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From the question Difference between the formula of Roger Cotes and Euler:

Euler: $e^{ix} = \cos x+i \sin x$

Cotes: $ix = \ln(\cos x + i\sin x)$

The answer (in full):

The problem is that the complex logarithm is multivalued under the current definition. Therefore Cotes' formula is not really true anymore, but it was when he got it.

This raises the question How has the definition of the complex logarithm evolved since Cotes?

I assume the answer suggests that log was single valued in Cotes' time, since it is multivalued now. Was it in fact of the same general form as it is now but with a single breakpoint in $\theta$?

Was it something like $ln(r) + i\theta$ and $-\pi \lt \theta \le \pi$ for example?

If not, please explain the change in the definition of logarithm from Cotes's time to now in such a way that someone with a basic working knowledge of complex numbers who's just read Wikipedia's Complex Logarithm could understand.

below: A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number z goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a branch point of the function. From here:

Riemann surface of log(z), projection from 4dim C x C to 3dim C x Im(C), color is argument.

Riemann surface of Log[z], projection from 4dim C x C to 3dim C x Im(C), color is argument. Source: Riemann surface log

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  • $\begingroup$ $\log(ab)= \log(a)+\log(b)+2ik \pi$ $\endgroup$
    – reuns
    Commented Aug 18, 2017 at 15:35
  • $\begingroup$ @reuns I'm asking how the definition of log has changed over time. It seems it was single-valued in Cotes' time, so that would not yet have applied then, correct? $\endgroup$
    – uhoh
    Commented Aug 18, 2017 at 15:40
  • $\begingroup$ I would be skeptical that Cotes' formula is actually $ix = \ln(\cos(x) + i \sin(x))$ -- it may be a translation into modern language, and things can be lost in translation. $\endgroup$
    – user14972
    Commented Aug 18, 2017 at 15:40
  • $\begingroup$ @Hurkyl I never said my math-history question was easy. I would not know where to begin, so I've asked for help here. $\endgroup$
    – uhoh
    Commented Aug 18, 2017 at 15:42
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    $\begingroup$ It might be worthwhile to ask this question on the History of Science and Mathematics Stack Exchange hsm.stackexchange.com. See for example a perhaps related question hsm.stackexchange.com/questions/4907/… and some answers and comments which mention Cotes's work. Somebody there might know something about the evolution of the idea since Cotes. (And probably they would understand your question better, too.) $\endgroup$ Commented Nov 5, 2017 at 4:55

1 Answer 1

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The emphasis on the single-valuedness of a function is a fairly recent phenomenon. Cauchy in his Cours d'Analyse (1821) deals in detail with multiple-valued functions for which he uses a double-symbol notation, such as $\sqrt{~}\!\!\!\!\sqrt{x}$ (to account for both roots in this case). If so, Cotes' formula was legitimate for his historical period.

Today we incorporate single-valuedness into our definition of a function following mid-19th century authors, so such a definition is no longer adequate.

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  • $\begingroup$ I've clarified the question a bit, I'm looking for literally how it changed. What was the definition of the complex logarithm then, and in what way has it changed. I've taken a guess in the question, but I'm not sure. Thanks! $\endgroup$
    – uhoh
    Commented Aug 18, 2017 at 15:28
  • $\begingroup$ I've decided to split off the more historical (rather than mathematical) aspects of the question to a separate (future) question in HSM SE. I think this covers the mathematical aspects quite nicely, thanks! $\endgroup$
    – uhoh
    Commented Oct 9, 2023 at 23:31

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