# Getting from a series to an underlying function

D

#### Don Lancaster

Jan 1, 1970
0
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

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

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

R

#### Robert Baer

Jan 1, 1970
0
Don said:
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

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.
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).

P

#### Phil Hobbs

Jan 1, 1970
0
Don said:
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

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.

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

K

#### Ken Smith

Jan 1, 1970
0
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

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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