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February 20th, 2013, 09:53 AM   #1
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imaginary part of a complex number

what is the imaginary part of
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February 20th, 2013, 11:51 AM   #2
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Re: imaginary part of a complex number







Since





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February 20th, 2013, 12:20 PM   #3
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Re: imaginary part of a complex number

thank you !
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February 21st, 2013, 11:37 AM   #4
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Re: imaginary part of a complex number

Quote:
Originally Posted by jks

Since
You have to justify this ...
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February 22nd, 2013, 09:21 PM   #5
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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 sub-sub 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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