[Math] Checking where the complex derivative of a function exists

complex numberscomplex-analysisderivatives

I have the following function:

$$f(x+iy) = x^2+iy^2$$

My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we use to check where a function is differentiable?

I know the Cauchy-Riemann equations, and that $u=x^2$ and $v=y^2$ here.

Best Answer

Being complex differentiable at a point is equivalent to the combination of

  1. Being real differentiable at that point, and
  2. Satisfying the Cauchy-Riemann equations

The real and imaginary parts of $f$ are $u=x^2$ and $v= ^2$. They are polynomials, so real-differentiable everywhere. The two Cauchy-Riemann equations take the form $2x=2y$ (from $u_x=v_y$) and $0=0$ (from $ u_y=-v_x$). The second holds everywhere. The first holds when $x=y$ and only then.

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