sqrtc

sqrtc denotes the square root of a complex number, commonly used to emphasize a complex-domain square root or to refer to a principal complex square root in programming libraries. In mathematics, the square root of a complex number z = x + iy is a number w such that w^2 = z, and there are two closely related values: ±w. The term sqrtc is often used to denote the principal value of this square root.

For z = x + iy, the principal square root sqrtc(z) can be written as a + ib, where

r = |z| = sqrt(x^2 + y^2),

a = sqrt((r + x)/2),

b = sign(y) sqrt((r − x)/2).

If y = 0 and x ≥ 0, sqrtc(z) = sqrt(x) (a real nonnegative value). If y = 0 and x < 0, sqrtc(z) is purely imaginary, equal to i sqrt(-x). The result is chosen so that a ≥ 0, and when y ≠ 0 the sign of b matches the sign of y.

Equivalent formulations exist using polar coordinates: z = re^{iθ} with θ ∈ (−π, π], then sqrtc(z) = sqrt(r) e^{iθ/2}. The principal branch typically has a branch cut along the negative real axis, ensuring continuity on the complex plane excluding that cut. The square root is multivalued, with the two values differing by a sign; trying to multiply square roots requires careful branch choice.

In practice, sqrtc is implemented in software libraries to compute the principal complex square root efficiently and stably, often via the algebraic formula or polar methods.