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Re: expected output from log of negative number


From: Ben Abbott
Subject: Re: expected output from log of negative number
Date: Fri, 13 Jun 2008 19:40:47 -0400


On Jun 13, 2008, at 6:26 PM, address@hidden 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: address@hidden

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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




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