The question
Let z = x + iy, with x and y real
If , find the value of the derivative of f at every point z where the derivative exists. Where is f analytic?
My attempt:
I used Cauchy Riemann sums as follows,
The equations hold when . This equals |x| = |y| which are intersecting lines through the origin of the x-y plane. So, the function has a derivative on these intersecting lines. If my understanding of analyticity is correct, this function is analytic nowhere since there's no epsilon neighbourhood where the function is differentiable.
My issue is, how do I find the value of the derivative? Do I use the definition of a derivative, or is there an easy way to find it using my result above?
Thanks.
Just restrict h to real values when applying the limit definition of the derivative. Since you know that f is differentiable at |x|=|y|, then the real-restricted limit must be equal to the complex limit. We end up getting
.
By the way, the reason we know f is differentiable on those lines is because all the first-order partial derivatives (1) satisfy the Cauchy-Riemann equations; and (2) are continuous. You mentioned (1) but don't forget that (2) is also required.