sqrtImz

sqrtImz is a mathematical function that denotes the square root of the imaginary part of a complex number. If z is written as z = x + i y with real x and y, then sqrtImz(z) is defined as sqrt(y). Interpreted through the principal branch of the complex square root, this means that for y ≥ 0 the value is the nonnegative real root sqrt(y), and for y < 0 the value is i sqrt(-y), a purely imaginary number.

Domain and range: the input is any complex number z, since every z has a real imaginary part y. The output is real and nonnegative when y ≥ 0, and purely imaginary when y < 0. Consequently, sqrtImz is not a real-valued function on the complex plane; its range consists of real nonnegative numbers and purely imaginary numbers, depending on the sign of the imaginary part.

Properties: sqrtImz is not a holomorphic (complex-analytic) function of z, because it depends solely on Im(z) and not on Re(z); as such it fails the Cauchy-Riemann equations except in trivial cases. It is a non-analytic, non-conformal example often used to illustrate functions that are continuous and even differentiable in a real-variable sense but not complex-differentiable.

Examples: for z = 3 + 4i, Im(z) = 4 and sqrtImz(z) = 2; for z = -2 - i, Im(z) = -1 and sqrtImz(z) = i.

See also: Im, the square root function, complex analysis, and non-analytic functions.