Integration video not up quite yet? I paused the video to watch all the gory details, but alas, I can't see it on the second channel.

@@Sinnistering It's just an integration by parts.

I assume you have already known that the decimal digit expansion of constant pi is absolutely identical with the decimal digit expansion of sin (0.00000........0000018) degree.

Let’s settle the issue of 0! and 1! first hey? Why are they both equal to 1

@@PetraKann Well, if you trust 2! = 2, and the recursion relation, then we want 2! = 2 × 1!, so 1! _must_ be 1. By the same reasoning, we want 1! = 1 × 0!, so 0! must _also_ be 1. Convinced?

I mean, technically they *were* the computer :D

I cried inwardly when Matt said "but, if you want a nicer one". As if the computer renderd one can compete with that hand drawn masterpice

what part of the video are you referring to?

@@kurzackd 7:06

I guess that begs the question: are the minds of thinkers pre-computer age, more elegant than those of today??

That sounds fascinating! Would you still happen to have your work around to show off? I'd love to see it!

What were you studying in hs to be messing with nth dimensional spheres? What is your degree in?

That turns out to be really important in quantum field theory. And the continuation of the gamma function to do FRACTIONAL numbers of dimensions is useful as a mathematical trick you can use to deal with divergent integrals during renormalization. It's called dimensional regularization.

That sounds like something 3blue1brown would use as a video concept... I really want to see a video about that, it sounds super interesting

@@jeremygalloway1348 The math is third-semester calculus. You just have to be really persistent.

graph: u n u n u n

@@RubyPiec Real world Registeel durability in [Indeterminate Time]

That sounds awesome! Would be interested to read that, care to drop us a link?

Search for the fractional langevin equation at utrecht university. I even won a thesis prize for it!

I suppose you could also do one where the height is the real component, and the color is the imaginary component.

@@PhilBagels that sounds cool as well, but if goes to infinity do you have to apply a logarithmic color scale?

Your research is fascinating! Fractional calculus is an interest of mine and I'd love to learn more about your research ⛰️

I had a similar thought as I was watching the video.

Chartreuse ????

@@Bean-Time Yes, that's a colour: RGB-Code #DFFF00

If we're using color anyway we _can_ make a sort of 4 d graph, just associate one spatial dimension with a range of colors. Essentially take one of the graphs of one part of the output and color it based on the magnitude of the other part.

@@Bean-Time My grandfather always says it's his favorite color so I grew up thinking it was a normal color that lots of people use. It's the color of tennis balls.

It could also represent the real or imaginary values of the input

using color might be better than an animation

@@gyroninjamodder It'd certainly be easier. But then, using colour and time you can have 5D graph.

@@spudhead169 You can use color to represent more than 1 dimension.

Or you could just look at an animation of how the inputs to the function move over to the output of the function. Seeing how the points of the complex plane transform under a function gives you a more visceral feeling for the behavior of the function on the complex plane. In my view, the fact that complex numbers can be represented using two real numbers in a particular order should be ignored as much as possible.

Unfortunately not quite how the factorial works, but I’ll let it slide.

@@TheBasikShow Right? It's so close, but it's such a good joke you kinda have to give it to him.

Only factorial joke I’ve ever heard :)

'definitely not the least of them" - that's awesome, I'll use this to describe athletes like Messi, Tiger, Jordan, when G.O.A.T discussions come up

Euler is up there with Gauss as a name that shows up absolutely everywhere.

@@MattMcIrvin He shows up CONSTANTly. :-/

Presumably a physicist and not a physician, no?

@@timotejbernat462 I think op is a French native speaker and that's why they made that mistake

Not an inaccurate label at all

Aah, exploded! I love it!

Yeah x! seems to grow faster than x^n and n^x where n is a constant.

I like my six exploded!

"My friend and I were one of the only ones to finish it." This is an interesting mathematical exercise in itself. The number of people who finished it was "only", and of those "only" people, you and your friend were "one". I suppose the fact that you and your friend were "one" means that you were the same person, and that you were just talking to yourself. I am still in the dark, however, about how many people comprise this mysterious number "only".

It works in Windows now!

blame jony ive and his ridiculous worship of ideals

@@michaeltajfel It's been working in Windows Calculator for 20 years. I tried that when I first learned what factorial is.

Those good times of high school and rabbit holes, sigh…

All math is real and all math is imaginary

"Is this a real value? Is this imaginary?"

I think this graph was made with pyplot, which uses 2D arrays to create plots, and creates the hue maps automatically. You can mess with them but it goes from a 2 minute job to a 10 minute job.

@@shearnotspear It's only 8 mins longer tho.

The Parker complex hue domain coloring

@@crackedemerald4930 Why call it that tho? It's not inherently flawed.

@@shearnotspear Yes my immediate thought was matplotlib (the parent library for pyplot) and I wondered why a standard color map was used (seems to be the default Viridis) rather than making hue based on imaginary values.

Every strategy has its own strengths and weaknesses as none of them give the full picture.

@@WindsorMason nah the colouring actually gives the full picture.

@@wizard7314 I should clarify that if you graph with both magnitude and angle as z and hue, then yeah, you have 4 dimensions being graphed. Just like when you have two side by side Re and Im plots. Numerically the full output is shown to you in both cases, but the "full picture" of understanding the behavior of the output plane is only partially glimpsed in each. There are so many different ways to squeeze those 4 dimensions into graphs and each one gives you a different view of the picture.

I mean I’m colourblind so, depending on the colour scheme, colour based graphs are kind of a nightmare for me

I don't know the answer. But when the square root of pi shows up, it reminds me of the integral of a Gaussian, and I DO know how the circle shows up there. So maybe we can transform the gamma function evaluation at these points into a Gaussian integral somehow.

@@MattMcIrvin Yes, it is because the square X² of a standard Gaussian random variable is a Chi-squared r.v., which is a Gamma distribution with parameters (1/2 , 1/2) hence the gamma(1/2). Now where is the circle ? It is in computing the bivariate Gaussian : the equivalence curves (ellipsoids) are circles x²+y².

Maybe the pillars that shoot up to infinity are circular

@@MarcusCactus that is awesome

The volume of a n-dimensional ball of radius 1 is pi^(n/2) / Gamma(n/2 + 1) (so with Matt's notations, pi^(n/2) / (n/2)!)

You get an 8D graph. ^_^

How about matrices?

The problem with quaternions (and also matrices) is that a^b is not clearly defined. since a times b is different from b times a, it means that there are actually 2 different powers. a "left power" and a "right power".

@@Tumbolisu Just use a^b=e^(In(a)*b) and taylor series for e^x. This is how we define complex number^complex number.

@@Anonymous-df8it Complex numbers have the property that a*b = b*a, so calculating a^b by using e^(ln(a)*b) is perfectly fine, since its also equal to e^(b*ln(a)). This is not generally true for quaternions or matrices, thus creating two possible definitions for the power.

I'm 99% sure you're writing the formulas (3/2)!, (5/2)!, etc., but that is the most confusing list of numbers I've ever seen. My brain can ONLY see [1, 5], [2, 5], [3, 5], etc. and I was trying to figure out what a factorial of a matrix would be. It took me multiple seconds to assume you must be from Europe, and you were just accidentally confusing the hell out of me with your comma shenanigans.

@@kindlin you think that's bad? In some cases I use comma in both decimal place and thousand separators I usually use a . for decimal point (like 5.2) And a , for thousand separators (1,000,000.2 = 1 million and a fifth) bit sometimes I use a , for decimal point (5,2 = 5 and a fifth) It's a matter of time before I accidentally do something like 1,000,000,2 and confuse the hell out of everyone

@@Xnoob545 1,234,567

i'd be interested in a graph/formula showing the inflection points.

Thinking of complex numbers in polar coordinates--absolute value and phase angle--is really useful. Multiplying complex numbers means you multiply the absolute values and *add* the phase angles (or "arguments"), so complex numbers with an absolute value of 1 (on the unit circle, that is) are useful for describing rotations in the plane, or anything that cycles or oscillates.

@@MattMcIrvin This is the core idea behind Euler's e^ix = cosx+isinx The absolute value of a complex number is just the magnitude of that number, same as any 2D vector. The distance between any two points [x1,y1] and [x2,y2] is just sqrt((x2-x1)^2+(y2-y1)^2), similar to the pythagorean of a triangle, c=sqrt(a^2+b^2). In fact, the distance formula in any number of dimensions is just the Pythagorean formula.

I saw him dancing without shoes, I almost vomited from bewilderment.

PS: There's an article by Professor Kent Conrad which collects many different proofs of the Gaussian integral. Really cool if you, like me, only knew the famous polar coordinates one.

what

Just another style of visualisation of 4D. I believe that is something 3blue1brown uses in some of his videos, but it's not really that intuitive. In 2D, the lines of the input plane just weirdly flip or spin around and overlap, after which you still don't have much but a cool animation. You definitely don't have a clear sense of what's happening, but that's hard to achieve with anything.

Yes indeed! And in these calculations, it's actually useful to (1) compute an integral on 4D spacetime by first computing the integral on 4D Euclidean space (which is a different thing entirely), and (2) continue the function into *fractional* dimensions, from 4D to (4 minus epsilon)D, to make some divergent integrals converge so you can get a handle on precisely how they are divergent during renormalization.

I had to search up the word Ubiquitous. And that made me realize just how little I know lol.

You probably wouldn't be able to do the 5 factorial one because only 1 person as far as we know (Jeanne Calment) lived to their 120th year.

Yeah, it's very unlikely, but not impossible, like say 6!

And the second channel doesn’t seem to have any (public) recent video! 😞

@@Agrajag22 I thinks that's why. Matt might not have uploaded it/made it public yet

yeah that made me wonder if you could express the same function as wave interference equations

Yes, noticed too it definitely looks like a sine wave and I was looking through the comments to find out about that. Are you sure it really _is_ a sine wave? Might be interesting to figure out why?

Do it.

Generalized method for generalizing - now there's a high-ranking idea.

Analytic continuation, maybe?

No, because you can do it in multiple ways. For instance, consider finite functions: if you have N points, you can calculate a unique polynomial of degree no larger than N that passes through all those points. A consequence of that is that you can add an additional point to get a new polynomial. And since this additional point can be anywhere, adding different points gives you different polynomials: in particular, if you take two candidate new points with the same X (say, (x, y1) and (x, y2)), the polynomial you'll get adding the first point must be different than the polynomial you get adding the second point (because p(x) can't be y1 and y2 at the same time). A simple example: take (0, 1), (1, 3), (2, 7). It shouldn't be too hard to see that p(x) = x² + x + 1 goes through all three points. Now add (-1, 4). You get a new polynomial, p1(x) = -0.5x³ + 2.5x² + 1, that goes through all four points. But if you instead add (-1, -2), you get p2(x) = 0.5x³ - 0.5x² + 2x + 1, which goes through _this_ extra point plus the original three. So both p1 and p2 are continuous functions that fit the original three points, but they are different functions.

@@joseville No. Analytic continuation is used to extend from an open interval of the real numbers to the complex plane. A discrete set of countably many points is not an open interval.

No. No such a generalized method exists. If you have a discrete function f : Q -> Q, then there are uncountably infinitely many ways to extend such a function to the real numbers.

If your function is to be analytic (differentiable over complex plane apart from discrete points like poles) then we can substitute x-->ix in your equation and it must still hold. f(ix) = f(ix-i)*ix = f(i(x-1))*ix Let g(x) = (-ix)! = Gamma(1-ix) So that g(ix)= x! = (x-1)!*x = g(i(x-1))*x. This comes pretty close to being your f. W can multiply by i when (x+1) by multiplying the whole function by exp{pi*ix/2} when the input is (ix). When (ix) is imaginary (for real x), this multiplies by i as (x+1). Also, g(x) = (-ix)! = (-ix - 1)!*(-ix) Define h(x) = (-ix)! * exp{pi*x/2}. We check: h(ix) = x! * exp{pi*i*x/2}. When x is real, this equals (x-1)! *x * i * exp{pi*i*(x-1)/2} = h(i(x-1))*ix So we have that h(ix) = h(i(x-1))*ix for real x. If h(x) is to be analytic then we can substitute x --> -ix and the equation would still hold. Giving h(x) = h(x-i)*x Which is your desired function. Hence f(x) = Gamma(1-ix) * exp{pi*x/2}. The Gamma(1-ix) part is simply a 90 degree imaginary rotation of the argument to the function. So the whole function graph is the same but turned 90 degrees. In fact if we plug in that rotated x into the function to see what it is apart from a rotation, h(ix) = x! * exp{pi*i*x/2} It's the same as x! = Gamma(1+x) except multiplied by that exponential. Along the real line of the factorial (remember the whole function is rotated though, so here 'real line of factorial' is actually on the imaginary axis) the function is oscillating in its complex phase. On the imaginary line of the factorial, it is exponential growth/shrink. What I wonder is whether the exponential growth in on one side substantially changes the topography of the function. Maybe it diverges to one side now, instead of dropping off to the sides?

The Gamma function is also continuous, smooth, and analytic. The domain is just not equal to C, as it has nonremovable singularities. Its reciprocal only has removable singularities, so it can be trivially extended to an entire function.

wait how

@@22tfortnitevevo negative values of n result in the same figures raised to the - 1 power. Decimal values can be converted to n/10, n/100, etc. Any number of terms can be achieved in a few ways, location based addition figures, location based multiplication on figures, series Binomials, or compacting into 2 terms using series substitution. Term coefficients can be varied by substitution down to 1S and then applying powers after the Pascal's expansion. In the case of complex numbers being used as coefficients its the same process but the final expansion will have nested Pascal figures. The real fun is for me is that the processes aren't that hard, I calculated 6 terms to the 6th power by hand in 45 minutes and wrote an excel that could handle up to 26 terms to any given power... Granted I didn't have enough rows to calculate past n=4 The real fun for me was figuring out that the result of multiplying polynomials of differing numbers of terms creates a partial Pascal's figure. Example ((a+b)^n1) ((a+b+c)^n2) will fit into the figure for (a+b+c)^(n1+n2) except the c terms will be truncated from the c-prime vertex by nsum-n1 sets. ((a+b)^2)((a+b+c)^2) will fit into the figure for ((a+b+c)^4) but will not contain terms with c^4 or c^3.

@@ffximasterroshi awesome dude you should make a visualization of it on a graphing software or smth

@@22tfortnitevevo I would love to be able to use a graphing program to quickly produce the figures, but math is only a hobby I get to play with on rare occasions. Plus my formal math training stopped at pre-calculus, so I don't even know how to write proofs.

@@ffximasterroshi damn that sucks, hopefully another commenter stumbles on this and is able to do it

Thanks Ben Kelly

You’re welcome Jeremy Johns

No, for me too. Watching on my Pi4 so I was in doubt if this was the cause....

That's nice! The recurrence relation shows that f(x)/Γ(x) has to be periodic; since it's analytic we can write it as a Fourier series using sines and cosines like in your example; but these will blow up in the imaginary direction.

f(x+1)=f(x)*(x+1) is also satisfied by f(x)=0. It doesn't contain the factorials though.

@@gdclemo Yes, what my comment actually proves is that if f satisfies this rule, is analytic on the strip in question, and is bounded on that strip, then f(x)/x! is constant. The constant is 0 if f is your constant 0 function, but 1 if f agrees with the factorial on the positive integers.

That they were

Is this a thing with people in England or is it just Matt's personal idiosyncrasy?

Matt can't be bothered to learn people's pronouns; he's too busy doing math!

@@gordoofdoom It's not an England thing. I think it's just a Matt thing. I don't know of anyone else that does it.

boring

They can be used as a singular third person pronoun, even in cases where the gender isn't necessarily ambiguous.

@@LofferLogge It can be, but its confusing and pointless and should not be used that way.

@@LofferLogge but why?

@@qupp75 Maybe it's a shield against getting cancelled.