/sci/ - Science & Math » Thread #12995225

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Anonymous Sun 18 Apr 15:44:21 2021 No.12995225 View Reply Original Report

Quoted By: >>12995403 >>12995633 >>12995756

Why is the fractional derivatives and integrals so uncommon in common core education? I studied calc 1, 2, multvariable calc, differential equations and I still haven't seen fractional calculus until a month ago.

Anonymous

Anonymous Sun 18 Apr 2021 16:19:27 No.12995371 Report

Quoted By: >>12995408 >>12995648 >>12995688 >>12995700

what is the half-derivative of a sine wave? what does that look like?

Anonymous

Anonymous Sun 18 Apr 2021 16:29:35 No.12995403 Report

Quoted By:

>>12995225
>common core
what?
they are uncommon because AFAIK I dont think they have much use at all to most people who take calculus in the first place

Anonymous

Anonymous Sun 18 Apr 2021 16:32:50 No.12995408 Report

Quoted By: >>12995474 >>12995529

>>12995371
Literally plug it into the definition given there with f(x) = sin(x)

$f^{\left( \frac{1}{2} \right)} (x)= \frac{1}{\sqrt{\pi}} \frac{d}{dx} \int_a^x (x - t)^{- \frac{1}{2}} f(t) dt<br />$
For any given x you can approximate the result to an abbitary percision using Tai's method.

Anonymous

Anonymous Sun 18 Apr 2021 16:47:27 No.12995474 Report

Quoted By:

>>12995408
are u terence tao?

Anonymous

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Anonymous Sun 18 Apr 2021 16:57:50 No.12995529 Report

Quoted By:

>>12995408
I tried plugging the taylor series for a sine wave into the equation written in green (the decimals are $\Gamma(2.5), \Gamma(4.5)$ etc.) and got this weird thing in wolfram

But if $\frac{d^{1/2}}{dx^{1/2}}$ is supposed to be consistent with $\frac{d^{1/2}}{dx^{1/2}}(\frac{d^{1/2}}{dx^{1/2}}(f(x)) = \frac{d}{dx} f(x)$ , I figured the half-derivative of a sine should just be shifted to the left 45 degrees.

Anonymous

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Anonymous Sun 18 Apr 2021 17:02:40 No.12995560 Report

Quoted By: >>12995611

and also got this using a different taylor series representation of sine

Anonymous

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Anonymous Sun 18 Apr 2021 17:11:14 No.12995611 Report

Quoted By: >>12995651 >>12995660 >>12995725 >>12995748 >>12995786

>>12995560
This is the plot that Maple gives.

Anonymous

Anonymous Sun 18 Apr 2021 17:14:19 No.12995633 Report

Quoted By:

>>12995225
What is common core? Some basic universal education for everyone? If yes, why would a cashier need to learn fractional derivatives?

Anonymous

Anonymous Sun 18 Apr 2021 17:15:23 No.12995640 Report

Quoted By:

because its only used in very specialised research applications to certain physics problems, teaching it to undergrads just because "it's cool lol" when they aren't going to do anything with it is pointless

Anonymous

Anonymous Sun 18 Apr 2021 17:16:46 No.12995648 Report

Quoted By:

>>12995371
>half
Is this common core intuition? lmao. America is fucked

Anonymous

Anonymous Sun 18 Apr 2021 17:17:14 No.12995651 Report

Quoted By: >>12995750

>>12995611
So it is just a 45 degree shift to the left? Also interesting distortion around x=0, what would cause it?

Anonymous

Anonymous Sun 18 Apr 2021 17:19:03 No.12995660 Report

Quoted By: >>12995681

>>12995611
what is the differenece between fresnelC and fresnelS

Anonymous

Anonymous Sun 18 Apr 2021 17:22:43 No.12995681 Report

Quoted By: >>12995716

>>12995660
$\mathrm{FresnelC}(x) = \int_0^x \cos \left( \frac{1}{2} \pi t^2 \right) dt \\<br />\mathrm{FresnelS}(x) = \int_0^x \sin \left( \frac{1}{2} \pi t^2 \right) dt<br />$

Anonymous

Anonymous Sun 18 Apr 2021 17:23:24 No.12995688 Report

Quoted By: >>12995711

>>12995371
$J^\frac{1}{2}(\sin(x))=\frac{1}{\sqrt\pi}\left[\frac{-\cos(x)}{\sqrt{x}}-\sqrt{2}\sqrt{{\pi}}\operatorname{S}\left(\dfrac{\sqrt{2}\sqrt{x}}{\sqrt{{\pi}}}\right)\right]$

Anonymous

Anonymous Sun 18 Apr 2021 17:26:09 No.12995700 Report

Quoted By: >>12995711

>>12995371
$J^\frac{1}{2}(\sin(x))=\frac{1}{\sqrt\pi}\left[\frac{-\cos(x)}{\sqrt{x}}-\sqrt{2}\sqrt{{\pi}}\operatorname{S}\left(\dfrac{\sqrt{2}\sqrt{x}}{\sqrt{{\pi}}}\right)\right]$

Anonymous

Anonymous Sun 18 Apr 2021 17:28:19 No.12995711 Report

Quoted By:

>>12995688
>>12995700
double dubs of obscurity
this clarifies nothing
shove it up your "S"

Anonymous

Anonymous Sun 18 Apr 2021 17:29:00 No.12995716 Report

Quoted By:

>>12995681
interesting so it's like 1/2*m*v^2 but not lol is this why it is involved in physics so much according to wikipedia

Anonymous

Anonymous Sun 18 Apr 2021 17:31:01 No.12995725 Report

Quoted By: >>12995763

>>12995611
does maple allow you to animate the function, with 1st derivative to 9/10, 8/10, 7/10.... derivative?

Anonymous

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Anonymous Sun 18 Apr 2021 17:35:42 No.12995748 Report

Quoted By:

>>12995611
Can you plot a complex plot like this in maple? Must be the software used for this plot right?

Anonymous

Anonymous Sun 18 Apr 2021 17:36:20 No.12995750 Report

Quoted By:

>>12995651
Maybe it really is the sin function shifted as that would make sense. If f(x)=sin(x) then
f'(x) = cos(x) = sin(x + pi/2)
f''(x) = -sin(x) = sin(x + pi)
f'''(x) = -cos(x) = sin(x + 3pi/2)
f''''(x) = sin(x) = sin(x + 2pi)
this suggests the s-th derivative is given by
f^(s) (x) = sin(x + s pi/2)

Anonymous

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Anonymous Sun 18 Apr 2021 17:37:33 No.12995756 Report

Quoted By:

>>12995225
It's too niche.

A niche PhD topic with too few applications.

>https://en.wikipedia.org/wiki/Fractional_calculus#Applications

Anonymous

Anonymous Sun 18 Apr 2021 17:38:56 No.12995763 Report

Quoted By:

>>12995725
I mean, you could just animate it phase shifting by (9/10, 8/10, 7/10...) * ?/2.
It seems like it's trivial to take fractional derivatives of anything with a fourier series

Anonymous

Anonymous Sun 18 Apr 2021 17:42:30 No.12995786 Report

Quoted By:

>>12995611
try fractional derivatives of bessel functions or hypergeometric functions

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