Hillarious FlyingGrandma You see this type of stuff very often in quantum physics. However, in other fields of physics, this would be irrelevant. You are safe as long as you do not study quantum. But also, it is not as scary as it looks. Once you get used to the definition, these types of derivatives become intuitive
#drpeyam WHOA! What is Spirit Science? It's a UA-cam channel that promotes New Age ideas, so in order to show that real science must be respected, you must search for the Spirit Science UA-cam Channel, and don't be afraid dislike every video on Spirit Science. Again, for real science, 𝗻𝗼𝘁 hate itself, because that can be a violation of the UA-cam policy concerning hate speech. : )
Just finished out differential equations today. I just want to thank you for helping me through it. At the beginning of this semester I commented on one of your videos that it was a shame you didn’t have any videos for diffy q, but then you directed me to your playlists and boy did those videos (along with Patrick JMT) save me in that class. You two youtubers helped me through my entire math career. Thank you.
5:18 how can you just write ...3*2*1 at the end there? if the whole point is that 'a' is not necessarily an integer, then (a-n) is not necessarily an integer, so it may never hit 3*2*1.
Since the Gamma function (or the Pi function for that matter) are arguably 'the best' we have as an extension of the factorial, particularly those essential discontinuities at zero and negative integers makes me wonder if the meaning of the function for z
This is one thing that is really cool about math; you can take something that is somewhat intuitive (derivatives are reasonably intuitive, since they are kind of like extreme cases of the point-slop formula), and then expand on that and show that it works for much more general situations.
It can be thought of as an "interpolation" between the integer derivatives (and antiderivatives, if we consider the "-1th" derivative to be the indefinite integral). This is because Γ is a continuous function in most of its domain, so small variations in the order of derivation will result in small variations of the resulting functions. Here's an animation: en.wikipedia.org/wiki/File:Fractional_Derivative_of_Basic_Power_Function_(2014).gif
Because why not? Jokes apart,it helps out a lot in analytic number theory and making sense of continuous products. By converting it into an integral for a generalized number, you open up a plethora of possibilities in analytic number theory, calculus, statistics and a lot more
There is also use in quantum gravity where space time experiences spectral dimensional reduction (has to do with brownian motion). In these cases you can have a 2.5, a pi, or anything else between 1.8 and 4.2 dimensions(depends on which theory you're looking at).
@@arpitdas4263 It's a history question rather than a YES and NO question. You don't say the emergence of the wave function is "why not", it has its own history, motivation for many years of classical quantum theory.
I really appreciate your channel, it's entertaining, educational, impressive, and unlike some people, you don't belittle minorities in the process! Thank you for making this videos, I appreciate you
I am a high school student taking Accel. Algebra 2 (That is covering half of geometry and all of Algebra 2) in my 10th grade year. Even know I BARELY know anything about differential calculus/calculus at all, I oddly enough understand what is going on in the video. And this is why I love stuff like this; stuff like gamma functions, pi functions and everything in between. I love to work out math equations on my spare time; finding factors of really big numbers like 99,999,999 (hint, there are three prime numbers that multiply to make a palindrome), gaps between primes, and so on. Math is really, REALLY beautiful if you have time for it.
Lol, i bought bis Book today And was surprised of this exercise and wrapped my Head Around this exercise, but now you explanined it😄 P.s:omg my hypothethis was Gamma of pi.😂
The gamma function is used probably because of a notational mistake but more usefully, because the gamma function appears as the output of transforms that use the Haar measure (which is very nice) Most notably the Mellin transform on the exponential decay function (e^-x)
This is a nice heuristic argument, but would be better by introducing the fractional operator as an integral transform and going from there to get Gamma(pi + 1).
Originally I am going to buy this book coz I also follow daily math IG. After reading your book review and also this video, I know my decision is defintely correct! There is a lot of fun in this book!
For any integer - d^n/dx^n [ f(x) = x^n ] = n(n-1)(n-2)... n times = n! = gamma(n+1) to allow non-integers. So, the answer is gamma(π+1) = *7.188082728976031*
Do you think you could make a video about hyperoperations? Specifically, I'm interested in a super-exponential function. For example, e^^x. The complicated thing is, there is really no known way to take a value to a non positive integer superpower (or at least that I know of), since the best definition we have for tetration is iterated exponentiation.
Another interesting thing that I'm not sure has been explored might be non integer operations? Or maybe that's just too much abstract bullshit at that point lol
Is the way you extend the n-th derivative of x^m from natural to real numbers unique? One could think even of analytically continuation to complex numbers by considering this as a function of n and m. But therefore you need either, that n and m are accumulation points in some area of the complex plane (thats not the case for natural numbers), or that every derivative matches in one point. Is the latter true for this situation?
I didn't get that last integral at the end, t^pi*e^-t dt which you set to approx 7.188. Evidently I missed some past video (can't find it). In this one, I couldn't make out what you said during the fast-motion sequence for this integration. How was it done?
i haven't fully grasped the idea of fractional/irrational derivatives ... so if (n-a) is a positive integer, would the answer just be 0? since at some point we will keep differentiating 0 over & over again?
Hey, can you do a "100 limit problems in one go" type video? I really need that. Your problems sets are good and not super easy ones like in most youtube videos.
My friend challenged me to find the πth derivative of x^π, at first I argued that it doesn't exist then I worked a little and found π!, but I thought that I was of course wrong but seeing this video makes me confident.
The (d^n/dx)(x^n) is simply n factorial. So the (pi)th derivative of x^pi is pi factorial right? And pi factorial is 7.188 (searched it on google) The answer is 7.18808272898...right?
The Gamma function is more popular because mathematicians have the extremely bad habit of what seems to be intentionally defining non-normalized shifted versions of the actual function we would realistically work with. For example, it happens with the definition of sinc(z), so much that non-mathematicians just use a different definition altogether. The same goes with the Fresnel integrals. It goes for a ton of different functions.
The Master Not if you define it via falling factorials. The falling factorial with negative exponent is well-defined, and is equal to a rising factorial.
Ben Evans Fractional derivatives are much like fractional powers. For instance, D^(1/2)·D^(1/2)[f(x)] = f'(x). In words, the half derivative of the half derivative is the first derivative. It also allows you to talk about definite integrals as derivatives to a negative power. The applications I am familiar with come from quantum physics. It can be particularly useful in solving differential equations.
A function equal to a number ? You might want to put (1) after each derivative to apply the function and get !n (Because the function x->d/dx^n(x^n) applied on 1 is n!)
Ha! I got it right! I treated it like a Taylor Polynomial. I remembered k(k-1)(k-2)... from Binomial series can be represented as k!/(k-n)! and thus represented the nth derivative of f(x)=x^pi as the following: [pi!/(x-pi)!]*x^(pi-n), then just inserted pi for n to get the pi-th derivative
Maybe I should watch the video about the gamma function first, but it looks like it's very similar to a Laplace transformation. Are there any correlations between the two?
Hermann Barbato The Gamma function is related to the Laplace transforms, but the fractional derivatives have absolutely nothing to do with Laplace transforms.
Hermann Barbato Let La[f(t)](s) denote the Laplace transform of f(t) with respect to s. If Re(s) > 0 & z in C\N-, then La[t^z](s) = Γ(z + 1)/s^(z + 1).
This is deeply connected to analytic continuation in complex analysis, so I don't see why the ith derivative wouldn't work but can't think of a formal proof or anything that it is indeed well-defined (i suppose we have to look at some dimensional regularization analysis and apply it to derivatives LOL)
When I was in school I looked at using fourier transforms to do these types of strange derivatives and one that I came up with was using (1/w)^w which is d/dx omega times. I think I called it the frequency derivative
In 1981 I asked my college algebra teacher if there was a 3/4 or 1/2 derivative, similar to x^3/4 or x^1/2 and he said: No! It doesn't make any sense. When was this concept originated?
you simply wont add that then. Because if a-n =0, a-(n-1) = 1. You've already got a! (constant) if you derivative it one more time, itll become 0. which is why a*(a-1)*(a-2)..(a-n) is the n+1th derivative of x^n = 0
I think the gamma function is more popular and stuff because of the gamma distribution in statistics. Having the gamma function inside the gamma distribution's probability density function's definition makes it nice and cute, but having pi inside gamma sounds weird :P
OK, for general case, you have this d^n/dx^n (x^a), this makes sense. But - suppose that a is an positive integer, and n is a non-integer greater than a+1. Meaning that using standard derivations, you will eventually reach zero, leaving only a fractional derivative of zero, which is also zero. However, when looking at the factorial representations, you run into something a bit more interesting. a! is a finite positive number, (a-n)! is also a finite number, as it's a non-integer negative factorial, and a-n is a finite negative number. Let's call a!/(a-n)! k for the time being, and (a-n) can be b. kx^b is a standard function, and due to b being negative, kx^b can never be zero! Since these two interpretations directly contradict, one of them must be false. Or possibly both. Which, if any, is considered the correct way to resolve this bit of calculus?
How do you know that a fractional derivative of zero is always zero? Interestingly, when working with integer derivatives, you don't run into this issue, because that would give a!/(a-n)!, where a-n is a negative integer. This is, using the Gamma function, infinity. Dividing by infinity is zero, so there's your zero. But with this interpretation, I am lead to believe that a fractional derivative of zero doesn't necessarily have to be zero. This makes me think of the +c of negative derivatives (integrals), however in case of f(x)=0, c is always 0. Huh. EDIT: now that I think of it, when n is negative (resulting in an integral), there should be some c's in there. Or maybe this only gives the principal antiderivative? But that then assumes c=0, so that's another problem in the reasoning above.
@@fghsgh This is only on the assumption of successive derivations being equivalent to taking the fractional derivative as a whole, which may be wrong. But when we're getting to the point where we've done a derivatives, this is, with respect to x, a!*x^0. The next derivative on the stack, then, multiplies in the zero, rendering the whole equation to zero. Further normal derivatives can't get rid of the zero, of course. But since the fractional derivative is also giving finite numbers, that also can't get rid of zero. Though if you throw in a negative integer into the pi function, you get the necessary infinity, but once you hit the point of f(x)=0 that could be any arbitrary power. Singularities are silly like that.
An intro like this should be included in calc 1 getting students to know derivatives is not just integer but it can take fractional derivative and can be think of as a function operated on an equation
Bro thats sounds like d^3/dx^3(x^3) with extra steps
did i say it 4 times?
It's the same answer anyway, he didn't need to go the extra step imo.
This comment was made by Engineering Meme Gang
Ulala, someone is gonna get laid in college
@@mountainc1027 lol
This is engineering thing
I used this book: amzn.to/2PpOJIX
That's one of those things that just looks absolutely impossible, but someone's solving it for 9 minutes.
Thanks!
Just the video-title on its own managed to make me feel really anxious. I am glad we physicists don't see things like that so often :D
Hillarious FlyingGrandma You see this type of stuff very often in quantum physics. However, in other fields of physics, this would be irrelevant. You are safe as long as you do not study quantum.
But also, it is not as scary as it looks. Once you get used to the definition, these types of derivatives become intuitive
WHOA!
This one vid really blew my mind.
Big fan you sir 😍🙏🏼
Who you resolve any question of calculation,i like your vídeos.
#drpeyam WHOA! What is Spirit Science?
It's a UA-cam channel that promotes New Age ideas, so in order to show that real science must be respected, you must search for the Spirit Science UA-cam Channel, and don't be afraid dislike every video on Spirit Science. Again, for real science, 𝗻𝗼𝘁 hate itself, because that can be a violation of the UA-cam policy concerning hate speech. : )
Make that like button blue if you subbed to my channel
I just found out that
Gamma(pi + 1) = 7.1880827...
Just finished out differential equations today. I just want to thank you for helping me through it. At the beginning of this semester I commented on one of your videos that it was a shame you didn’t have any videos for diffy q, but then you directed me to your playlists and boy did those videos (along with Patrick JMT) save me in that class.
You two youtubers helped me through my entire math career.
Thank you.
5:18 how can you just write ...3*2*1 at the end there? if the whole point is that 'a' is not necessarily an integer, then (a-n) is not necessarily an integer, so it may never hit 3*2*1.
nathanisbored yes you are right!!!! I should have done it as multiplying the top and bottom by (a-n)! that would have been better.
I will pin this for other ppl to see. Thanks Nathan.
you rock nathan!
Since the Gamma function (or the Pi function for that matter) are arguably 'the best' we have as an extension of the factorial, particularly those essential discontinuities at zero and negative integers makes me wonder if the meaning of the function for z
Yooo nathanisbored I love your videos
This is one thing that is really cool about math;
you can take something that is somewhat intuitive (derivatives are reasonably intuitive, since they are kind of like extreme cases of the point-slop formula), and then expand on that and show that it works for much more general situations.
How do you even interpret the pi-th derivative geometrically?
Gottfried Leibniz You don't.
Not everything has a geometric interpretation in mathematics.
It can be thought of as an "interpolation" between the integer derivatives (and antiderivatives, if we consider the "-1th" derivative to be the indefinite integral). This is because Γ is a continuous function in most of its domain, so small variations in the order of derivation will result in small variations of the resulting functions. Here's an animation: en.wikipedia.org/wiki/File:Fractional_Derivative_of_Basic_Power_Function_(2014).gif
Wasn't it your invention?
@@thorbynumbers5368 no it was Newton's
There’s a video on my channel about the geometric intuition behind it, I think it’s the Grunwald-Letnikov formula or something
Got out of surgery a few hours ago, so I was very happy to see a new video from you in my recommended. Nice work :D
2:26 Ah yes, my favorite kind of numbers. Negative whole numbers!
Also,
Ah yes, the floor is made out of floor!
What's the motivation behind the expanding of the definition of derivative into the real number order?
Because why not?
Jokes apart,it helps out a lot in analytic number theory and making sense of continuous products. By converting it into an integral for a generalized number, you open up a plethora of possibilities in analytic number theory, calculus, statistics and a lot more
@@arpitdas4263 It also has uses in fluid dynamics.
There is also use in quantum gravity where space time experiences spectral dimensional reduction (has to do with brownian motion). In these cases you can have a 2.5, a pi, or anything else between 1.8 and 4.2 dimensions(depends on which theory you're looking at).
@@arpitdas4263 It's a history question rather than a YES and NO question. You don't say the emergence of the wave function is "why not", it has its own history, motivation for many years of classical quantum theory.
You should be having a billion subscribers coz studying Mathematics with you is really really Fun, love these kinda videos, Fantastic work sir!
OMG, thanks for the shout-out!!! ❤️
Dr πm now you might calculate πm th derivative of x^πm
Here’s the man! Mr. Dr. Peyam!!!
I really appreciate your channel, it's entertaining, educational, impressive, and unlike some people, you don't belittle minorities in the process! Thank you for making this videos, I appreciate you
I am a high school student taking Accel. Algebra 2 (That is covering half of geometry and all of Algebra 2) in my 10th grade year. Even know I BARELY know anything about differential calculus/calculus at all, I oddly enough understand what is going on in the video.
And this is why I love stuff like this; stuff like gamma functions, pi functions and everything in between. I love to work out math equations on my spare time; finding factors of really big numbers like 99,999,999 (hint, there are three prime numbers that multiply to make a palindrome), gaps between primes, and so on.
Math is really, REALLY beautiful if you have time for it.
What about the i-th derivative of a function? (Yes, i^2=-1)
Does it make any sense at all?
I think u r the only teacher who makes me so curious about maths
Your writing style is very good
Some great theoretical math!... Any practical science or engineering applications?
According to the fundamental theorem of engineering, I got 6.
oh, that's a familoar form! Before shifting to the gamma function, that's just P(a,n) * x^(a-n). Neat!
"at the end we'll get a really nice answer"
later at the end: "~2.188"
I truly was hoping that you'll show an exact answer like e^(pi/4) or something
José Ignacio Cuevas Barrientos He did give an exact answer. It is π!
7.188*
1.488*
@@alexanderskladovski no
e^(pi/4) = ~2,193. lol intuitively?
Hi! At 5:52 how do you make sure a
You are so fun and kind!
Thank you.
If a and n are both integer and a < n the answer should be 0 right?
Lol, i bought bis Book today And was surprised of this exercise and wrapped my Head Around this exercise, but now you explanined it😄
P.s:omg my hypothethis was Gamma of pi.😂
Ups, it' s gamma(pi+1)😂
I discovered that use of derivatives that gives you pattern of factorial almost 2 years ago and never got to post it anywhere in the internet.
The gamma function is used probably because of a notational mistake but more usefully, because the gamma function appears as the output of transforms that use the Haar measure (which is very nice)
Most notably the Mellin transform on the exponential decay function (e^-x)
No notational mistake!
This is a nice heuristic argument, but would be better by introducing the fractional operator as an integral transform and going from there to get Gamma(pi + 1).
For Γ(-1):
Using (integral = ζ)
Γ(t) = ζ[0,infinity] [ (e^-t)/t ] dt
=> Γ(-1) = -ζ -(e^-t/t)dt
=> Γ(-1) = -E1(t) |(0,infinity)
=> Γ(-1) = -E1(infinity) +E1(0)
=> Γ(-1) = -0+ infinity
Taking limit from x=1_ becomes + Infinity
Taking limit from x=1+ becomes - infinity
.•. Γ(-1) is divergent, there’s asymptotes for each negative integer.
I want to subscribe you million times .
This looks amazing ..
But I have a problem :
Can we inverse a lot of functions in one function ?
So in order to differentiate we just need to calculate some integrals? That sounds really useful xD
for the value of pi factorial you will need the following video:
ua-cam.com/video/L4Trz6pFut4/v-deo.html
When he takes the blue pen, either math turns into a crime, or it becomes really, really dangerous.
The book you showed in the video should be used for which level (calc 1 ,2,3 or adv )
Plz tell !
Bhupinder Kaurhut a strong understanding in calc 2 will help.
Originally I am going to buy this book coz I also follow daily math IG. After reading your book review and also this video, I know my decision is defintely correct! There is a lot of fun in this book!
Thank you!
@@hamza_alsamraee Welcome!
WOAH ... you are almost at 400k
Yea! I am very happy about it! Especially I set that goal in the beginning of this year!!
Congrats for the 400000! (
jagatiello thank you!!!!!!
For any integer - d^n/dx^n [ f(x) = x^n ] = n(n-1)(n-2)... n times = n! = gamma(n+1) to allow non-integers. So, the answer is gamma(π+1) = *7.188082728976031*
Guys, we did it! blackpenredpen reached 400K subscribers! Let's celebrate with #YAY !
The Kremlin clock chimes 12 times at the beginning of 2020...
Yes!!! Thank you!!!! I am running a giftaway for Christmas. See my community post and let me know what you think.
Do you think you could make a video about hyperoperations? Specifically, I'm interested in a super-exponential function. For example, e^^x. The complicated thing is, there is really no known way to take a value to a non positive integer superpower (or at least that I know of), since the best definition we have for tetration is iterated exponentiation.
Another interesting thing that I'm not sure has been explored might be non integer operations? Or maybe that's just too much abstract bullshit at that point lol
Is the way you extend the n-th derivative of x^m from natural to real numbers unique? One could think even of analytically continuation to complex numbers by considering this as a function of n and m. But therefore you need either, that n and m are accumulation points in some area of the complex plane (thats not the case for natural numbers), or that every derivative matches in one point. Is the latter true for this situation?
I didn't get that last integral at the end, t^pi*e^-t dt which you set to approx 7.188. Evidently I missed some past video (can't find it). In this one, I couldn't make out what you said during the fast-motion sequence for this integration. How was it done?
That is exactly why math is beautiful!
i haven't fully grasped the idea of fractional/irrational derivatives ... so if (n-a) is a positive integer, would the answer just be 0? since at some point we will keep differentiating 0 over & over again?
Sir you relate your videos pretty well!
Seeing 3B1B as one of your patrons made me smile
Thank you!!!
Hey, can you do a "100 limit problems in one go" type video? I really need that. Your problems sets are good and not super easy ones like in most youtube videos.
Can you do a video on laurent series?
Always cool and great extension of my HS maths.
Hi, Can you give a concise proof of higher order derivative test for Max/min
Great work, now what about a πth order differential equation?
Fascinating!
My friend challenged me to find the πth derivative of x^π, at first I argued that it doesn't exist then I worked a little and found π!, but I thought that I was of course wrong but seeing this video makes me confident.
what is sum of 1/r! upto infinity
The (d^n/dx)(x^n) is simply n factorial. So the (pi)th derivative of x^pi is pi factorial right? And pi factorial is 7.188 (searched it on google)
The answer is 7.18808272898...right?
How did you solve the integral at the end?
What is geometrical interpretation of radical order derivatives
Can you also get the pi'th antiderivative?
What's the name of the background music piece? Interesting video.
The Gamma function is more popular because mathematicians have the extremely bad habit of what seems to be intentionally defining non-normalized shifted versions of the actual function we would realistically work with. For example, it happens with the definition of sinc(z), so much that non-mathematicians just use a different definition altogether. The same goes with the Fresnel integrals. It goes for a ton of different functions.
I saw the thumbnail and I went and tried it on my own. I got the answer but I used the laplace transform!
Could be a fun a follow up video.
what is the actual physical relevance of something like pi factorial ?
Wouldn’t this definition of the nth derivative for the power rule have issues for n
The Master Not if you define it via falling factorials. The falling factorial with negative exponent is well-defined, and is equal to a rising factorial.
how do you intepret fractional derivatives and do they have any applications?
Ben Evans Fractional derivatives are much like fractional powers. For instance, D^(1/2)·D^(1/2)[f(x)] = f'(x). In words, the half derivative of the half derivative is the first derivative. It also allows you to talk about definite integrals as derivatives to a negative power.
The applications I am familiar with come from quantum physics. It can be particularly useful in solving differential equations.
A function equal to a number ? You might want to put (1) after each derivative to apply the function and get !n (Because the function x->d/dx^n(x^n) applied on 1 is n!)
Ha! I got it right! I treated it like a Taylor Polynomial. I remembered k(k-1)(k-2)... from Binomial series can be represented as k!/(k-n)! and thus represented the nth derivative of f(x)=x^pi as the following: [pi!/(x-pi)!]*x^(pi-n), then just inserted pi for n to get the pi-th derivative
8:39 love it!
Shouldnt it be a-n+1 in 4:25 ?
What an idea! Sir
Excellent !! vow !!
finally fractional calculus!!
うっわ･･･ めっちゃくちゃ簡単にガンマ関数まで繋げてくれた
見るの1分かからなかったよ
老師は教える天才やな
UCBもええとこなんやろな わしローソンの夜勤バイトしか知らんねん
Hi BlacpenRedpen ... can u help me for solve this integral ?
int from y to 0 sinx² dx ?
I like all the derivatives of pi - raspberry, chocolate, beef, chicken... the list goes on.
Can we use.... x^π = e^(π. ln x)... Relation?
Maybe I should watch the video about the gamma function first, but it looks like it's very similar to a Laplace transformation. Are there any correlations between the two?
Hermann Barbato The Gamma function is related to the Laplace transforms, but the fractional derivatives have absolutely nothing to do with Laplace transforms.
@@angelmendez-rivera351 Thank you. Then how are they related?
Hermann Barbato Let La[f(t)](s) denote the Laplace transform of f(t) with respect to s. If Re(s) > 0 & z in C\N-, then La[t^z](s) = Γ(z + 1)/s^(z + 1).
This is deeply connected to analytic continuation in complex analysis, so I don't see why the ith derivative wouldn't work but can't think of a formal proof or anything that it is indeed well-defined (i suppose we have to look at some dimensional regularization analysis and apply it to derivatives LOL)
When I was in school I looked at using fourier transforms to do these types of strange derivatives and one that I came up with was using (1/w)^w which is d/dx omega times. I think I called it the frequency derivative
5:27 what if n is greater than a
6:03 * uses gamma function instead of permutation(nPr) *
Permutation: am i a joke to you?
In 1981 I asked my college algebra teacher if there was a 3/4 or 1/2 derivative, similar to x^3/4 or x^1/2 and he said: No! It doesn't make any sense.
When was this concept originated?
When you add a-n to the denominator, doesn't that mean you have to add a rule a=/=n, to avoid div.by.0?
you simply wont add that then. Because if a-n =0,
a-(n-1) = 1.
You've already got a! (constant)
if you derivative it one more time, itll become 0.
which is why a*(a-1)*(a-2)..(a-n) is the n+1th derivative of x^n = 0
Thank you
Derivative function derived from integration.
Amazing pi th derivative of x^π
What happened to your previous video about the series?
I messed up again... lol
I think the gamma function is more popular and stuff because of the gamma distribution in statistics. Having the gamma function inside the gamma distribution's probability density function's definition makes it nice and cute, but having pi inside gamma sounds weird :P
2! is 2*1, 1! is 1, we divided by 2. thats why 0! is 1, we divide by1, which means (-1)! is 1/0 since we divide by 0.
Which is the background song’ s name?
Please
The Entertainer by Scott Joplin
The Entertainer
Yup. Download for free on YT audio library : )
Always reminds me of 'The sting' movie...you follow? Hahaha
What about imaginary order of derivatives? Like the ith derivative of x^2?
OK, for general case, you have this d^n/dx^n (x^a), this makes sense.
But - suppose that a is an positive integer, and n is a non-integer greater than a+1. Meaning that using standard derivations, you will eventually reach zero, leaving only a fractional derivative of zero, which is also zero.
However, when looking at the factorial representations, you run into something a bit more interesting. a! is a finite positive number, (a-n)! is also a finite number, as it's a non-integer negative factorial, and a-n is a finite negative number. Let's call a!/(a-n)! k for the time being, and (a-n) can be b. kx^b is a standard function, and due to b being negative, kx^b can never be zero!
Since these two interpretations directly contradict, one of them must be false. Or possibly both. Which, if any, is considered the correct way to resolve this bit of calculus?
How do you know that a fractional derivative of zero is always zero?
Interestingly, when working with integer derivatives, you don't run into this issue, because that would give a!/(a-n)!, where a-n is a negative integer. This is, using the Gamma function, infinity. Dividing by infinity is zero, so there's your zero. But with this interpretation, I am lead to believe that a fractional derivative of zero doesn't necessarily have to be zero. This makes me think of the +c of negative derivatives (integrals), however in case of f(x)=0, c is always 0. Huh.
EDIT: now that I think of it, when n is negative (resulting in an integral), there should be some c's in there. Or maybe this only gives the principal antiderivative? But that then assumes c=0, so that's another problem in the reasoning above.
@@fghsgh This is only on the assumption of successive derivations being equivalent to taking the fractional derivative as a whole, which may be wrong.
But when we're getting to the point where we've done a derivatives, this is, with respect to x, a!*x^0. The next derivative on the stack, then, multiplies in the zero, rendering the whole equation to zero.
Further normal derivatives can't get rid of the zero, of course.
But since the fractional derivative is also giving finite numbers, that also can't get rid of zero.
Though if you throw in a negative integer into the pi function, you get the necessary infinity, but once you hit the point of f(x)=0 that could be any arbitrary power. Singularities are silly like that.
I was hoping for you to explore derivatives of fractional order, don't know why. Maybe some other time, keep up the quality vids please, take care.
Amazing Man
8:45 shouldn't it be t^(pi+1) ???
You can actually arrive to the same conclusion using the spectral derivative, using the Laplace transform.
reminds me of the time when we do the Taylor non stop.
An intro like this should be included in calc 1 getting students to know derivatives is not just integer but it can take fractional derivative and can be think of as a function operated on an equation
Amazing!