in the complex field C, rather than the real numbers R. Many objects in science and engineering are naturally complex-valued, including signals, state amplitudes, and transfer functions.

A complex-valued function f: X -> C can be written as f(x) = u(x) + i v(x), where u and v are real-valued functions representing the real and imaginary parts. The magnitude is |f(x)| = sqrt(u(x)^2 + v(x)^2) and the argument is arg(f(x)) = atan2(v(x), u(x)). Complex numbers multiply and add as usual, and conjugation flips the sign of the imaginary part. The polar form f(x) = r(x) e^{i theta(x)} is often convenient.

In analysis, a complex-valued function may be studied for continuity, differentiability, and analyticity. A complex-valued function that is differentiable in a neighborhood of every point is called holomorphic.

In linear algebra and functional analysis, vectors and matrices can be complex-valued, and inner products typically involve complex conjugation, for example ⟨x, y⟩ = sum x_i conjugate(y_i). This places the complex field as a natural setting for many problems in physics and engineering.

A real-valued object is a special case of complex-valued, where the imaginary parts are zero. Complex-valued concepts underpin complex signal processing, quantum mechanics, control theory, and many numerical methods.