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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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