what is the infinite tetration of i
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- Опубліковано 28 лис 2019
- What is i^i^i^... ? This is a super fun math because we will need the Lambert W function.
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i to the i to the i tatatatatatata
Calm Down hahahahaha yea
TATATATATATATA YEA
How is this marked 2 weeks ago?
Wtf black magic "2 weeks ago"
Wtf how is this posted 2 weeks ago
* serious voice *
*Here is the fish*
Do you know that e to the i to the e i 0 is e to the wau to the tau wau wau?
I know the reference!!!
Waut the fauk are yau taulking abaut?
Me reading the comment inside: Hmm, that sounds familiar. Where have I heard it? Oh! Vihart!
Need some pow wow chow to grok the tau wau wau!
Why did I read this in vihart’s voice
It is interesting to plot i, i^i, i^i^i, etc. on the complex plane to observe the process of convergence.
im assuming it just spirals on to whatever that is equal to
It does so in shrinking and rotating triangles
Do you like fish?
Hazlo con ratas por favor.
Attention BPRP Subscribers:
International Mathematical Union (IMU) just approved new standards, as follow:
1. "i over i over i ta ta ta..." can be described as this:
ᵢ ᳟
ⁱ
i
"ta-ta-ta" replaces any kind of continuity;
2. Lambert-function-on-Fish (no, it is not a seafood recipe) definition:
🐟
W(🐟 e ) = 🐟
where 🐟 can be a function or variable, real, imaginary, ta-ta-ta. It replaces all recursive functions and variables of any kind;
3. Any color including purple are allowed (why not?), even when there is a restriction of colors, such as "black and red pens only", in all Mathematical works (paper, classes, exams, ta-ta-ta) from now on;
Why are all of your fish evil? Are you building an army?
🐟 >:)
PlutoTheSecond
Hahahhahaha
Sorry I meant
Tatatatatatatatata
Now, what is: W(i)
W(i) :)
A question that I have always had is: When to put and when not to put: +2n(pi)i
When staying with the exponent of the exponential form in complex numbers.
This guy is so damn good that he makes me see how far I still have to go to answer any question in math.
Huge respect!
is there a way to calculate the Lambert function? I have no idea what W(-ipi/2) is, nor a single a idea of close to what value it is
You can approximate it numerically: www.wolframalpha.com/input/?i=Lambert+w+function+of+%28-+i*+pi+%2F2%29
The Lambert W Function can be implemented in your calculator using Newton's Method. To find W(A), the equation to solve is:
f(x) = x*exp(x)-A = 0 (1)
f”(x) = (x+1)exp(x)
And f(x)/f'(x) can be written as:
(x-A/exp(x))/(x+1)
Finally, the recursive formula for Newtons Method is:
X = x-(x-A/exp(x))/(x+1))
From (1) we have x*exp(x) = A or ln(x) + x = ln(A); x = ln(A)-ln(x); x ~ ln(A); ln(A) is a good initial value for the Newton Method when A>1, if this is not the case we use A. If your calculator can handle complex numbers ln(A) is a good initial value too.
You can find that W(-i*pi/2) ~ 0.5664173303 - i0.6884532271 and
exp(-W(-i*pi/2) ) ~ 0.4382829367 + i0.3605924719
you can compute i^i^i^i... a few times in your calculator and verify that it converges to this number.
You can use Newton-Rapson
@@jcbuchin You wrote f(x)=
f''(x)=
Its a small typo but it may confused people. (Just saying)
@@theseeker7194 I'm pretty sure that the infinite tetration of i converges as i is within the region of convergency of the infinite tetration.
Also, you are doing it wrong, you are collapsing the entire exponential from the bottom instead of the top, which is why the answer you gave it's not correct.
The tetration of i should converge to 0.4...+i*0.5... iirc.
I'm loving these i videos!
Thanks!!!
Yay it's the fish power tower hahahaha! It's happy that Chirayu also did it after your unlisted video!
Do i'th root of the i'th root of the i'th root..... of the i'th root of i
that's the same thing as i^(1/i)^(1/i)^(1/i)^(1/i)^..., which would be equivalent to saying i^-i^-i^-i^-i^-i^-i^... (as 1/i = -i). So, do the same thing for -i (-i^-i^-i^-i^-i^... = z), find z and calculate i^z.
@Yusuf Onur, Nope, sorry, that's not correct. What he is asking, is not a tetration (a.k.a. "power tower"). Instead, he is just asking about an expression of the form (..((((i^a)^a)^a)^a)...)^a with a being equal to 1/i = -i. This simplifies to i^[ a*a*a*a*...*a ] , which will be non-convergent.
i'th root of the i'th root of the i'th root ... of the i'th rooth of i =
= ⁱ√( ⁱ√( ⁱ√( ⁱ√( ⁱ√( ... ⁱ√( i )..)))))
= (..((( i^[1/i] )^[1/i] )^[1/i] )^[1/i]... )^[1/i]
= (..(((((( i^[-i] )^[-i] )^[-i] )^[-i] )^[-i] )^[-i] )...)^[-i]
= i^[ (-i)*(-i)*(-i)* ... *(-i) ]
which is either i^(-i) , or i^(-1) , or i^(i) , or i^(1) , and hence isn't convergent.
In other words: it's the limit of the sequence
a[n+1] = (a[n])^(1/i) ,
starting from a[0] = i .
But as it turns out,
z = lim{n-->+infinity} a[n]
doesn't exist.
@@usuyus 1/i isn't -i my dude
@@nostalgiafactor733 expand the fraction with i, so you'll get this:
1/i = i/(i*i) = i/-1 = -i
Hope that clarifies
@@yurenchu ah i see... You are right
I think you haven't found all solutions, because if you replace i^i^i^.... such i^i^z, you will get another. Maybe I don't have right. But I want to know if other solutions are exist.
lambert w function has multiple branches. answer shown is the principle value
I did it a slightly different way and got the answer of 2iW(-ipi/2)/pi which ends up being equal to what you got in the video! Pretty cool stuff
No matter how much you try, the trio of e, π, i will always find a way to be introduced in your proof x')
Hell yeah, ln is purple for me synesthetically
Pretty difficult question to think:
tetration of i to the i
i↑↑i
:P
If I am correct, complex tetration was generally solved in 2018.
Otherkin SW Can you show me a link for complex tetration being solved?
As far as I know, it has been shown that there exists a unique piecewise smooth monotonic function satisfying f(a + 1) = b^f(a) for all complex a provided that 1/e^e < b < e^(1/e), consistent with Schröder's equation and Abel's equation, which is agreed to be the standard definition of tetration. However, that such an f exists does not mean there is closed-form expression for it. i^^i may or may not be defined, but if it is, then it, for all practical intents and purposes, cannot be simplified or calculated other than with some numerical algorithm for computers. As far as I am concerned, I would say we are far from solving the problem of tetration.
@@angelmendez-rivera351 I guess you are right, the article is far beyond my skills, that's why I wasn't sure that I understood it :0
link.springer.com/article/10.1007/s10444-018-9615-7
As a person who likes math and do good in my college math as engineering student. . You are very good and talented Big up 👍👍👍
Thanks for the great video! I have a quick question: Since i is imaginary and i^i=e^(-pi/2) and i^i^i = i^(a real number), each iteration of the power tower makes the result alternate between real and non-real. Would the result you got still be valid, or would the limit of the power tower not converge? Thanks again for releasing great videos in your spare time!
PlasmaCrab _ The power tower does converge, so his result is valid. However, proving it does converge is quite a pain in the ass, and probably outside the scope of this comment section. You might wanna check Knesser's result, though. If you search that, you might find something satisfactory.
That’s not quite true. The series doesn’t alternate, and i^i^i^i isn’t real. In fact, I believe that the only real answer is i^i
@PlasmaCrab, As @MrSaxobeat already remarked, your claim is not true. The imaginary part of i^i^i is not a multiple of pi/2 , hence neither the imaginary part nor the real part of i^i^i^i will be 0.
@@yurenchu Ah that makes sense, I got too ahead of myself. Thanks for the explanation!
@PlasmaCrab , No problem! I'm glad to have helped.
Attention BPRP Subscribers:
International Mathematical Union (IMU) just approved new standards, as follow:
1. "i over i over i ta ta ta..." can be described as this:
ᵢ ᳟
ⁱ
i
"ta-ta-ta" replaces any kind of continuity;
2. Lambert-function-on-Fish (no, it is not a seafood recipe) definition:
🐟
W(🐟 e ) = 🐟
where 🐟 can be a function or variable, real, imaginary, ta-ta-ta. It replaces all recursive functions and variables of any kind;
3. Any color including purple are allowed (why not?), even when there is a restriction of colors, such as "black and red pens only", in all Mathematical works (paper, classes, exams, ta-ta-ta) from now on;
Alex de Moura Hahahhahahahha! Love this
Sorry I mean
Tatatatatatatata
@@blackpenredpen can you answer my question? I have a genuine doubt. Pls reply yes if you read this.
Is it correct to say that ln(i)= i pi/2? Complex log has infinit values and if we change what is inside Lambert function value will change too. So i ^i tatata has many different values, hasn’t it?
Thanks for your video 😍
W(x) and ssqrt(x) are closely related so we can write the answer in terms of ssqrt(x)
i^z = z
i = z^(1/z)
1/i = 1/z^(1/z)
-i = (1/z)^(1/z)
1/z = ssqrt(-i)
z = 1/ssqrt(-i)
why you put extra s at the beginning of suare root
@@jameeztherandomguy5418 ssqrt stand for super square root not square root
@@armax6452 what ??
@@jameeztherandomguy5418 you dont know about super square root?
@@armax6452 no and i cant even find anything about it anywhere lmao
can you manipulate an infinite series agebraically like that?? Im not sure unless that infinite series converges but then you would either have to know that beforehand or just make the assumptions and have a potentially incomplete answer at best no?
Can't _imagine_ doing this kind of work
Try integrating e^(-x^2), but use 1.9999999 instead of 2 to see if it prevents having to use the error function
Going by the intro music, I was looking for some Entertainment in this video, and I wasn't disappointed.
Too great to watch maths at this level you make it easier.
What classes will you be teaching next semester?
Channel named blackpenredpen, 0:40 of course we’ll use a purple pen
MESTER, COULD YOU TALK ABOUT LA CONSTANT DE INTEGRACION POR LO METODO DE SUSTITUCION?
When I first started watching your videos I was in high school and didn't understand any of it. Now I am in college and understand most of it.
These fishes would invade the entire humanity........
#tatatatatatatata army
Wouldn't it be easier to introduce an "inverse tetration" (superroot) as a special function instead of Lambert W function?
How do we compute the exact(or approx) value of Lambert w function of a constant without wolfram alpha?
by using newtons method:
suppose you want to calculate this value, which is W(-iπ/2)≈ W(-1.57i)
The Newton method for reverse engineering functions goes as follows. You make a guess on what the answer should be (any guess will do, but a better guess will take you fewer steps)
my first guess g will just be 1, which will be used to get a better value g1, which we will use to get g2 ...
because I am using a calculator the initial guess doesn't have to be good at all, but if you want to do it on paper, then maybe take some value inside the complex unit circle, that's usually a good place to start when dealing with i
For newtons method we need the derivative of the function, in this case xe^x, which is xe^x + e^x
g1 = g - (ge^g - iπ/2)÷(ge^g + e^g)
If you are confused, generally newtons method looks like this:
g1 = g - (function(g) - inputValue)÷(derivative(g))
If your calculator has an Ans button, you can do the following: Enter 1 to save it as an answer
Then input Ans - (Ans • e^Ans - iπ/2)÷(Ans • e^Ans + e^Ans)
I got the following values for g0=1
g1 = 0.5 + 0.289i
g2 = 0.43 + 0.719i
g3 = 0.58 + 0.679i
g4 = 0.566 + 0.688i
g5 = 0.566 + 0.688i
...
Ok, after writing all this, I noticed your comment is 2 years old, so sorry for being in your inbox if you already know how to do this, and if not, you're welcome
God of mathematics 😍🔥🔥🔥
you’re a genius
Man, I wish I could handle equations like that...
I was trying to figure out how to figure out the parameters of the binomial distribution formula (like when you have P(X>k), n and p and need to figure out a k) and failed miserably...
Excellent video friend
Why is it always e and pi when I solve something
I tried really long to find a set where x -> i^x is a contraction, so it must be a mapping onto itself and the imaginary part must be greater than ln(pi/2)/(pi/2) (where |f'|
It is possible to find that, if you also restrict re(x), but then i is not in it, so you have to find a i^i^... ^i thats in there...
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いやーーー お見事ですねーー
W関数便利すぎる チートやん
高校の授業って、Excelで出来ることをそろばんでやらされてるようなものやな
What is this W function? I'm taking complex analysis and I haven't seen it yet
Lambert W function
- en.wikipedia.org/wiki/Lambert_W_function
- mathworld.wolfram.com/LambertW-Function.html
This stuff is way more advanced than the basic "complex analysis" course.
that thumbnail made me click :D
nice!!!
but does it converge??? (love the video and the clever algebra)
Hey, I was wondering, since i=e^[i(Pi/2)], could you subsitute the i in the exponential by e^[i(Pi/2)] ?
So you'd get: i= e^[e^[i(Pi/2)](Pi/2)]
And most importantly, if you repeat this an inifinite amount of times, is it still equal to i ? And since i never appears on this infinite exponential, does it become a real number equal to i?
This might be very poorly explained, and Im sure I made some mistakes here and there, feel free to ask me more and correct my errors.
Adeal's TASes i = e^(πi/2) = e^[πe^(πi/2)/2] implies that there is a sequence to consider here. Namely, a(n + 1) = πe^(πa(n)/2)/2, with a(0) being the parameter that one need to be careful about. If a(0) = 1, the sequence diverges. In fact, the sequence only converges if |a(0)| < 1, or if a(0) = i or a(0) = -i. Assuming the sequence converges, one can obtain the limit by setting πe^(πz/2)/2 = z, and then solve for z. However, in the theory of divergent sequences, one can find the value of infinite expressions without regard for their convergence. In this case, it would be possible for the expression (e^π/2)^^♾ to be i.
What in blazes is the intuitive meaning of putting something to the power of an imaginary number?
I also did this one a month ago :o, and went to the same answer, but my interpretation of that is: if well it's a valid answer, in the sense of convergence and algorithm process it's not. I mean, there are infinite numbers equal to i^i^i^i^..., when the process of tetration is just one, there is no limit but a function of double infinite solutions (by iπ/2 which can be generalized, and by W function which has other subfunctions). Very mindblowing.
Also, exponentiate i to itself represents a twist in the complex plane, each time you do it. So, doing it infinitely results on an infinite twisting, without a limit but 0 as a possible convergent answer (because 0 is indifferent to the act of twisting).
Allain Cumming No, that is incorrect. The sequence (1, i, i^i, i^i^i, ...) converges, and there is a theorem in complex analysis that directly states this.
@@angelmendez-rivera351 Really? What theorem is it? I'm interested.
PD: Hablas español?
@@angelmendez-rivera351 And the convergence is just the principal value of the solution? Why there is different answers by applying different revolutions to ln(i)?
Allain Cumming Each element in the sequence is multivalued, so the limit of the sequence, which is the infinite power tower, is also multivalued. i^i itself is already multivalued. Anyhow, I do not know if the theorem has a name, but the theorem states that if 1/e^e < |a| < e^(1/e), then the sequence (1, a, a^a, a^a^a, ...) converges, and the limit, which is equal to a^a^a^•••, is -W(-ln(a))/ln(a).
Looking forward to this sh@t :)
Bien video plumanegraplumaroja.
I love your videos but there are so many concepts I don't understand. Will you make an introduction to Calculus for beginners. Your videos are of such a high quality and you are a great teacher but for people like me who only got a D in maths but want to learn it is difficult to get started.
Professor Leonard has good introductory math courses, and MIT OCW (Gilbert Strang etc.) is also good
i love the tibees t-shirt :D
It’s ≈ 0.43828 + 0.36059i for anyone wondering
thumbnail goals
I put i, i^i , i^i^i , i^i^i^i, i^i^i^i^i... and got on with the next things on the graph with Geogebra to view approximation of i^i^i^i^i^i^....(this video shows) and it looks like a spiral. What does it mean?
Complex powers represent a combination of rotation and scaling in the complex plane. The successive rotations and scalings converge (namely to the value demonstrated in the video), and in order to converge they have to diminish sufficiently fast. Thus they formal a spiral that funnels down toward the point of convergence. This is often what convergent sequences look like in the complex plane.
What happens if you use this lambert w function solution for a power tower of a positive number like 2 or 3?
My logic says you should get z=infinity for a number like 2 or 3, but wolfram alpha gives me something imaginary instead...
Is infinity imaginary...
any13th No. The problem is that those sequences are divergent. If you assign any actual value to the infinite expression, then that value cannot be equal to the limit of the sequences. In other words, the sequence is discontinuous at infinity. They become maps from R U {+♾, -♾} -> R.
I love that thumbnail
How do you know it converges in the first place?
So close to a nice answer
"blackpenredpen" ... Nowhere does it say "purple pen!"
I still dont know what the answer is
What does W(iπ/2) equal
I love this beautiful channel and the beautiful math within. But am I the only one who is left a bit unsatisfied when we start making sense of i^i^i^... And then hear "and that's it" when we reach an even more comprehensible expression?
Lambert would be proud.
I was imagining about this as an IIT Bombay student
The i's (ayes?) have it. Now I got it, too!
When I tried this on my own I did Z^i = Z then when I took the ln of both sides got ilnZ=lnZ which got i=1. What went wrong?
exp(-W(-(i*PI)/(2))) and Wolfram says "no roots exists". Hmmm, what does it means? Can not be quantified?
please master give me an a link for your video which show the fish method.. 🐟🐟🐠🐠🐠🐠🐟🐠🐠🐠🐠🐠🐠🐟🐠🐟🐟🐠🐠
Another equivalent representation would be : z= W(-alpha)/(- alpha) ; where alpha= i pi/2
What do you think about Patrick jmt? We're you inspired by him?
lone wolf
He is great! As I mentioned in my 100 calc 2 problems video, he and khan were the OG math UA-camrs and of course I was inspired by them.
What if... log in base i? Is there something special you can tell us about log with complex bases?
LogiX (log x to base i)= Inx/Ini , In i =iπ/2, In x *2/πi, 2Inx/πi, Rationalizing -2i*Inx/π
Its 3 am in india ..what is the time there sir??? Right now??
Chirayu covered this in his last to last to last video i guess
akshit chodhary who is he ?
Yes he did but I uploaded this video (unlisted) on Nov. 9th. That's why you can see the comment from 2 weeks ago.
Yup, I did
@@MathswithMuneer I am Chirayu Jain, I also have math channel, Please subscribe to it
Very interesting! Greetings from Germany
I have no idea how to aproximate it
Something is annoying me... what if you replace i=e^(i*pi/2) for each i in the infinite tower... we'll have something like e^(e^(e^(...*pi/2)*pi/2)*pi/2), and I can see only real values here...
Just wondering, is x ↑↑i = π, possible to solve?
Thank you a lot!!( from Korea
This is the power of requiem
After watching....
Me : uh.....so fish is delicious. Right?
can you do x-ln(x)=ln(45)
has two solutions but i dont know how to get them :(
Wouldn't we need to first show that the complex sequence defined by U0=i and Un+1=i^Un for all integer n converges?
Otherwise :
2*2*2*...=x => 2*x=x => x=0
And then 0=2*2*2*...=+∞ is not solution
If someone know how to show the existence of a solution (or to show that exp(-W(-ipi/2)) effectively works) it would be great :)
youssef yjjou The statement 2*2*2*2*••• = +♾ is not correct, as ♾ is not a number.
Also, it is a well-known theorem in mathematics that z^^n as n -> ♾ converges if 1/e^e < |z| < e^(1/e). However, proving it is way beyond the scope of this channel, as it requires some really high-level mathematics. Nothing wrong with taking the theorem for granted.
@@angelmendez-rivera351 Thanks a lot for your answer, with such a theorem my problem vanishes and it reassures me ^^
(I was refering to lim[n→+∞](2^n) for 2*2*…=+∞ )
I will still search for a proof :)
youssef yjjou lim 2^x (x -> +♾) = +♾, this is correct. However, you have to be careful. In some contexts, it may be useful to say 2*2*2*2*••• = lim 2^x (x -> ♾). However, in the theory of sequences, as well the theory of extensions of the real numbers, we tend to talk about those two expressions as different quantities. Think of it this way. If there exists a quantity that is bigger than all real numbers, called M, and if we consider f(x) = x |-> 2^x : R & {M} -> R, then f(M) =/= lim 2^x (x -> M). This is just a formalism, but the point is that the limit alone doesn't allow you to say the expression cannot take on a finite value.
-1?
guessed because sqrt(2)^sqrt(2) infinitely is 2 so i guessed it would work the same with other numbers although i have no idea what i’m on about
Is e^e^e^e^e... a solution for ln(x) = x ?
That's an incredible question.
What's is a decimal answer?
I just was wondering about this question not 2 hours ago and I find this video!? What luck!
You only found partial limits, you're doing bad because people might misunnderstand. Other solutions are i and -i
How do you know that i^i^i.... doesn't diverge?
blackpenredpen yay!
2:34 „here is THE FISH”
Blackpenredpen uses a purple pen.
*Me- Wait that's illegal!!!!!!*
Ok, can you do it without the W function in the answer?
Interesting.
For which values of X given an infinite power tower x^x^x^... does it converge??
If we regard the positive real number line, it would be for x between (and including) 1/e^e = 0.0659880358... and e^(1/e) = 1.44466786... . The corresponding values of the infinite power tower at these boundary values would be respectively y = 1/e and y = e .
See also the article by R. Arthur Knoebel, which is referenced (with a clickable link to an online copy of the article) at ua-cam.com/video/DmP3sFIZ0XE/v-deo.html
I don't know about convergence outside the positive real number line. (Apparently there is also convergence when x = -1 .)
So the result is approx 0.44 + 0.36i. Wow. But what does it mean ?
Pffft no check for convergence
Please somebody tell me how some comments were posted 1 or 2 weeks ago😂😂
I uploaded this on nov 9th, kept it unlisted in my playlist.
nice video
What is w fanction?
What is the w function?