Timeline for History of the definition of complex derivative
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 6, 2022 at 21:13 | history | edited | KCd | CC BY-SA 4.0 |
added 3 characters in body
|
| Oct 25, 2021 at 11:47 | comment | added | Rodrigo de Azevedo | Shouldn't it be $$ f'(z_0)=\lim_{z \rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$$ instead? | |
| Oct 25, 2021 at 11:46 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
edited title
|
| Oct 24, 2021 at 4:58 | comment | added | Conifold | 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. | |
| Oct 24, 2021 at 0:31 | vote | accept | Penelope | ||
| Oct 23, 2021 at 14:35 | answer | added | Alexandre Eremenko | timeline score: 5 | |
| Oct 23, 2021 at 13:53 | comment | added | Penelope | 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. | |
| Oct 23, 2021 at 8:22 | comment | added | Conifold | 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. | |
| Oct 23, 2021 at 6:25 | comment | added | Conifold | 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". | |
| Oct 22, 2021 at 23:45 | history | asked | Penelope | CC BY-SA 4.0 |