Mathematica 9 is now available
Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2010

[Date Index] [Thread Index] [Author Index]

Search the Archive

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

In[2]:= step1 =
 FullSimplify[ComplexExpand[(Cos[x] + I*Sin[x])^n],
  Assumptions -> {Element[n, Integers], Element[x, Reals]}]

Out[2]= Cos[n Arg[E^(I x)]] + Sinh[n Log[E^(I x)]]

In[3]:= step2 = step1 /. {Arg[Exp[I*x_]] :> x, Log[Exp[I*x_]] :> I*x }

Out[3]= 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.
>
>



  • Prev by Date: Re: Scrolling text: Changing text color, etc.
  • Next by Date: Re: Scrolling text: Changing text color, etc.
  • Previous by thread: Re: The formula of Abraham Moivre
  • Next by thread: Re: The formula of Abraham Moivre