       Re: The formula of Abraham Moivre

• To: mathgroup at smc.vnet.net
• Subject: [mg106819] Re: [mg106792] The formula of Abraham Moivre
• From: Leonid Shifrin <lshifr at gmail.com>
• Date: Sun, 24 Jan 2010 05:41:14 -0500 (EST)
• References: <201001231236.HAA16448@smc.vnet.net>

```Here is a 2-step process which does it:

In:= Clear[n, x, step1, step2];

In:= step1 =
FullSimplify[ComplexExpand[(Cos[x] + I*Sin[x])^n],
Assumptions -> {Element[n, Integers], Element[x, Reals]}]

Out= Cos[n Arg[E^(I x)]] + Sinh[n Log[E^(I x)]]

In:= step2 = step1 /. {Arg[Exp[I*x_]] :> x, Log[Exp[I*x_]] :> I*x }

Out= Cos[n x] + I Sin[n x]

The second step is manual and is correct under an assumption that -Pi<x<=Pi,
which we can safely take given that the original function is periodic.

Perhaps there are shorter and completely automatic ways based only on
built-in rules  but I did not find them.

Regards,
Leonid

On Sat, Jan 23, 2010 at 4:36 AM, Arnold <sender999ster at gmail.com> wrote:

> How by means of Mathematica to transform (Cos [x] +I* Sin [x]) ^n in Cos
> [n*x] +I*Sin [n*x]?
>
> Thanks.
>
>

```

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