How do you use the Taylor series for curve...

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How do you use the Taylor series for curve fitting when it requires knowing the curve in the first place?

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She is a beauty, but she do lack curves.

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>>7767309

because you don't do taylor series for curve fitting

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>>7767309

I thought you could only use taylor to approximate a function only near a certain point..? Can you use it to approximate an entire function? Correct me if I'm wrong /sci/

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>>7767349

the more terms you add to the taylor sequence, the broader your approximation gets.

the term of order 0 is just the value of the function at 0 for example.

if you include the term of order 1, you take into account the neighborhood of 0 as well. This will be a line that approximates best f around 0.

if you include the term of order 2, you start also taking into account the curvature, so your approximation gets more precise further from 0.

And so on and so forth.

(all of this works well only for functions that are equal to their taylor series of course)

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>>7767309

I would imagine you could do so because you can assume that a taylor series converges to your function for neigborhood around x, and apply Taylor's formula to deduce that a partial series will adequately represent the data that you have. Then it's a matter of simply finding the coefficients. But really, Taylor Series aren't used for curve fitting.

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>>7767352

so, with the goal of getting a perfect approximation near that point, you end up with a perfect approximation of the whole function?

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>>7767421

and also I guess this is its main difference from fourier transform (apart from the fact that you use cos/sin only now); in this case you don't "start" from a single point but you're already looking from -oo to +oo

is this correct?

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>>7767421

>>7767422

Power series expansions only work around the points you calculate them at.

And Fourier transforms only work for continuous functions and smooth functions.

If you want a goal for power series expansion, it allows you approximate the behaviour of a complex function with a few easy ones. In engineering this is usually done to simplify the equations around the points that physics allows.

Resistance for example is not actually R= V/I

It is a sigmoidal function, but a linear function approximates it well enough for usage.

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She's kinda cute :3

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>>7768218

>And Fourier transforms only work for continuous functions and smooth functions.

Fourier transform works for every L^1 function and can be extended to work with L^2 functions too.

You can even further extend it for all tempered distributions.

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>>7768218

>Power series expansions only work around the points you calculate them at.

so did the other anon lie to me?

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>>7768249

No, what he said is true for analytic functions.

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>>7767309

>curve fitting

You're thinking of polynomial interpolation. https://en.wikipedia.org/wiki/Polynomial_interpolation

Also, Taylor series just require knowing the function and all its derivatives at a point, and there's different methods for different functions/situations.

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>>7767349

Some functions can be represented as a power series, i.e. the "analytic functions" Some of these series converge for all complex numbers. These are called "entire."

Some functions f have Taylor series that do not converge to f even locally, say when all derivatives at a point are zero yet the function is not constant in any neighborhood of the point.

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Let's say I draw a random function, and lets say that I somehow am able to measure all the derivatives I want.

Would the power function make any sense?

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