Re: simple Sin

*To*: mathgroup at smc.vnet.net*Subject*: [mg89900] Re: [mg89859] simple Sin*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Tue, 24 Jun 2008 03:27:26 -0400 (EDT)*References*: <200806230646.CAA00151@smc.vnet.net>

Narasimham wrote: > It is surprising a bit, by numerical computation Sin[ t] has a > different series representation, is not even an odd function of t ! > Looks like has a different chemistry. > > NDSolve[{y''[t] + y[t] == 0, y'[0] == 1, y[0] == 0 }, y, {t, 0, 2 Pi}] > sin[u_] = y[u] /. First[%] > Plot[sin[t] - Sin[t], {t, 0, 2 Pi}] > > Series[{sin[t], Sin[t]}, {t, 0, 8}] > > Also, why do we not get an expansion of sin as Series[Im[Exp[I*t]], > {t, 0, 8}] ? > > TIA > > Narasimham The starting point for solving a differential equation numerically is probably not the best place to do a series expansion. For a contrast, try: NDSolve[{y''[t] + y[t] == 0, y'[-2*Pi] == 1, y[-2*Pi] == 0}, y, {t, -2*Pi, 2 Pi}] sin[u_] = y[u] /. First[%] Plot[sin[t] - Sin[t], {t, -2*Pi, 2 Pi}] Series[{sin[t] - Sin[t]}, {t, 0, 8}] // N Actually you can end at 0 (instead of 2*Pi) and still recover what appears to be a viable result from the series expansion. Daniel Lichtblau Wolfram Research

**References**:**simple Sin***From:*Narasimham <mathma18@hotmail.com>