SqrtI

SqrtI, commonly written as sqrt(i), denotes the square root of the imaginary unit i, where i^2 = -1. There are two square roots: z = ±(1/√2)(1 + i). The principal square root is z0 = (1/√2)(1 + i).

A standard way to derive this is to let z = a + bi with real a and b, and require (a + bi)^2 = i. This gives the system a^2 − b^2 = 0 and 2ab = 1. From a^2 = b^2 and ab = 1/2, one obtains a = b = ±1/√2. Thus the two roots are z = ±(1/√2 + i/√2) = ±(1 + i)/√2.

Each root has magnitude 1, since |i| = 1 and taking a square root preserves magnitude in this case. In polar form, i = e^{iπ/2}, so the square roots are e^{iπ/4} and e^{i5π/4}, i.e., ±(cos(π/4) + i sin(π/4)) = ±(√2/2)(1 + i).

The principal value is e^{iπ/4} = (1/√2)(1 + i). In the context of the principal branch of the complex square root function, sqrt(i) = exp(1/2 Log i) with Log i = iπ/2, yielding sqrt(i) = exp(iπ/4).

SqrtI is used in solving equations involving complex numbers and in fields such as complex analysis and engineering, where explicit square roots of imaginary units arise in computations and transformations.