Talk:De Moivre's formula

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HI De Moivre's formula is actually true for all complex numbers x and all real numbers n, but this requires careful extension of several functions to the complex plane. <-- Loisel I'm not so sure this makes any sense<math>.</math>

Thanks for pointing that out; n has to be an integer. x can be complex however, since Euler's formula

e^(ix) = cos(x) + i sin(x)

works for all complex x. --AxelBoldt

Is the function described as a [multivalued function]? in fact a periodic function?

Yes it is. It is 2π periodic, because sin and cos are. --AxelBoldt

Is the function described as a multivalued function in fact a function? If so, what is the domain of this function? z,w being fixed chosen numbers, it is is some sense rather a multivalued constant, isn't it? (I mean, a constant can be seen as a nullary function, but this would be even more nonsense hair-splitting... and again, I can't see on what domain it is defined - don't say the empty set, because it is well known that there is only one universal function defined on this set, and it definetely has not multiple complex values.) — MFH: Talk 19:46, 10 May 2005 (UTC)


Contents

[edit] proof

Is the proof in the article the original proof of the formula? I mean, is it the way DeMoivre obtained or proved it?

Since DM's formula is a lot easier to prove using Euler's formula, I see no other reason for this proof to be in the article (except curiosity)... -- Euyyn (March 26 2005, 2346 GMT)

[edit] first proof

I don't think the first proof is correct, because we are assuming without proof that the exponential law is true also for imaginary numbers.

[edit] cos(x+i)

I think you guys have the parentheses in the wrong places, don't you?

cos(x+i) sin x is not the same thing as cos x + i sin x, and I'm pretty sure the latter is what you mean. When someone who knows how to edit HTML fixes this, feel free to delete my comments here on the talk page. --anon

You are right, there was a bug in the proof. Fixed now. Oleg Alexandrov 19:33, 3 August 2005 (UTC)

[edit] de moivre formula is valid for all n

i dunno why this article says only integers, but i can prove its true for i. all i did is the hyberbolic trig functions and i proved that cis(ix)=e^-pi or something, my point is its valid for all complex n. is there something wrong with what i did? --anon

The big problem is that for nonintegers n, the n-th power is not defined uniquely. If you try to use De Moivre's formula then, you run into trouble. See here:

-1 = \cos \pi + i \sin \pi\,

but also

-1 = \cos 3\pi + i \sin 3\pi\,

By De Moivre's formula:

(-1)^{1/2} = \left(\cos \pi + i \sin \pi\right)^{1/2} =\cos \frac {\pi}{2} + i \sin \frac{\pi}{2}= i\,

but also

(-1)^{1/2} = \left(\cos 3\pi + i \sin 3\pi\right)^{1/2} =\cos \frac {3\pi}{2} + i \sin \frac{3\pi}{2}= -i\,

so you get two different answers which is not good. Does this help? Oleg Alexandrov (talk) 05:51, 16 October 2005 (UTC)

[edit] TeX formatting/Expert plea

I did a lot of formatting with the TeX and the general organization. Since I'm not an expert in the topic, I really hope I didn't mess anything up. So, if anyone who is an expert can give it a check, I'd be very grateful. Foxjwill 19:52, 5 March 2007 (UTC)

[edit] Picture

The picture has the cube roots of unity as (-1/2 +/- i*sqrt(3/2)), where they should be (-1/2 +/- i*sqrt(3)/2).