sqrtz2

sqrtz2 denotes the principal square root of z squared, written as sqrt(z^2) with a fixed branch of the complex square root. In this usage, sqrt is the principal square root, defined for a complex number w by sqrt(w) = exp(1/2 Log w), where Log is the principal logarithm (branch cut along the negative real axis). Thus sqrtz2(z) = sqrt(z^2) uses the standard principal branch of the square root.

For real arguments, sqrt(z^2) equals the absolute value of z: sqrt(z^2) = |z|. This makes sense because z^2 ≥ 0 on the real axis and the principal square root returns the nonnegative real root.

In the complex plane, the value of sqrt(z^2) depends on the chosen branch. With the principal branch described above, sqrt(z^2) typically equals z for points in the open right half-plane (and certain boundary points), and equals -z for points in the open left half-plane; crossing the imaginary axis induces a discontinuity. The imaginary axis acts as a branch cut for sqrt(z^2) under this standard convention, so the function can flip between z and -z as one moves across that line.

Key properties include that (sqrt(z^2))^2 = z^2, while sqrt(z^2) is not generally equal to |z| when z is complex. This distinction between real and complex arguments is a central feature of multivalued square-root behavior, made explicit by fixing a principal branch.

See also: square root, principal value, multivalued functions, complex analysis.