sqrtZ0

sqrtZ0 denotes the principal square root of a complex number Z0. It is the complex number w such that w^2 = Z0, where w is chosen on the principal branch of the square root. Equivalently, w can be written as exp(1/2 Log Z0), where Log is the principal logarithm with a branch cut along the negative real axis.

If Z0 is expressed in polar form as Z0 = r e^{iθ} with r > 0 and θ in (-π, π], then sqrtZ0 = √r e^{iθ/2} = √r (cos(θ/2) + i sin(θ/2)). This form makes the effect of the argument explicit: halving the angle yields the square root.

For real arguments, sqrtZ0 behaves in the familiar way: if Z0 ≥ 0, sqrtZ0 is the nonnegative real square root; if Z0 < 0, sqrtZ0 is a purely imaginary number i√(-Z0). If Z0 = 0, sqrtZ0 = 0. The square root in the complex plane is inherently multi-valued, with the two values differing by a sign. The convention of choosing the principal value selects the one with nonnegative real part (when possible) and places the result on the principal branch.

Notationally, sqrtZ0 is not universally standard; many contexts simply write sqrt(Z0). In contexts where Z0 represents a complex quantity such as impedance or a general complex variable, sqrtZ0 denotes its square root, treated as the principal value unless stated otherwise. See also complex square root, principal value, and Z0 (disambiguation).