Raise complex number using De Moivre - integer only?

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This is probably a silly question, but it is not really clear to me whether De Moivre's theorem of raising a complex number to the nth power only work if n is an integer value?
E.g. if I try to raise (2-2i) to the power of 3.01 then my manual calculation get a different result than my calculator, but for all integer values it works.
 

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  • #2
cepheid
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This is probably a silly question, but it is not really clear to me whether De Moivre's theorem of raising a complex number to the nth power only work if n is an integer value?
E.g. if I try to raise (2-2i) to the power of 3.01 then my manual calculation get a different result than my calculator, but for all integer values it works.

Well, at least according to the Wikipedia article on the subject, 'n' must be an integer:

http://en.wikipedia.org/wiki/De_Moivre's_formula
 
  • #3
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>>Well, at least according to the Wikipedia article on the subject, 'n' must be an integer:
argh, I missed that. Thanks
Do you know how I can raise a complex number by a real number?
 
  • #4
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I may be wrong but in mathematics, n is used to denote an integer by convention.
 
  • #5
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>>I may be wrong but in mathematics, n is used to denote an integer by convention.
I think you are right.
 
  • #6
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Do you know how I can raise a complex number by a real number?

You can raise it to any power by using this method:

[itex]z^w = e^{ln|z^w|} = e ^{w \cdot ln|z|}[/itex] and simplify from there. From the top of my head, I believe this formula gives an infinite number of solutions depending on how you choose your branch of the logarithm.
 
  • #7
HallsofIvy
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No, the power does not have to be an integer. DeMoivre's formula is very commonly used, for example, to find roots, the "1/n" power.
 
  • #8
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After some more Bing'ing I found this
http://www.suitcaseofdreams.net/De_Moivre_formula.htm

For r = 1 we obtain De Moivre’s formula for fractional powers:
(cos+i sin)p/q = cos((p/q))+i sin((p/q)). (1.27)


So the decimal power should work as well, right?. I will try and convert the decimal to a fraction, but I dont see why it would make a difference.

When I tried it worked correct for quadrant I and II, but I could not get it right for III and IV.

I will keep playing with this.
 
  • #9
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I played some more with this and it was all just a matter of adjusting θ depending of the quadrant.
For quadrant II
θ=(atan(im(z)/re(z))+pi)*n, where n is any number
And for III
θ=(atan(im(z)/re(z))-pi)*n
 

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