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.
