what is i factorial?
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- เผยแพร่เมื่อ 27 ม.ค. 2019
- What is the factorial of i? Yes, the imaginary unit i. Does i factorial actually work? Yes, we will have to use the extension of factorial, namely the Pi function or the Gamma function. And we will also have to know how to deal with (a real number)^(complex exponent).
Pi & Gamma function, extending the factorial 👉 • extending the factoria...
sqrt(2) factorial 👉 • factorial of sqrt(2)?
Euler's Formula, deal with complex exponent 👉 • Euler's Formula (but i...
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Do u factorial? Comment and let us know!
Yes, i!
of course i factorial, factorialing is my favorite pastime, i think I'm the best factorialer that i know, i dont know do you think you can factorial better than me?
Let’s just jump!!!! Right into this!
Yes, i factorial!
But can you prove that (((...(((i!)!)!)!...)!)!)! converges to 1?
sure math is magic, but the way he switches markers is another level
Bobby JCFHv Lichtenstein thanks!!!
Honestly it's so true!
Board skills are tough! I've had to do presentations for some of my math classes and am always shocked how much better the professors are at it. Seems like it should be just like writing on a page, but it's way harder.
blackpenredpen is VERRRY skilled!!
Yeahh
Me: you know how many ways I can arrange √(-1) quarters?
Friend: what?
Me: You can arrange it 0.498015... - 0.1549498... _i_ ways
Friend: *what?*
Me: what?
Scott Maday I laughted
Me: whaaaaaaaa?
you can arrange it an unreal number of ways
This reminds me of the question about cats who genetically altered to be very happy: they have purr mutations
@@muttleycrew 😂😂😂 fuck that was good👌
*MINDBLOWN: i! rotated 180deg is still i!*
TM Fan lollll
And rotated 90deg it is an equal sign.
Is this geometric algebra?
Blue Blue no，equal dont have 4 disconnected parts
TM Fan, so you think, I won’t get the fields medal for my discovery?? Suppose you are right, topology would never agree, too.
But my HP41 displays the equal sign as two dots over a line... can HP be wrong?
Blue Blue is that what ur pc show when u press the = button on pc？
Non-closed-form answers are so unsatisfying. :(
Wow, I just left this comment almost word for word.
True. Unfortunately, you run into a lot of those in engineering - sometimes you just have to get "close enough" to build your widget. Numerical methods might give a good enough answer, but they rarely tell you "why" it is so.
@@jesusthroughmary yea just saw it lol
If you take the magnitude of the complex number i! (√(a^2+b^2)), it gives
π*cosech(π)!!😊
∫ is considered closed
It’s so sad when you can’t express an integral in the exact form(
This is so sad. Alexa play despacito
When he dropped those decimals, I sighed. I guess it was too much to hope for a closed form (other than the integral itself.)
You can remain happy, just demonstrate that the real and imaginary parts of i! are irrational
@Kaname Locon Sure you can: (exp(ix)-exp(-ix))/2
@@Tomaplen But how do we know it's not in terms of nth roots? logs? sines? e? pi? :(
Here's a really cool fact: The *absolute value* of i factorial (aka its distance away from 0 on the complex plane) is actually √( π/sinh(π) ). That's crazy
Pie is everywhere
The absolute value of i factorial is one.
@@Nigelfarij |i!|, not |i|!
Nice! What is the angle?
I always find it funny how such things as i! or i^i or the antiderivative of x^i are not particularly special numbers. In fact, they almost seem arbitrary, given how fundamental i is to math. Honestly complex numbers fascinate me and I'm irritated at myself that I didn't truly appreciate their beauty until relatively recently.
What the factorial
GD Lennyvania lol
What the fuktorial!!
1:40 the “for you guys” here really gets me good
Engineers be like:
i!=1/2
Elsword Maniac spanish be like : it’s an exclamative sentence
I don't get it
Alex Bot The joke is that engineers make approximations like pi=3 or e=3 and stuff like that when making calculations.
@@JoBo0209 != means unequal in programming languages
Sakura Hikari Ok? That had nothing to do with what I said
Can you do the derivate of n! and explain the digamma function
Great idea!
I would love to see that
By ‘derivate’, do you mean ‘derive’ or ‘differentiate’?
@@Angel33Demon666 'derivative' makes more sense.
Yes as a gamma function
Real fans watched the unlisted version first
Real fans just watched the thumbnail.
But real fans aren't allowed
Loved the video! And honestly one of my favourite channels on TH-cam.
K thank you!
Can you prove that the real and imaginary parts of i! are irrational? Ive asked this to quora but nobody answered me
Their irrationality still unknown
@@icespirit really? No proof exists?
@@user-hy6cp6xp9f No one
No proof exists for the irrationality of e+pi lol
ViperDaniel are there two irrationals that add to a rational?
Great video. At first sight I could not image how to go through this, the use of the Gamma function is brilliant. Thank you.
If x! = i
Find x
no
(no!)^i
i was waiting for this one for a long time !! thank you so much !! now show us pi factorial...:P
Can you do W(i) (Lambert W of i)? It also gives a cool complex number.
#yay from Costa Rica :D
ze^z=i => z=? :)
Your big fan mr. Blackpenredpen
Very, very, very, very interesting. One of your best
3:22, when he says "lemme take that to the front" but what he actually does is taking that behind the integral.
I watched the entire video.
I only have one question.
What?
Whenever i click your video it makes me happy 🙂 you cover mostly interesting topics
Amazing,I have never thought of that
Man!! These titles are getting funnier and funnier day by day..!!
can't believe i'm watching math videos cuz i cant sleep😂 love your videos!
Your presentation is better than I have seen by others. However or in addition every time you say “fomula” I can’t help myself smiling.
I am very confused but I can't stop watching.
as someone who has not learned any calc yet, but has learned polars, this really reminds me of rcistheta
I like the way u teach
kiran mk thanks
Excellent !!!
If you take the magnitude of the complex number i! (√(a^2+b^2)), it gives
π*cosech(π)!!😊
I plugged that into a calculator, I got 0.272029054982 for πcsch(π) and 0.521563973414 for that magnitude of i!. Did I do something wrong?
@@jacobschaumann I'm sorry... I just checked and it's not correct... I don't even remember why I wrote this almost two years ago...
Maybe you gotta use the double factorial of πcsch(π)...
Do the inverse factorial of x, that would be wild
Yes !
Thanks for your enthusiasm on this specific topic! I got frustrated learning stuff on wikipedia... (wikipedia should just link to your video, instead of making us suffer their comprehensive answers).
Brilliant!
This is the only channel in the world who can explore an alien using maths
Cool ,pls keep doing math videos .
Bringing the heat!
The best math teacher ever
never even thought of this.....ammmaaaaazzzziiiingg
I wish my calculus professors brought out interesting ideas like this.
Finally the comment section won't be filled with that.
Icestrike Cubing yes!!!! Thank god!!!
Right
I am out of the loop. Filled with what if I may ask?
@@nevs0917 i!
Was just wondering about the integral of the gamma function.. hehe
The bizarre thing is that I never realised that I *_needed_* to know what i factorial is until YT recommended me this vid...
Great video as always, but can you turn off the feature of your video setup that constantly adjusts the exposure? It gets dim when your black hoodie leaves the frame!
Ninja of math ....🏅
I love this channel, but how often would you run into these problems in real life as mathematician, scientist or engineer.
Video idea: explain how elliptical integrals are approximated!
This is fun to look at
leonardo navarrete thanks!
All your videos including calculus start to make me not hate what they tried to teach me at A-Level (which kinda made me give up on higher maths)
"The baby fonction " u re a legend
Thanks! I was waiting for this, but what's about... n!! and Super-, Hyper factorial?
lol the title before watching tthe video
1:40 Well that woke me up nicely.
cool.... great idea...
There is a nice analytic solution for the modulus of i! if I remember correctly. The phase is the ugly part.
It really helps that Γ(n+1) = n! , when n is literally any number.
can you prove the integral formula for subfactorial please ?
#yay from France
Eliott Morgensztern I will have I have time in the weekends.
According to dcode it’s Γ(1+i)
Verry good
I was hoping you would make a substitution early on so that you could do i factorial by doing u factorial
Do (Black pen Red Pen)!
I factoreo
Eye Factorio
Oreo in the eye
Factoweo
Here, its blackpenredpengreenpen
Thanks i asked u it!
1 minute in:
Yeah, I think I’m going to go back to my times tables...
I double factorial, if you know what I mean.
Martiniano Faure I know!!!
OMG, you actually answer my comment. I love your videos. You are one of my favourite math channels.
Martiniano Faure thanks!!
_i_
bonus visual -> wolfram alpha: plot (cos(theta) + i*sin(theta))! theta= -pi:pi
Shoot a video about what is t : a^b = b^a*t
blackmarkerredmarkergreenmarker is a nice plot twist
Have you done laplace transforms yet? Laurent series and can you explain calculus of variations?
He has done Laplace transforms.
I'm just a sophomore, this blows my mind. I understand none of it, but it's still really cool.
Wow!
I left a comment on the last video of a problem I was wondering and I guess I'll ask once more. Can you derive a general formula for ln(a+bi)?
It would just be 1/(a+bi) m8
@@sgurdmeal662 I'm not asking for the integral, I mean a general formula that you can compute easily for ln(a+bi) without going through so much with moving the powers around and all that
Benjamin Brady
I actually have an unlisted video on ln(-2), th-cam.com/video/IX_23EWpF5U/w-d-xo.html
I will work out ln(a+bi) later.
Benjamin Brady you just use the fact a+bi can be expressed as re^(theta*i) where r is the distance from 0 to a+bi, so it will be (a^2 + b^2)^1/2,
and theta is the angle a+bi makes with the positive x-axis, if you draw it out you’ll see that the opposite and adjacent are b and a respectively. now we know tan(theta) is b/a so theta is inverse tan of b/a (i’ll use tan-1 to write it
plugging these in we get
a+bi = (a^2 + b^2)^1/2 * e^(tan-1(b/a)*i)
so ln(a+bi) = ln[(a^2 + b^2)^1/2 * e^(tan-1(b/a)*i)]
separate it into two logs
= ln[(a^2 + b^2)^1/2] + ln[e^(tan-1(b/a)*i)]
the ln and e cancel and the ^1/2 can be brought outside the log
= 1/2*ln(a^2 + b^2) + tan-1(b/a)*i
and thats it, hope this isn’t too excruciating to read
@@benjaminbrady2385 There IS NO way to calculate it without moving the powers around. Consider a complex number z = Re(z) + Im(z) i. Then ln(z) = ln|z| + i arg(z) = ln SqRt(Re(z)^2 + Im(z)^2) + i atan2(Im(z), Re(z))
Do the integral from 0 to x of floor(t) dt
Does using the reduction formula for partial integration help simplifying these integrals?
idk
They asked "What?"
Me asked "How?"
Can you use feynman integration or Complex Analysis or both to get exact answers for the real and imaginary parts of i factorial?
im gonna try
Nice video. Question: will the fact that t^i is multivalued cause any problem? Also, I was expecting an evaluation of the improper integral using contour integral so it was a little anti climatic at the end.
"Also, I was expecting an evaluation of the improper integral using contour integral so it was a little anti climatic at the end."
If only all film reviews were like that.
Sir please 🙏🙏🙏 make a video on imaginary logarithmic functions
Wow I really hoped this was gonna be like e^-pi or something
I have a challenge for you:
Derive a formula for simplifying √(x+yi) in terms of x and y using only algebra (no trigonometry)
BTW: I know the formula; this is just a challenge for you
write √(x+yi=ai+b
square both sides
compare coefficients of real and complex part.
Get a and b.Am i right
I have a challenge. Find a general formula for the nth root of a+bi
Tip: rewrite in polar form.
@@lok7396 (a+bi)**(1/n)
@@orlandomoreno6168 nope.
@@lok7396 there is De Moivre's formula
The 'dope version' my gosh I am dying XD
Thumnail needs an award,,
Priyank Sisodia thank you!
That's a very interesting definition of a "very nice number" xD
Tres bien prof même ne connais pas l'anglais ,et connais les regles mathematique merci boucoup prof deuxieme fois
Can you do this WITHOUT a numerical approximation? That is, symbolically solving the integral and plugging in values?
yes, I do!
What is the floor function for( i ) please make a vedio to solve it
There is another nice way to get i factorial. Watch Dr. Peyam's proof that integral frac(tan(x)) dx from 0 to pi/2 = 1 + i/2*ln(gamma(1-i)/gamma(1+i)). From Euler's reflection formula: gamma(1 - i)/gamma(1+i) = (pi* csch(pi))/((gamma(1+i))^2). Substituting back and expanding the logarithmic term we get 1 + i*ln(sqrt (pi*csch(pi))) - i*ln|i!| + arg(i!). Now from the fact that the integral has to be a real number (the terms with i must be equal), we get that |i!| = sqrt(pi*csch(pi)) and arg(i!) = (the integral) - 1. If this isn't awesome, then I don't know what is!
Is there any intuitive or visual explanation for this value/calc?
I was hoping the answer wouldn’t be so anticlimactic. Great video nonetheless! Great job, thanks!
i! Can be easily calculated by using euler definition of gamma function and euler reflection formula....
When ur a freshman and u dont understand this but your next year course in calc bc so ur so hyped cuz u ll be able to understand this
And here I was thinking that the factorials of real-valued fractions were weird.
FINALLY
Does anyone know where to find a rigorous proof for the convergence of that integral around zero?
It is pretty basic using Lebesgue integration, since the absolute value of the integrant is less or equal than exp(-t) which is integrable from 0 to infinity with a finite result.
hmm... z! equation looks very similar to Laplas transformation, exactly F(p) = integral f(t)e^(-pt)dt from 0 to ∞. Is there any explanation of that?
There are other hyper-real numbers. Like a+u*b, with u^2==0 and a, b to be reals. I first thought that you meant something like that. BTW, series with these numbers are quite easy to solve. I'm sure u know it. 😉
That is not a hyper-real number, that is a dual-complex number, a type of complex number different from the standard-complex numbers. The hyperreal numbers include infinite integers and infinitesimals in halos around real numbers. Very different thing.