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Getting from a series to an underlying function

Discussion in 'Electronic Design' started by Don Lancaster, Mar 12, 2007.

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  1. It is usually a fairly simple matter to get from a trig or exponantial
    function of one sort or another to its underlying series form.

    But how can you get from a known accurate series expression to a
    nonobvious and crucially esoteric equivalent function?

    Specifically, the "raw" power series

    [-1517.83 5094.6 821.18 -29457.7 61718.9 -61268.8 30448.6 -4770.84
    -269.684 -2892.14 3300.63 -1460.88 213.578 78.8959 -49.2164 12.3083
    -1.74731 0.149743 -0.00245142 0.103691]

    where 0.103691 is the x^0 term, -0.00245142 is x^1 etc...

    The equivalent McLauran Series (or Taylor about zero) is found by
    dividing each term by its factorial. 0.103691/1! , -0.00245142/2!...

    ... may be of extreme interest in finding a closed form expression
    that involves trig products and possibly exponantials. The range of
    interest is from 0 to 1.

    The function appears continuous and monotonic with well behaved derivatives.

    The trig angle of 84.0000 degrees is also expected to play a major role
    in the solution. As is the trig identity of cos(a+b) = cos(a)cos(b) -
    sin(a)sin(b). As is a magic constant of 0.104528. Everything happens in
    the first quadrant.

    Sought after is a closed form determnistic solution that accepts the 0-1
    value, the 84 degree angle, and the magic constant that evaluates to the
    above series.



    --
    Many thanks,

    Don Lancaster voice phone: (928)428-4073
    Synergetics 3860 West First Street Box 809 Thatcher, AZ 85552
    rss: http://www.tinaja.com/whtnu.xml email:

    Please visit my GURU's LAIR web site at http://www.tinaja.com
     
  2. Robert Baer

    Robert Baer Guest

    Well, if the series can be expressed like one sees in the Chem Rubber
    handbook then one needs to have a good idea as to what each "common"
    series looks like and use the massively parallel computer in a pattern
    matching experiment.
    Many times, the first guess is correct (providing you do this a lot,
    especially on a consistent basis as the ability does fade away over tie).
     
  3. Phil Hobbs

    Phil Hobbs Guest

    What you've given us is a polynomial, which is a very well behaved
    analytic function. What's not to like about it?

    You seem to think that there is some simple trig expression underlying
    this, but you haven't said why. Where did the coefficients come from?
    Is this a fitted function, or a series expansion of some analytic thing?
    There also seems to be some confusion--have you posted the values of
    the derivatives at zero, or the polynomial coefficients themselves?

    There's no unique way to translate that into a trigonometric-type
    function, because you can give the higher coefficients any values you
    like. If you can give a parameterized family of functions, you can fit
    the parameters so as to give the same polynomial coefficients.

    Cheers,

    Phil Hobbs
     
  4. Ken Smith

    Ken Smith Guest

    If the power series stops at 20 terms, the sin() and cos() family are very
    unlikely to be an exact fit.

    There for a minute I thought I had something but it didn't pan out.
     
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