复变函数问题请证明：arg(z)在原点和负实轴上不连续。请说的专业点，尽量严格。

Question

复变函数问题请证明： arg(z)在原点和负实轴上不连续。请说的专业点，尽量严格。

Answer 1

z = amplitude * exp(i * alpha) 1. amplitude != 0 when alpha -> pi+ arg(z) -> pi while alpha -> pi- arg(z) -> -pi the two limitations do not converge, thus, arg(z) jumps in the region around negtive real axis 2. for example, exp(i*alpha) = i, z = i * amplitude when amplitude -> 0+, arg(z) -> 0.5*pi when amplitude -> 0-, arg(z) -> -0.5*pi they do not converge, so it is not continuous.