In the complex plane, there can be a multitude of functions with a branch point at 0. If, for example, the real axis is chosen as the branch cut in C, then the function becomes explicit:
Is it differentiable everywhere in the complex plane? Explain why or why not.
For real numbers, we often write a = √4 = 2. If we write a complex number in the form z = re^(iθ), where r is a positive real number and θ ∈ [0, 2π), then
√r · exp(iθ/2)
with n = 0 or 1.
Write f(z) = √z = √(x + iy) in real and imaginary parts u(x,y) and v(x,y). Thus, in the form f(z) = u + iv.