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]
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 josephp...@ieee.org wrote:
Hi,
Am i doing something silly here in
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
On Mon, Jan 30, 2012 at 11:43 AM, 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
As they say, the error is between the keyboard and the
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
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 =
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
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