sqrtv

sqrtV denotes the principal square root function applied to a variable V. In its common real-valued form, sqrtV(V) is defined for V ≥ 0 and returns the nonnegative number y with y^2 = V. Thus, sqrtV maps the nonnegative real numbers to themselves; for negative real inputs the real-valued sqrtV is undefined, while a complex-valued extension exists.

Properties: Domain is [0, infinity); range is [0, infinity); sqrtV is continuous and strictly increasing on [0, infinity). It satisfies sqrtV(V)^2 = V and sqrtV(0) = 0. For V > 0, the derivative of sqrtV with respect to V is 1/(2 sqrtV(V)).

Complex extension: For complex V, a principal branch of the square root is used to define sqrtV; this yields a single-valued function on the complex plane with a branch cut typically along the negative real axis. The value depends on the chosen branch, and the equation sqrtV(V)^2 = V continues to hold in the appropriate sense.

Computation and notation: sqrtV can be computed as V^(1/2) or by specialized algorithms such as the Newton-Raphson method. In programming libraries, a function named sqrt typically implements the principal real square root for nonnegative inputs and a complex square root function handles complex inputs.

Applications and relationships: In statistics, if V denotes variance, sqrtV(V) is the standard deviation. In physics and engineering, V may represent quantities that are nonnegative, making sqrtV a value with units that are the square root of the units of V. The term sqrtV is therefore a concise way to refer to the square root of a variable V in mathematical expressions.