{-
	Code transcribed from M. Douglas McIlroy,
	Power Series, Power Serious, JFP 1998 Functional Pearl.
-}

default (Integer,Rational,Float)

ps0, ps1, x :: Num a => [a]
ps0 = 0 : ps0
x = 0 : 1 : ps0

ps1 = 1 : ps1

infix 7 .*
(.*) :: Num a => a ->[a]->[a]
c .* (f:fs) = c*f : c.*fs

instance Num a => Num [a] where
   fromInteger c = (fromInteger c) : (fromInteger 0)
   negate (f:fs) = -f : -fs
   (f:fs) + (g:gs) = f+g : fs+gs
   (f:fs) * (g:gs) = (f*g : 0) + (0 : f.*gs + g.*fs) + (0 : 0 : fs*gs)

instance Fractional a => Fractional [a] where
   (0:fs) / (0:gs) = fs/gs
   (f:fs) / (g:gs) = let q = f/g in
      q : (fs - q.*gs)/(g:gs)

compose (f:fs) (0:gs) = f : gs*(compose fs (0:gs))

revert (0:fs) = rs where
    rs = 0 : 1/(compose fs rs)

deriv (f:fs) = (deriv1 fs 1) where
    deriv1 (g:gs) n = n*g : (deriv1 gs (n+1))

integral fs = 0 : (int1 fs 1) where
    int1 (g:gs) n = g/n : (int1 gs (n+1))

expx = 1 + (integral expx)

sinx = integral cosx
cosx = 1 - (integral sinx)
tanx = revert (integral (1/(1+x*x)))