What happens when you take the Taylor expansion of a straight

>>7969667
Nothing interesting. Why don't you try calculating it yourself?

>>7969667
Well if you took the Taylor series of a line (ie. y=mx+c) then the series would just terminate. For a Fourier series you'd still return an infinite series, all be it over a specific domain.

Taylor expansion of a straight line would just be the first two terms (a constant and a linear function), i.e. all further terms 0.

>>7969676
>Why don't you try calculating it yourself?
This
You could've even asked wolfram alpha to do it for you.

>>7969667
Polynomials are their own Taylor expansions

>>7969692
If centered at 0

The Taylor series of [math]x^2[/math] centered at 1 is [math]1 + 2(x-1) + (x-1)^2[/math]

>>7969914
try simplifying that expression

>this thread

I hope you're being ironic OP

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>>7969914
Jesus fucking Christ.

>>7969667
Do you understand what a Taylor series is supposed to be? Or a derivative for that matter?

>>7969941
>simplifying a taylor series
That's kind of missing the point.

>>7969667
fucking retard

>>7970046
Is this bait, or are you just retarded?

>>7970083
Is this?

Typical /sci/ autist
>>7969941
>>7970083

Guy who understands math is a art
>>7970046

Of course it could be replaced by [math]x^2[/math], we knew that going in, you don't find the Taylor series of an analytic function just to replace the series with the closed form you started with which you ALREADY KNEW it would equal.

It's like if I was given [math]e^x[/math] and derived the Taylor series [math]\sum_{n=0}^{\infty} \frac{x^n}{n!}[/math] and you spouted I could just replace it with [math]e^x[/math]
Dumbass.

>>7970584
R u retarded?

>>7970588

I'm just trying to point out why this post >>7969692 has the wrong idea since technically everything analytic function is its own Taylor Series - that's not the point

>>7969667
What is a Fourier transformation? I'm not high enough in my maths classes to learn it yet but I'm using it for my NMR work and it seems cool as hell.

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>>7970608
decomposition of a function into a frequency space .

>>7970613
Okay so essentially the wave like function generated by the NMR instrument is collapsed into discrete probabilities for the chemical states?

>>7970617
If your function already has a wave structure, then a fourier transform would tell you with what ratio each frequency appears in the function. Basically it's the difference between measuring the light that comes off a lamp as "energy absorbed per second" and actually using spectroscopy to decompose the light into it's component wavelengths.

I'm not really familiar with the NMR instruments output (only the quantum theory responsible), but I'm sure that it's analogous.

Fourier transforms may also be used to solve problems in a space where the solution is more immediate, then transforming back. See Laplace transformations if you're interested in this. They're beautifully used to transform differential equations into algebraic ones. It's actually one of the coolest fucking things I've ever seen in mathematics.

>>7970584
Literal retardation.

>>7970631
Ahh now I get it. Thanks for the help, I've been teaching myself the advanced NMR stuff and it's pretty fucking cool.

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>>7970639
Fourier-Transformed spectroscopy might also gather your interest. In condensed matter physics, the development of the scanning tunneling microscope has allowed us to actually see the wave nature of electrons on the surface of conductors. Pic related is a density of states map of an arrangement of iron atoms on a lower copper lattice. the little waves you see are actually electrons exhibiting quantum behavior (called Bloch waves).

We can find this since the density of states is proportional to the differential conductance which can be directly measured by the microscope.

If you took a 2D Fourier transform of the data, you could extrapolate key information about electron scattering off the iron atoms and the momentum states that they are likely to be in while propagating along the surface of the copper lattice.
This technique is being used to confirm the existence of novel states of matter right now. I worked on this technique briefly as an undergrad researcher.

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>>7970653
pic related:
No idea what sample this is, but on the left you have the density of states resolved spatially. Looks like noise or chaotic wave interference. On the right you have an FT map resolved in momentum space. Geometry emerges!

Probably a hexagonal lattice.

>>7970663
Also, with a set of energy resolved FT maps, you can reconstruct the band-structure of the material instead of using angle-resolved photo-emission spectroscopy (ARPES). I don't know if this technique (called FT-STS) is spin-resolved though...


Math is cool.

>>7970617
no. does your post even make sense?