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MathGroup Archive 1998

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Re: Triangle function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14424] Re: Triangle function
  • From: John Doty <jpd at w-d.org>
  • Date: Wed, 21 Oct 1998 03:32:21 -0400
  • Organization: Wampler-Doty Family
  • References: <70dctu$22k@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Andrew Strobel wrote:
> 
> Hello,
> 
> Has Mathematica got a triangle function where I can choose the frequency
> and the amplitude?
> 
> The function is necessary to describe a voltage source. It must be an
> analytical function.
> 
> like this:
> 
>   /\    /\
> -/--\--/--\---->t
> /    \/    \

Your function isn't analytic at the turning points: it has no second
derivatives.

The simplest expression is probably:

2/Pi*ArcSin[Sin[t]]

However, this is not very tractable.

More conventional, and more tractable, is the Fourier series:

8*Sum[Cos[n*t]/n^2, {n, 1, Infinity, 2}])/Pi^2

replace Infinity with a suitably large odd integer, depending on the
bandwidth of your voltage source.

You can also transform the infinite series into a rather inscrutable
"closed form" by evaluating the following (try it):

<<Calculus`FourierTransform`
InverseFourierTransform[
  FourierTransform[Sum[8/Pi^2 Cos[n t]/n^2, {n,1,Infinity,2}],t,w],w,t]

This may be good for some purposes.

To adjust frequency and amplitude, multiply "t" by the angular
frequency, and the whole expression by the peak amplitude.

-- 
John Doty		"You can't confuse me, that's my job." Home: jpd at w-d.org
Work: jpd at space.mit.edu


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