R help - Jan 2012 - Euler identity with complex exp

Hi,

Am i doing something silly here in expecting Euler's
formula to be handled by exp? exp( ix ) = cos x + i sin x.
The first example below follows this, the others not.

Thanks for the education!

 > exp( complex(real = 0, imag = 2*pi) )
[1] 1-0i
 > exp( complex(real = pi, imag = 2*pi) )
[1] 23.14069-0i
 > exp( complex(real = pi/2, imag = 0) )
[1] 4.810477+0i


	[[alternative HTML version deleted]]

Seems fine to me:

exp(pi + i*2pi) = exp(pi) * exp(i *2pi) = exp(pi) * (cos(2pi) +
i*sin(2*pi)) = exp(pi) *(1+ 0i) = exp(pi) ~ 23.14
exp(pi/2) ~ 4.81

What would you expect?

Michael

On Mon, Jan 30, 2012 at 10:37 AM, Joseph Park <josephpark at ieee.org>
wrote:
> Hi,
>
> Am i doing something silly here in expecting Euler's
> formula to be handled by exp? exp( ix ) = cos x + i sin x.
> The first example below follows this, the others not.
>
> Thanks for the education!
>
> ?> exp( complex(real = 0, imag = 2*pi) )
> [1] 1-0i
> ?> exp( complex(real = pi, imag = 2*pi) )
> [1] 23.14069-0i
> ?> exp( complex(real = pi/2, imag = 0) )
> [1] 4.810477+0i
>
>
> ? ? ? ?[[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

Not sure why you think the formula does not hold... but am guessing
you think that sin(x) and cos(x) are have values in [-1, 1]? Well that
only holds for real x. If you have a complex x, sin(x) and cos(x) are
unbounded - indeed, if you can write x=iy and y is real, you can show
(up to my own ignorance of possible signs) cos(x) = cosh(y), and
sin(x) = -sinh(y) simply by expressing (from the formula you wrote)
cos(x) and sin(x) as

cos(x) = ( exp(ix) + exp(-ix) )/2
and sin(x) = ( exp(ix) - exp(-ix) )/2

In any case, plug any complex number into
exp( ix )
and
cos x + i sin x

in R and you will get the exact same answers.

HTH,

Peter

On Mon, Jan 30, 2012 at 7:37 AM, Joseph Park <josephpark at ieee.org>
wrote:
> Hi,
>
> Am i doing something silly here in expecting Euler's
> formula to be handled by exp? exp( ix ) = cos x + i sin x.
> The first example below follows this, the others not.
>
> Thanks for the education!
>
> ?> exp( complex(real = 0, imag = 2*pi) )
> [1] 1-0i
> ?> exp( complex(real = pi, imag = 2*pi) )
> [1] 23.14069-0i
> ?> exp( complex(real = pi/2, imag = 0) )
> [1] 4.810477+0i
>
>
> ? ? ? ?[[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.