Thanks Gentlemen. Now I see the disconnect.
I was misusing exp( i x ) and expecting to get exp( i x ) = cos x + i sin x, which is Euler's formula. Since it is a mapping of a real number onto the unit circle in the complex plane, any answer it gives must have a magnitude of 1 and the argument to it must be purely real. It seems the notation of i in exp( i x ) is used to indicate the complex mapping of the real x into the complex plane. Thus my error in passing a complex argument to exp. Rather I should stick with: expi(x) = complex( cos(x), sin(x) ) when working with phasors based on Euler's formula. Just as a note: I don't think this is an expression of Euler's formula: exp(x + iy) = exp(x)*(cos(y) + i sin(y)) The argument is complex and will in general not map onto the unit circle. On 01/30/2012 02:13 PM, R. Michael Weylandt wrote: > 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<josephp...@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<josephp...@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@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. [[alternative HTML version deleted]] ______________________________________________ R-help@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.
