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February 20th, 2013, 10:53 AM  #1 
Senior Member Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0  imaginary part of a complex number
what is the imaginary part of 
February 20th, 2013, 12:51 PM  #2 
Senior Member Joined: Jul 2012 From: DFW Area Posts: 609 Thanks: 83 Math Focus: Electrical Engineering Applications  Re: imaginary part of a complex number Since 
February 20th, 2013, 01:20 PM  #3 
Senior Member Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0  Re: imaginary part of a complex number
thank you !

February 21st, 2013, 12:37 PM  #4  
Math Team Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions  Re: imaginary part of a complex number Quote:
 
February 22nd, 2013, 10:21 PM  #5 
Senior Member Joined: Jul 2012 From: DFW Area Posts: 609 Thanks: 83 Math Focus: Electrical Engineering Applications  Re: imaginary part of a complex number
I'm glad that you asked for justification as it gave me cause to study complex exponentiation and complex logarithms further. If the justification that I give below is inadequate, then I will require, and appreciate, help to make it correct and complete. I think that you might have two main objections. The first may be that as this reference points out, for a real number, b, to complex powers u and z, then in general, is not valid (see the subsub section "Complex exponents with positive real bases" down below the sub section with the same title; I could not figure out how to link there directly since the titles are the same). The second objection may be that I used the principal value of but I did not specifically state that I was doing so. Further down on the same reference, a complex number, w, to a complex power, z is defined as . Also, this reference gives the logarithm of the principal value of a complex number in polar form, as: (which I assume extends to any value of ). Using these definitions, then: with . Taking the principal value with gives , and from the definition given above: (which is the next step after the 'Since ...' in my derivation above). Of course, this is equivalent to: and if we take the log, we get: Just for grins, taking a non principal value: I hope that this is correct, and that the justification is adequate. 

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