
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 20th, 2013, 09:53 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, 11:51 AM  #2 
Senior Member Joined: Jul 2012 From: DFW Area Posts: 614 Thanks: 83 Math Focus: Electrical Engineering Applications  Re: imaginary part of a complex number Since 
February 20th, 2013, 12:20 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, 11:37 AM  #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, 09:21 PM  #5 
Senior Member Joined: Jul 2012 From: DFW Area Posts: 614 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. 

Tags 
complex, imaginary, number, part 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Real part of complex?  archer18  Complex Analysis  3  March 2nd, 2014 12:46 PM 
Polynomial with Complex/Imaginary Roots  Schmidtacus  Complex Analysis  2  February 20th, 2013 09:12 AM 
Imaginary number problem  skarface  Algebra  6  March 10th, 2012 10:01 PM 
Complex equation decomposition to real/imaginary  silvetobristol  Complex Analysis  0  April 28th, 2010 01:50 AM 
Lots of imaginary number problems  Malgrif  Algebra  5  September 17th, 2009 03:26 PM 