>2.71828...
>e -- useful in calculus; equals
> lim (1+x)^(1/x) = lim (1+ 1/x)^x
> x -->0 x --> infinity
> and (here's the kicker...) diff(e^x, x) = e^x
lets see how good my memory is:
i like e... ;-)
i = (-1)^(1/2) (imaginary unit)
e^(ix) = cos x + i(sin x) (so e^(iPi) = -1)
e can also be expressed as the summation of 1/n! index n=0 limit infinity
e^x is the summation of x^n/n! index n=0 limit infinity
therefore if you take e^(ix) the real terms will be only the even n's
imaginaries will be all the odd terms.
real (e^(ix)) = summation of ((-1)^(n+1)(x)^(2n))/(2n)! (index n=0 and limit
infinity)
imag (e^(ix)) = summation or ((-1)^(n)(x)^(2n-1))/(2n-1)! (index n=0 and
limit infinity)
cos x = summation ((-1)^(n+1)(x)^(2n))/(2n)! (index n=0 limit inf)
sin x = summation ((-1)^(n)(x)^(2n-1))/(2n-1)! (index n=0 limit inf)
and that is how i choose to define sine and cosine functions...
~amir~
~the great tinkerer