[seqfan] Re: Does (e i)^(pi i) have an imaginary part?

[Andrew N W Hone]

Dear Robert, 

Indeed you are right. I was being rather cavalier in working only with the principal branch of the logarith (n=0), in order 
to recover the output from Mathematica, and had not noticed that you had already given a much more accurate account 
in an earlier message.

Best wishes,
Andy 


________________________________________
From: seqfan-bounces at list.seqfan.eu [seqfan-bounces at list.seqfan.eu] On Behalf Of Robert Israel [israel at math.ubc.ca]
Sent: 08 August 2010 18:39
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Does (e i)^(pi i) have an imaginary part?

On Sun, 8 Aug 2010, Andrew N W Hone wrote:

> Hi -
>
> It's best to learn how to unravel this without a computer if you can, as with mathematica or maple
> one can always worry about whether zero really means zero!
>
> I would suggest reading an introductory text on complex analysis: the complex exponential is usually one of
> the first examples of analytic functions in any such book.
>
> The multi-valuedness of the logarithm may take a bit longer to understand than the exponential.
>
> However, if you know how the ordinary (real) exp and log functions work, then you should quickly get the hang of the
> complex case.
>
>> From the basic rules of exponentiation,
>
> (a b)^c = a^c b^c

Stop right there.  This is not valid in general for complex numbers.
The basic rules are
     a^b = exp(b log a)
     exp(a + b) = exp(a) exp(b)
     if exp(a) = exp(b) then a - b = 2 pi n i for some integer n

So (a b)^c = exp(c log(a b))
while a^c b^c = exp(c (log a + log b))
and log(a b) = log a + log b + 2 pi n i
so (a b)^c = exp(2 pi n c i) a^c b^c

> means that
>
> (e i)^(pi i) = e^(pi i) i^(pi i) = - i^(pi i).
>
> Also, exp and log are inverses of each other, and one of the basic rules of logs is that
>
> log (a^b) = b log a,

Again not true in general: log(a^b) = b log a + 2 n pi i.

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada


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