[R] Euler identity with complex exp

[R. Michael Weylandt]

This is off-topic for R-help, but we might as well finish what's been started:

Take a closer look at exp(i*x). If x is real, i*x is a pure imaginary
number, not a complex number so the formula you are using doesn't hold
in general.** The general Euler result for complex (= mixed real and
imaginary) numbers looks like this:

exp(x + iy) = exp(x)*(cos(y) + i sin(y))

That is, the real part gives the modulus and the imaginary part goes
solely to the argument. What's often surprising about this is that

exp(2 + 2*pi*i) = exp(2) = exp(2+4*pi*i) = exp(2 - 2*pi*i)

because the trig functions which get applied to the imaginary part are
periodic.

Take a closer look at what you wrote:

complex( real = cos(2*pi), imaginary = sin(2*pi) )
exp( (complex( real = 2*pi, imaginary = 2*pi) ) )

The number in the first line is not what gets exponentialed in the
second! You'll get the expected (by you) behavior if you actually use
the same number for both calculations:

complex( real = cos(2*pi), imaginary = sin(2*pi) )
exp(complex( real = cos(2*pi), imaginary = sin(2*pi) ))

or

complex(real = 2*pi, imaginary = 2*pi)
exp(complex(real = 2*pi, imaginary = 2*pi))

If you work out the second like I did for exp(pi + 2*pi*i) in my first
email, you'll get the correct answer.

All in all, R is definitely correct in it's "interpretation" of
Euler's formula. There's only one way to parse this relationship that
gives mathematical consistency and it's what Peter and I have set out
for you.

Michael

** Not actually true, if x is complex, it of course works out
correctly as well, but you wind up having to use the more general
expression I give to get there.

On Mon, Jan 30, 2012 at 2:43 PM, 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
>
> Not this:
>> exp(a)
> [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
>
> That would not work in Michaels expansion, the answer must
> be 1 + 0i.
>
> Which seems to suggest that exp( ix ) and cos x + i sin x (as
> written above) are different interpretations.
>
>
> On 01/30/2012 12:47 PM, Peter Langfelder wrote:
>
> 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.