I know that the exponential function satisfies the condition of Cauchy-Riemann and is differentiable but how to tell where the function is differentiable specifically? Sufficient Conditions for Differentiability in this case are: If the real functions u(x,y) and v(x, y) are continuous and have continuous first-order partial derivatives in some neighborhood of a point z, and if u and v satisfy the Cauchy-Riemann equations at z, then the complex function f(z) = u(x, y) + iv(x, y) is differentiable at z

  • $\begingroup$ Just the phrase "f is differentiable" with no conditions means it is differentiable where? $\endgroup$ – user247327 Jul 15 at 15:03
  • $\begingroup$ Do we have tell the values of x and y at which it differentiates or something else? $\endgroup$ – Arham Qadeer Jul 15 at 15:10

That function can be naturally expressed as the composition of two differentiable maps (the exponential function and $z\mapsto z^2$). Therefore, it is differentiable. You don't need the Cauchy-Riemann equations here.

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