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It is stated that there are infinite number of ways to approach any given point of complex plane where function is defined, not just «left» anf «right» for the case of real variable. That is where necessity for having the same limit no matter which direction is regarded comes into play and the next step is to deduce the Cauchy-Riemann equations from this necessity.

Would not it be more appropriate to compare the case of complex variable with case of two real variables? Here one has x — y plane and different directions as well, but no eqivalent of C.-R. equations, one starts considering partial derivatives instead. Why is that? What is the qualitative difference between pairs (a; b) and (a; ib)? And what about split-complex numbers, where i^2 = +1 from the viewpoint of differentiability of their functions?

Sorry, if this thread is redudant and my question has already been answered in another one. Just point me to the latter in this case, I failed to find it myself.