[R] Euler identity with complex exp

[Peter Langfelder]

On Mon, Jan 30, 2012 at 11:43 AM, Joseph Park <josephpark at ieee.org> wrote:
> Thanks Michael & Peter.
>
> Michael's expansion makes sense.
>
> This is what I expected:
>
>> a = pi + 0i
>> complex( real = cos(Re(a)), imaginary = sin(Im(a)) )
> [1] -1+0i

As they say, the error is between the keyboard and the chair. You
cannot drop parts of a - in your formula above you dropped the
imaginary part of a in cos and the real part of a in sin. In this case
it doesn't make a difference but in general it will.

>
> Not this:
>> exp(a)

you need exp(ia), not exp(a):

i = complex(real = 0, imaginary = 1)
exp(i*a)
[1] -1+0i


> [1] 23.14069+0i
>
> Is this not an implementation of Euler's formula:
>> complex( real = cos(2*pi), imaginary = sin(2*pi) )
> [1] 1-0i
>
> And that is a result Michael depends on in his
> expansion, yet if we pass this argument to exp:
>> exp( (complex( real = 2*pi, imaginary = 2*pi) ) )
> [1] 535.4917-0i

Again, you are not using the formula correctly. Remember x = 2*pi, so you need
exp( i * 2 * pi)

and you get the same result as complex( real = cos(2*pi), imaginary =
sin(2*pi) )

Peter