Blame Newton. This inventor of "calculus" was fond of doing everything with power series, so complex numbers inadvertently got dragged into the picture. Post-Newton, engineers also found power series to be incredibly useful, so they defined "calculus" as "stuff you can do with power series". Most of my MIT undergraduate engineering courses were heavily wedded to power series & complex analysis. It took *fractals* to finally move engineering a bit away from power series (and topics like the Newton's (!) N-body problem, where the power series may exist, but are useless because they converge almost nowhere). At 07:16 AM 9/24/2018, Andy Latto wrote:

Why do so many people think the false "theorem" is true? One reason is that when we consider functions from the complex numbers to the complex numbers, rather than from the reals to the reals, the theorem is true. The statement that a function from C to C is differentiable is a much stronger statement than the statement that it is differentiable as a function from R^2 to R^2. The latter statement says that the function is well approximated locally by a function in the family of linear functions from R^2 to R^2, a space with real dimension 4. The former statement says that this function is one of the ones that corresponds to multiplication by a complex number, a linear subspace of dimension 2. The statement that a function is differentiable from C to C doesn't just say it's smooth; it says it's conformal, a much more rigid criterion. The fact that the theorem is true in complex analysis has little bearing on the kind of curve-fitting we are talking about here, since there is no meaningful extension of the functions in question to complex arguments and values, and no reason to expect that the resulting function would be complex-analytic if we did extend it.