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>
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