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Almost all of modern complex analysis (Cauchy residue theorem, analytic continuation, etc) depend on the definition of a complex derivative.

That definition requires the derivative at a point $z_0$ is the same no matter which direction the limit is taken to that point.

$$ f'(z_0)=\lim_{z \rightarrow z_0} \frac{f(z-z_0)}{z-z_0}$$

This definition is quite strong, and leads to the Cauchy Riemann conditions.

Question: Why did this definition become the standard one?

Why didn't, for example, a derivative that depends on direction become standard?

If other options are possible, do they lead to useful but different complex analysis?

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    $\begingroup$ It literally transplanted the standard definition from real analysis and led to the rich results of complex analysis. Derivative without the Cauchy-Riemann conditions amounts to treating the function as a function of two real variables. Which is, of course, useful but the results are much weaker because it does not take advantage of the complex structure, and so hardly deserves the name "complex derivative". $\endgroup$
    – Conifold
    Oct 23 at 6:25
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    $\begingroup$ As for history, Euler in his textbooks applied calculus rules to both real and complex functions without much ado (as did others), which produces answers consistent with the limit definition. Cauchy explicitly wrote it down in Cours d'Analyse (1821), and linked it to complex integration in Resume (1823). Both were highly influential in establishing modern notation and terminology. $\endgroup$
    – Conifold
    Oct 23 at 8:22
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    $\begingroup$ hi @Conifold thanks for the useful comments. I'm trying to work out why history didn't go down the "function of two variables" path and instead took the path it did. What was the motivation? Was it actually an accident of history where, naively, it was assumed the limit was the same regardless of direction (because rules of calculus were applied to complex functions naively), and later this just became standardised. $\endgroup$
    – Tariq
    Oct 23 at 13:53
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    $\begingroup$ But it did go both ways. Analysis of functions of several real variables was developed during the same period. There was just nothing there to single out 2 as opposed to 3 or more, or to link it to complex structure specifically so it would go under that label. There was nothing to assume, taking complex derivatives analytically did not go through taking directional derivatives. And there was no fork in the road for choosing or motivation to do the choosing. It did not happen in a way for the question to make sense. $\endgroup$
    – Conifold
    Oct 24 at 4:58
  • $\begingroup$ Shouldn't it be $$ f'(z_0)=\lim_{z \rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$$ instead? $\endgroup$ Oct 25 at 11:47
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In general, definitions in mathematics are not arbitrary. Useful definitions are those which lead to interesting and important objects. This particular definition is good because it leads to an interesting and important class of functions.

A function $f(z)=u(x,y)+iv(x,y), z=x+iy,$ of one complex variable is the same as two real functions of two real variables. So partial derivatives, directional derivatives etc. can be defined, but you do not obtain anything new with these definitions, in comparison with real calculus of functions of several real variables. Since this theory (real calculus) exists for functions of any number of real variables, one does not obtain anything new.

The definition of complex derivative uses the field structure on complex numbers in a very essential way: it involves division. There is no "division" of real vectors. And this definition leads to a remarkable and useful class of functions: complex analytic functions.

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  • $\begingroup$ hi Alexandre - this is a very helpful reply. I'd like to understand your final paragraph better before I mark it as "solved". Is it possible to explain what "field structure" is and your point on "division"? If it is not easy to explain, perhaps a link to resources I can read? $\endgroup$
    – Tariq
    Oct 23 at 21:20
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    $\begingroup$ A field is a set on which addition, subtraction, multiplication, and division are defined and have certain properties. Cf. this question on Mathematics Stack Exchange $\endgroup$ Oct 24 at 0:20

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