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Does calculus for complex numbers exists? did any one ever tried to make research on this?

What would it mean to have complex numbers calculus? for instance: What is the meaning of $$\lim_{x\to z} f(x) = L \\ for \ complex \ values \ z, L, \ and \ the \ complex \ variable \ (x)$$

What is the meaning of $$\frac {dy}{dx}=\sqrt {-1}$$

Or what's the meaning of $$\int_{z_2}^{z_1} F(z) dz = \sqrt {-1},\ for \ complex \ values \ z, z_1, z_2$$

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    $\begingroup$ You may want to learn [complex analysis][en.wikipedia.org/wiki/Complex_analysis]. $\endgroup$
    – Feng
    Aug 17, 2019 at 7:41
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    $\begingroup$ Yes, it's called complex analysis and it has been studied for a couple of centuries alongside its real counterpart. $\endgroup$
    – user239203
    Aug 17, 2019 at 7:42
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    $\begingroup$ Yes - in fact the question of the integral is more interesting, because if we are integrating between two complex numbers we have to specify a path in the complex plane along which to integrate and different paths can give different values for the integral. The theory is rich and there are some surprising results. $\endgroup$
    – Mark Bennet
    Aug 17, 2019 at 9:26

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