expected output from log of negative number

[Ben Abbott]

On Jun 13, 2008, at 6:26 PM, heimdal at aracnet.com wrote:

> Rob Mahurin wrote:
>
>> On Jun 13, 2008, at 2:40 PM, E. Joshua Rigler wrote:
>>> If I type log10(-1), I get a complex number back whose real part is
>>> the log of the absolute value of the argument, and whose imaginary
>>> part is always equal to 1.36438.  What's more, I get similar
>>> behavior with a natural log, but the imaginary part is always equal
>>> to Pi.
>>
>> If you want exp(ln(x)) == x for negative x, you need this behavior.
>> You can play with Taylor expansions to see that
>>
>> 	exp(ix) = cos(x)+i*sin(x)
>>
>> (look for "Euler equation"), and so
>>
>> 	exp(i*pi) = -1,
>>
>> which suggests
>>
>> 	i*pi = ln(-1).
>>
>> Some people are intrigued that -i*pi, 2i*pi, etc. work just as well.
>> Intrigued enough to write books.
>>
>> Cheers,
>> Rob
>>
>> -- 
>> Rob Mahurin
>> Dept. of Physics & Astronomy
>> University of Tennessee 	phone: 865 207 2594
>> Knoxville, TN 37996     	email: rob at utk.edu
>>
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>
> Hi Rob,
>
> Interesting. Could you recommend a book?
>
> Scientia est potentia,
>
> John

Wikipedia has some nice pages on the subject

	http://en.wikipedia.org/wiki/List_of_complex_analysis_topics

	http://en.wikipedia.org/wiki/List_of_complex_analysis_topics#Special_functions

	http://en.wikipedia.org/wiki/Exponential_function

Ben