Given the function: f(x) = im(x). So, it takes the imaginary part of x.
On the real line, this function always returns 0. However, in the imaginary direction, it returns real values and has a slope of 1.
With the definition of the derivative, you look in the real direction only and miss that there's a slope in the imaginary direction. Wolfram Alpha gives 0: http://www.wolframalpha.com/input/?i=derivative+of+f%28x%29+%3D+im%28x%29+at+x+%3D+5
So, is that indeed correct then? Do you have to ignore the slope in imaginary direction for a derivative? Shouldn't the answer be something like 1, or -i?
My concern is that this derivative is not useful for Newton's method to find zeroes of complex functions, e.g. the above example would always return 0 even though there's a slope. I'm also wondering how to define slope in a 4D space (2D input, 2D output).
Edited by Lode, 27 April 2014 - 04:55 AM.