Symbolic calculation on Sage 4.7.1 using Napier's constant

code.sage

## Problem: Do not use symbol 'e'.

## see http://homepage1.nifty.com/herumi/diary/1110.html#14

var('x t a b c d g') # we can not use symbol 'e' instead of 'g'.
assume(t > 0)
f = ((exp(x) - (a + b*x + c*x^2 + d*x^3 + g*x^4))^2).integral(x,0,t)
ans = solve([f.diff(a), f.diff(b), f.diff(c), f.diff(d), f.diff(g)], [a,b,c,d,g])

## work around
ans1 = [eq.subs(e=exp(1)) for eq in ans[0]]

## check, limit to 0
for eq in ans1:
  print eq.limit(t=0)

for eq in ans1:
  print eq.left(), '=', eq.right().subs(t=log(2)/4096).n(500)