2011-06-04, 05:06 AM
z is a complex variable, so f(z) is not really a scalar plane. It's also a vector plane (a,b), where a is the real part and b is the imaginary part (otherwise known as a+bi). I don't really understand the theory behind this, but even though they're different components, Euler's formula for the expansion of exp(iθ
still holds so we don't get a vector divided by a vector when we do this.
I did a bit more searching. Apparently, this is linked to ∇arctan(y/x), so your prediction was correct. I found on Wolfram that the complex argument function, arg, is defined as arg(x+yi)=arctan(y/x). Unfortunately, I have no idea how this can help, but it is interesting to note that the deformation theorem works for ∇arctan(y/x), and 2π is the winding number. Similarly, I should obtain 2πi for ∮1/z dz.
Good luck on your finals, by the way.
still holds so we don't get a vector divided by a vector when we do this.I did a bit more searching. Apparently, this is linked to ∇arctan(y/x), so your prediction was correct. I found on Wolfram that the complex argument function, arg, is defined as arg(x+yi)=arctan(y/x). Unfortunately, I have no idea how this can help, but it is interesting to note that the deformation theorem works for ∇arctan(y/x), and 2π is the winding number. Similarly, I should obtain 2πi for ∮1/z dz.
Good luck on your finals, by the way.