what is i factorial?
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- Опубліковано 27 січ 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
*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？
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👌
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.
1:40 the “for you guys” here really gets me good
What the factorial
GD Lennyvania lol
What the fuktorial!!
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
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
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 UA-cam.
K thank you!
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=? :)
Great video. At first sight I could not image how to go through this, the use of the Gamma function is brilliant. 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?
Your big fan mr. Blackpenredpen
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
I watched the entire video.
I only have one question.
What?
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.
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 like the way u teach
kiran mk thanks
Whenever i click your video it makes me happy 🙂 you cover mostly interesting topics
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!
Amazing,I have never thought of that
I am very confused but I can't stop watching.
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).
Excellent !!!
Thanks! I was waiting for this, but what's about... n!! and Super-, Hyper factorial?
1:40 Well that woke me up nicely.
The best math teacher ever
Was just wondering about the integral of the gamma function.. hehe
Do the inverse factorial of x, that would be wild
Yes !
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!
never even thought of this.....ammmaaaaazzzziiiingg
can you prove the integral formula for subfactorial please ?
#yay from France
Eliott Morgensztern I will have I have time in the weekends.
I wish my calculus professors brought out interesting ideas like this.
Cool ,pls keep doing math videos .
Brilliant!
Bringing the heat!
as someone who has not learned any calc yet, but has learned polars, this really reminds me of rcistheta
Video idea: explain how elliptical integrals are approximated!
The bizarre thing is that I never realised that I *_needed_* to know what i factorial is until YT recommended me this vid...
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.
Ninja of math ....🏅
This is fun to look at
leonardo navarrete thanks!
Have you done laplace transforms yet? Laurent series and can you explain calculus of variations?
He has done Laplace transforms.
This is the only channel in the world who can explore an alien using maths
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)
What is the floor function for( i ) please make a vedio to solve it
There is a nice analytic solution for the modulus of i! if I remember correctly. The phase is the ugly part.
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?
lol the title before watching tthe video
Sir please 🙏🙏🙏 make a video on imaginary logarithmic functions
I was hoping you would make a substitution early on so that you could do i factorial by doing u factorial
Is there any intuitive or visual explanation for this value/calc?
I love this channel, but how often would you run into these problems in real life as mathematician, scientist or engineer.
Do the integral from 0 to x of floor(t) dt
Tres bien prof même ne connais pas l'anglais ,et connais les regles mathematique merci boucoup prof deuxieme fois
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
cool.... great idea...
Verry good
"The baby fonction " u re a legend
Does using the reduction formula for partial integration help simplifying these integrals?
idk
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), ua-cam.com/video/IX_23EWpF5U/v-deo.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))
i! Can be easily calculated by using euler definition of gamma function and euler reflection formula....
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 (Black pen Red Pen)!
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!
According to dcode it’s Γ(1+i)
Around 5:00, why are you starting to evaluate at the limits before integrating?
Please give a vedio on zeta function theorams proof
Can you do this WITHOUT a numerical approximation? That is, symbolically solving the integral and plugging in values?
bonus visual -> wolfram alpha: plot (cos(theta) + i*sin(theta))! theta= -pi:pi
Here, its blackpenredpengreenpen
I factoreo
Eye Factorio
Oreo in the eye
Factoweo
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_
Just a general qn. Is it possible to not use euler's formula? Instead he cld combine it and become e^(iln(t) - t) wld this not be more straight forward?
Wow!
Shoot a video about what is t : a^b = b^a*t
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.
Is there an inverse function for the factorial? Because I couldn't find one.
Thanks i asked u it!
Could you integrate by parts instead of Euler’s formula?
indefinite integral or a proper definite integral would work. you could use by parts but the limits would pose a problem
Is there a brief analytical answer or are we stuck with numerical approximation?
yes, I do!
Is there an i double factorial and can i factorial be expressed as a product of double factorials?
Thumnail needs an award,,
Priyank Sisodia thank you!
Is there any significance to the polar form of the answer. ?
What happened for the sum of n to the i n goes from 1 to inf ? Do the ramanujan formula work ?
Wow I really hoped this was gonna be like e^-pi or something
It really helps that Γ(n+1) = n! , when n is literally any number.
FINALLY
I'm just a sophomore, this blows my mind. I understand none of it, but it's still really cool.
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
How come you didn't need to antidifferentiate the 2 integrals at the end?