Strange Math You've Never Seen
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- Опубліковано 19 чер 2024
- In this video I will show you a book that contains tons of really cool math. The book focuses on special functions, which are functions that you might see in certain physics or engineering courses. If you take a course like partial differential equations you will also see some special functions. In this video I talk a little bit about the book and then discuss infinite products. I actually do an example where we find an infinite product. I picked an example that hopefully you can understand if you have had some calculus. The pre-reqs for this book are a full calculus sequence(calc 1 through calc 3) and a course on differential equations. This book was used to teach a course of advanced undergraduate students and graduate students. The book is called Special Functions and it was written by Larry Andrews.
This is the book
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0:00 Discussing the Book
3:29 Defining the Infinite Product
6:25 Computing an Infinite Product
12:01 Finishing Up
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Thank you:)
At 5:30 it's "and" instead of "or"!
Oh wow yes can’t believe I did that. Gonna pin this comment. Thank you.
That makes much more sense, thank you!
Also, writing something like "P < \infinity" as a way of actually meaning that a limit exists and is finite is a bit sloppy too...
@@TheMathSorcerer Hahaha
@@alexfekken7599 I don't think that's sloppy at all. It's just a shorthand, and when you consider extended real valued functions it means exactly what it says, that it is finite.
Ah, Bessel functions (and spherical harmonics) were part of my second year in my physics degree. The lecturer spent about two full hours talking about spherical harmonics only for me to completely not understand even the vague idea of what we were doing. In the end, a group of us asked him outside of a lecture to show it again but slowly. Being the great guy and great lecturer that he is, he spent a further hour with us in his own time. He also said that the moment anyone gets lost, just stop him and ask him to go over it. I think after that hour we all had a better understanding than most of the rest of the cohort.
+1 on the Bessel functions......who new I would rediscover them in RF design?
@@jimsmith6937 As far as I remember, I never used them again but maybe some other students that later specialised in theoretical physics or solar physics or something like that did need the knowledge. I reckon, at least for me, we were taught this stuff as a way of getting at least some practical application of being to handle infinite sums of sinusoidal components, and then that lead into the next module which was largely about Fourier methods.
Even then, I didn't have to do any Fourier transforms after the exam for that module. I 'specialised' (for want of a better word) in experimental physics simply because that was the default and I couldn't pick a specialism. I ended up doing a masters and PhD in volcanology but in both of then, my main focus was experimental exploration of material properties.
My professors would literally just say "theres this book called nvfjnvjfrnvfr, look it up, I have better things to do now".
Watch someone poke a water balloon in slow motion.
@@jimsmith6937 Surprisingly, modified bessel functions of the 1st kind show up in the analysis of strong non-linearities of junction transistor amplifiers
This course would weed out 50% of the engineering majors when I was taking my degree. This for people who had already done Calc 1-3+ courses. I'd often describe it as the math that even mathematicians don't do.
It depends on the discipline really. Not all of these equations are for all disciplines. Maxwell's equations, for example, are probably only going to be used for electrical engineering. Whereas something like numerical integration will be found in physics engines and physics emulators (and maybe even in stock market AI), Differential equations is a wide field and it has applications just about everywhere.
@@BitwiseMobile I saw some yt video on QM and maxwell's equations are apperently still used there cause of particle wave duality, describing QED and fields.
@@mb2776 Maxwell's equations pop up in QED, but they are contained in the field tensor. Basically it all surrounds the idea of guage invariance.
The jargon is you minimally couple a U(1) guage field to some field theory by introducing a covariant derivative and identify the connection with the vector potential. We can make this vector potential dynamical by introducing a guage invariant term, being the field tensor!
I had to use the confluent geometric functions to solve a Schrodinger eq. Every last one of those functions pops up in physics, most in your undergrad. Many representations of these functions also pop up as integrals when calculating propagators. I've seen them in numerical analysis, and other ones, for interpolation and quadrature.
@@BitwiseMobile Although in general I agree with you, I disagree on numerical integration. Numerical integration is found in a lot of general science/engineering and data processing, and extremely useful knowledge to have.
This reminded me of a book in my library. When I worked at NASA JSC in the early 70's they had a technical book store where employees could buy books at discounted rates. I bought "Handbook of Mathematical Functions with Formulas, Graphs, and MathematicalTables" by Abramowtiz and Stegun. It was published by the US Department of Commerce, has a total of 1046 pages and all this before hand calculators. Still has the original price tag at $12.65
Ah, yes. AMS 55. I have it in hardcover and softcover. Best thing since Jahnke und Emde. You can get it cheap from Dover Publications. Good choice!
Thanks for the infinitive product example and simplification approach you used. By the way, the GAMMA function is one of my favourites. Thank you for showing this book. _(And yes, it is very well made... and I can see the layout is very readable, clear and uncluttered.)_
I have this book and I know Dr. Andrews. And he knows me. When it comes to higher level mathematics he was probably one of the best math teachers I've known. I have three of his books, the other two are Mathematical Techniques for Engineers and Scientists, and Elementary Partial Differential Equations.
If you want to see some harder problems, look up the gamma function....
Actually, I used this gamma function to solve a real world problem in diffusion in 2D quantum wires. This particular problem also involved Legendre polynomials, a Heaviside function, a Fourier Series, all buried inside of a differential equation which was buried inside of an integral which came in two parts.
It was fun. It took me six months to figure it out, but it was fun.
Wow that's awesome. Thank you for this comment!
Only 6 months? Nyyyyce
What the hell are you talking about
It makes me so happy to see I recognise all those topics... Proud of how far I've come as a Physics student...
And excellent review! Thanks!
The book by W W Bell is also an excellent reference on tue topic special functions nearly all of these functions generally arise out of a study of well-known differential equations from physics
👍
Yeah, Bell is available from Dover.
This is a great book. Larry Andrews is an emeritus professor at UCF CREOL. He's very nice guy. I knew one of doctoral students Olga Korotkova.
Very nice. I appreciate you working out the problem as an example. I have not worked these types types of problems since 1977 when I was in a DE class. It brought memories. Thank you
You are welcome!
This textbook gives me flashbacks of doing applied maths and chemical engineering in the 80s ;)
ps. The student/textbook version often only had 'answers' to about 5%-10% of the problems so that students could be assigned Qs that they couldn't look up the answer. If you want all the answers there was often a 'teacher's manual' version of the textbook that provided answers to every exercise Q. Might be hard to find a copy though ;)
Oh man, some of those chapter titles bring me back to my engineering and physics classes. A lot of them we wouldn't actually calculate ourselves, rather we were encouraged to buy a book of tables (Schaum's Mathematical Handbook of Formulas and Tables, to be specific) with solved general forms and the object would basically be to finagle the problem into something resembling one of the forms and use that to solve things like Bessel functions.
...at least until we got to Math Methods, which I could totally see this being a textbook for.
Abromowitz and Stegun or Gradshteyn and Ryzhil!
Yes broken question, the final part of the video with the question wrong, with first 1/2 × 3/2 with brackets, goes 3/2,8/3,15/4, he cancelled cross multiply out of the original brackets with the above formula, k= 2/1 + 0/1 is added with no balance in the Infinite formula. K=2/2
I'm sometimes amazed humans have attained such levels of higher knowledge. I think we take some of it for granted since it almost seems commonplace. But the people who worked it out and passed it to the next generation are brilliant.
"Strange math you've never seen" aka "what I see when I look at the exam"
Lol
It's nice to see some familiar faces in these books! I did floating point implementations of Beta, incomplete Beta and Error functions for the Forth Scientific Library. But for my own compiler I did a lot more. I especially like Gamma functions - especially the "weird" ones, like Ramanujan and Cristinel Mortici approximations.
I also like Gamma functions and thet have some very interesting properties as mellin transforms, which is rarely taught in topics of special functions. For some time, ive also been pondering about combinatorial idenities being expressed as gamma functions with the use of the gamma factorial.
Awesome video! I subbed forever ago when I was in Calc 3, and since then the number of videos I watch has dropped, but this one has peaked my interest yet again. I forgot how cool Calc 2 was!!
I appreciate how you kept in every step of the solution. My teachers back in school would always skip a bunch and only the nerds would be able to keep up
A reason to only allow positive factors is so that the infinite product is equivalent to an infinite sum of logs of the factors. Allowing negative factors only adds non-essential complications. A corresponding reason to call a limit of 0 divergent is that it corresponds to the log sum diverging to -infinity.
Makes complete sense, since you are able to develop convergence criteria of products by using and reinterpreting those for series.
Of course!
I can’t see no other essential reason why the limit zero is divergent, are there others?
Great video! To hold your book open I recommend using a binder clip if you havent tried it!! I use a big binder clip to hold my textbook pages open when I put them on my stand :)
I went to the University of Central Florida where Dr. Andrews taught. I never had him as a professor, but I did hear that he was an amazing teacher. I heard that he wanted to make sure his students understood what he was doing on the chalk board. I have a copy of his Partial Differential Equations with Boundary Value Problems book. That book is also well written.
Nice !
LETS GO KNIGHTS what did u major in?
That is a cool book.
And yeah, I've heard of those, but that's because I watch some other math channels here, not (strictly) because of my education. Thank you for sharing!
Great video. Would love to see you do some of the more complicated stuff in that book.
I actually took this class with him at UCF. I wish I had him sign it. His lectures were like fine wine. That was a general consensus
UCF?? Small world
When you do a PhD in theoretical physics and use maple to solve some model equations it usually spits out all those functions at once. ;)
I haven't done calculus in decades. I didn't do so well in calculus when I took it in college. Yet I was able to follow along quite easily...Very well explained!
My first sight of the Higher Transcendental Functions was glimpsed in Part II of the textbook: "A Course of Modern Analysis" by Whittaker & Watson, published by the Cambridge University Press. It's title is somewhat cryptic now considering it was first published way back in 1902; it can appear rather archaic now using "Shew" instead of "Show", but it is a veritable treasure trove of all the advanced functions. I believe it is still in print, on Amazon as a paperback, as it was/is a real Classic! I gather both Profs were contemporaries of the superstar G.H.Hardy, whose own book "A Course of Pure Mathematics" is another abiding classic, still in print.
As some bore once opined: "Don't read the Books about the Books; instead, read the Books!"
Physics major here (Junior currently). I took a Mathematical Physics course and a large section of the class was special functions. It does come up a lot but the sad part is we didn't have a book for the class. Hard to study and practice when there isn't a textbook to skim through.
I was a sophomore learning this, so it was a shock when you mentioned that high undergrads and/or beginner grads learn this stuff (Save me :( ) Definitely interesting.
Modern Analysis by Whittaker and Watson is also a book which includes several special functions and in general it can be called a legendary book as it way more information than a standard analysis book.
I think I still have this textbook !!
My favorite was the Bessel and Gamma Functions and integrals, and I believe they did some Fourier Analysis in this text [ though I had a separate text just for Fourier ] ....
Strange...I actually enjoyed those topics 🙂
❤️
I am using Bessel functions of the second kind, in a research study that I'm conducting.
Spherical harmonics have a lot of important applications in computational materials science. Thank you for sharing your take on this book!
To condition a high quality bound volume, you need to condition the spine! Stand the book on the spine and open the two covers. Then holding the pages up vertically, begin from the outer pages on both sides and begin paying them flat a few pages at a time. Press the pages down at the binding, and repeat, working a few pages at a time from the outside to the center. Repeat this process until the binding becomes supple. Hopefully you've not broken the spine already.
We're lucky we don't need the paper knives anymore
Haven't watched the video yet but for a second I thought you were correcting some of the weird math in the book. "ah yes, bound volume, some sort of amplifier function *nods sagely*"
A dying art form!
@@Oceloctopus Lol I thought that too until the second sentence
Had old Soviet book written by another author but named exactly like this (though in Russian). Needless to say the content is the same and even the sequence is somehow similar (but hyperheometric function was explained at first place and used further on e.g. in Gamma function explanations). Good old times of studentship...
Was it the one by N N Lebedev? The contents looked very similar. I worked through much of it in school and found it very useful.
@@kdub1242 as far as I remember - yes, it was Nikolay Lebedev's book, issue of 1962.
@@user-xv9fe4eo1b An American math professor named Richard Silverman translated a whole bunch of great Russian language books into English and that was one of them. Fond memories. I studied physics rather than math, but many of the books were extremely useful for applications.
@@kdub1242 oh wow, never thought they were appreciated as highly! Funny enough, one of my handbooks was written by a guy with a name you'd expect to be Russian or Ukrainian, and it came as a surprise for me that the book was translated from English and was first published in New York. The book is called "Vibration problems in Engineering" and it's main author Stepan Timoshenko (who was actually born near Chernigov and graduated in Saint Petersburg) was also one of the key early contributors to ASME code development.
@@user-xv9fe4eo1b Oh yeah, there were a whole bunch of applied mathematics books that became classics in the US. And for undergrad physics, the very famous "Problems in General Physics" by I.E.Irodov has, despite its boring title, entertained and challenged young physics students worldwide for decades. And for grad level physics, I need only mention Lev Landau. When it comes to math and physics (and probably a lot of other subjects too), you just can't beat the Russians and Ukrainians.
There is a certain joy in successfully working a problem.
That partial product demo was super-cool !!!!!!
Excellent! This should be a thing for all your book reviews. You should pick an interesting problem from each book and work through it with us!
Ok!
👍
@@TheMathSorcerer yes please..very nice suggestion....make your minds prepared for future ..will wait for this 🙏
Yes I really enjoy doing these videos. I feel like people learn something after watching them and it’s worth it👍
Physics major here. I enjoyed solving the exercises of Whittaker & Watson's "Course of Modern Analysis" when I was a student. :)
I loved the example! Thank you^^
That table of contents takes me back to graduate school. Bessel functions - the horror.
I have this book. Takes me back to my engineering days in the mid 2000's
Cool
Enjoying these combo book review and problem solving vids.
😀
The problem at the end reminds me of a beginner’s Real Analysis problem. I found Real Analysis to be overwhelming and difficult
I love this problem. Nicely done!
Also you can use induction if you prefer.
Base case: 3/4=1/2(1+1/2).
Induction: 1/2(1+1/k-1)(1-1/k^2). Which is equal to 1/2(1+1/k) after some algebra.
There are two editions to this book:
1st - 1986, and
2nd - 1992. The 2nd edition also has a reprint by a different publisher.
The 1st edition is discussed above.
Love your videos professor 😍
I joined your channel membership today itself and i got many perks
Thank you!!
Some good books on special functions are the ones by Olver (Nist handbook), Butkov (Mathematical Physics), Watson's Treatise on Bessel Functions, Prudnikov (table of functions and integrals) all five volumes, Gradshteyn (table of funtions and integrals)
I just purchased a "Engineering controls and control systems" from 1957. Im in my controls class now and its weird. Root lucust plots are crazy
11:40 "You just have to..." Open MS-Excel, fill in the index column (2, 3, 4...), type in the formula, add another column for the result-so-far, then pull it down the screen (duplicating rows) until it converges, examine it and then state, "Looks like about one-half." Pull it down some more, "Yep, converging to 0.5." Very likely competitive in time to the analytical approach. 🙂
I've heard of most of those things, but only because at one point I skimmed a lot of reference books at the library for cool functions to use in Four 4s. I'll have to check if my library has that book!
Do you know where one could find that exact version?? i love those covers and would love to add this to my collection of books i skim and hope one day to need in my research :p
I have books like these; gifted to me by dear friend. Now I'm so thankful. For anyone who needs to have comprehensive knowledge and written works for E.E this will help if you're in college or university or even if you're on the job learning
I have an exam on exacly those things in a couple of weeks😬
One of the most difficult courses I had up to now, at least for me
Great video! One thing to note on the practice problem if you the viewer want insight that applies to other problems: one way to do the cancellation is to split the fractions between two partial products, re-index one of the products to start at, say, n=1, and then identifying what cancels when comparing terms with the same index. Visual pattern recognition is great with simple problems, but refining techniques of manipulating partial sums or products to get terms to cancel will eventually take you a lot further.
Great content and delivery. I was wondering if you can recommend a book or books that really highlight the amazing things you can don with math. Practical applications. If not, you should write one.
Courant & John vol.1&2 is a classic for Calculus. There’s one from Lax that has the same spirit and is recent.
I would love to recommend you more math books, but the problem is that many interesting things that can be done with math require several undergraduated courses, even those of linear algebra applications or operational research topics. My discrete and geometry math knowledge is very limited, but you can find fun things and even practical applications in those areas that are easy to understand
Well, it's pretty standard collection for applied mathematicians - we learned all those functions in university. I couldn't comprehend them at the time but I still remember the names.
Вы в советском Союзе как мой отец занимались более продвинутой математикой. Американцы до этого не дотягивается.
Я немножко сожалею потому что сам по математике не понимаю но хочу заниматься Инженером по электричестве. Если у вас есть какая-то рекомендация или совет по книгам пожалуйста дайте знать.
@@daviddavid-up1jc Да нет, обычный ВУЗ в Сибири. В 2016-ом закончил. Лучшая книга это та, которую вы можете понять.
perfect example of the meme: “i know some of these words”
WOW you saw the math from a book called “Functions for… applied mathematicians” when you were studying applied mathematics at university!? Whoever would have guessed? Phenomenal! With top minds like yours, this Ukraine thing should be wrapped up in no time.
I have come across so many of these! Very interesting
Love all this! Why I am relearning after taking it all 40 years ago!
Very interesting book!! I’ll be arriving at infinite series soon in my self study of math.
Awesome !
let ,set,define,consider,compute and finally solutions i have touched learnt all kinds of mathematics from simple to complex from high school tech to physics to ME.i miss that time.
I have plenty of books from the 90s that I kept from school that were well made. Maybe they don't make them well anymore, but I have two Calculus textbooks and one Algorithms in C textbook that have survived the years relatively unscathed (aside from some minor notes and highlights in the text). I even have the book from my first C programming class. It was in softcover and it's still 100% intact. The binding is still solid and there are no loose pages. I guess they just don't make them like they used to.
3 seconds in and I knew this would be one of those "making sense of any and every thing in the book is left as a trivial exercise to the reader"
Like 80% of this book plus 40% more stuff is covered -- better IMO -- in Arfken and Weber. We used that textbook for my undergrad mathematical methods course and I still use it as a reference.
Cool little example . . . I love the ones anyone can follow, 30 years after their diff eq class. ;)
Also, how surprising that an applied math/engineering math book lacks comprehensive solutions. Wow. :)
Okay, granted the problem you did was not that difficult, but it still felt great being able to follow along step-by-step. Especially considering I haven’t taken a math class since the late-1970s.
that's awesome! I picked it so people who had some background might understand, I am glad you could:)
Until chapter 8 it looked pretty much as the syllabus of my Mathematical Physics course. We had also Fourier series and transforms together with functions of a complex variable. Good reference man!
Very cool!
Hypergeometric functions were the best part of my degree, looking forward to what you do with them.
The book is fantastic and beautifully explained.
I haven't studied math in over 14 years and was average at best in high school but somehow got the feeling it was converging to 1/2 a few mins before you finished :0
This video is attracting two groups of people that share only a modest overlap: advanced mathematicians, and people into book binding.
I feel so much better after watching this video because I struggled like a dog with these problems in my undergraduate studies
If you like this book you would like Courant & Hilbert, Methods of Mathematical Physics. My teacher for an applied math course was a Dr. Brown who was a grad student of Courant who was a grad student of Hilbert. History matters.
Special people will always tend to special people after all
Really nice. Thank you for making this video
This was fun to watch, thanks
Thanks for introducing this book, it looks like a good one.
Everything pretty standart to be fair. Nice book, should get a copy :D
I wonder if there are tests for convergence of infinite products like there are for series, and what the connection between them is
There is another resource that allows calculus to be learned easily by hand, literally. Level N of Kumon Math (no, not marketing for Kumon) has worksheets that allows one to learn by working from simple to complex differential equations in incremental steps over 200 worksheets organised in 20 sets of 10 worksheets for each level of difficulty. Hand-eye-mind connection is attained with little sweat and much achievement for learners as young as 15 years old.
Am currently going through Larry Andrews / B. Shivamoggi's "Integral Transforms for Engineers" and try to work through the exercises; hit a brick wall every now and then. Currently in a rut on part (b) of Q16 in Exercise 3.5. Wish there was a solutions manual. Well written book. Bought a softback edition in India for Rupees 250. Its kept me entertained for some 4years now - slow going.
Excellent video. A detail: Missing is the justification of why you can't say an infinite product converges if it approaches zero.
Dear Sir
Thank you for introducing this book. It is so good, I always like book of this kind, strange enough, deep enough, wide enough, yet very readable. I always try to find book of this kind for any other topics in maths.
Would you also kindly please advise if there is any similar book type for probability and statistics? not necessary for introductory type, it can be very advanced.
Your kind assistance is highly appreciated. Best Regards.
The schaums for probability and statistics are pretty good.
@@TheMathSorcerer Dear Sir, Thank you for the recommendation. I already got it. Schaums books are always the very learner friendly type.
I always have some
strange tendency in mind that I forgot probability and statistics is applied mathematics, I habitually tends to find sort of pure maths type of approach on probability and statistics. Maybe I am a bit abstract principle minded instead of down to earth peripheral application minded.
I like your channel very much, very very rich maths materials. I wish I could know you earlier.
By the way, I also have a question on topology, I ask you in your topology book recommendation. Thank you.
Sir Make videos on Geometry . And also give some tips regarding Geometry.
Brother your works is so great as infinity I am proud of you I have one suggestion see if it is good to make a video please make a video srinivasa ramanujan notebooks
This is the favorite book I’ve never owned. There is an empty spot in my bookcase where I sometimes go to appreciate the fact that I do not now and shall never own it.
Legendre is pronounced "leh-jeah-dra."
I took a year of courses in applied mathematics, we spent an entire week on the gamma and beta functions, a week on wavelets (discrete fourier analysis and alternate methods of harmonic analysis), and an entire month on various families of orthogonal polynomials and numerical quadrature. I didn't get to the courses on dynamical systems but I imagine that they covered the remainder of the functions in this book during that year.
You will *definitely* see these functions if you plan on working in statistics, numerical approximation, or dynamical systems.
What a clear, useful and welcoming video....Thank you !!!
You are welcome!
Back in the day, (statical engineering) several function/ regression modeling was quite the directive in terms of demo of the star-wars tech back in the early 1980s! This saved my bacon more than one time!
Proper way to do this is to do an integral about a pole in the complex mapping of a function. As you circle around and calculate the residuals you will find the convergence and the terms. See Churchhill Complex variables.
You should see most if not all of this in a physics undergrad and maybe a math minor in the US. Some undergrad physics classes may handwave you through the more rigorous parts of the math, though for me, it was usually left as an exercise to be turned in later, on one occasion left to be worked out as a question on the final exam. I was a dual degree math and physics and physics set me up for an easy A in most of my mathematics classes because I was seeing everything in physics first then in math a semester later.
I always had the feeling the math classes are lagging behind and then at some point they became years ahead (e.g. residual thm to calculate integrals only popped up during grad courses for me)
Nice book not only for engineers but for applied physics as well
I've seen it, at 7:30am MWF in the 90s, on the opposite side of campus from all my other classes, my apartment, and any available parking. 🙃
Didn't you just LOVE those classes! Ah, fond memories.
It's a good coincidence UA-cam showed me this video. I have a case with Legendre polynominals involved.
yes finally the mathematic formulas I needed to organize my sock drawer
Large rubber bands can hold the pages. The lack of worked out answers is deliberate so the professor can "earn" his way - after all that who the book is marketed to.
There's no restriction in only looking for positive infinite products. If you want a negative number, you tack the minus sign at the end. If the terms of the product alternate in sign, you have an ill defined product, since the sign of the product depends on the parity of the n-th partial product. If you want a complex number, it can be a bit more subtle, but in the end you only need e^ (i*final phase) of the product: if your terms are complex, you end up with a series for the final phase, and an infinite product of the magnitudes of each term, which are all positive numbers.
Quantum mechanics textbook: Hold my beer 🍺
General relativity textbook: Hold MY beer.
The infinite product expansion is very reminiscent of the classic infinite product expansion of sinc(x). Since you don't want the n=1 term of that expansion, we see that sinc(pi*x)/(1-x^2) has the infinite product expansion of yours but with x->1. Taking the x->1 limit, we get 1/2. This is one explanation of why the product you showed has such a simple answer.
I’m self studying Bessel functions right now in my free time so that whenever I encounter them again in physics they don’t feel like their coming out of nowhere and I’m familiar with all their properties. I’ve only made it through the first chapter but so far Introduction to Bessel Functions by Frank Bowman is pretty good. I’m reading it more for the applications but I’ve surprisingly been able to understand all the proofs so far. They probably could be more rigorous but they work for me.
I used to do research on generalizations of Bessel functions. You should check out Watson, a 700 page behemoth all about Bessel functions.
That pencil is iconic
I feel like one final statement is missing after solving the limit. I truly believe my math professor would have taken a point away 24 years ago for not stating something like: “since P = 1/2 and P < infinity AND P != 0, we can say that the infinite product converges to 1/2”
Unless your math professor was also a comp sci professor he should've taken marks away from himself for using P != 0 rather than P ≠ 0.
@@lightyagami1752 I think even then marks would have been taking off. I was just typing quickly ! But yes, one should not Solve a mathematical problem with a programming language :-)
We have Unicode now. Why shouldn’t modern programming languages accept “≠” as an operator?
@@lawrencedoliveiro9104 the C and C++ standards do not define the that symbol as an operator.
@@johnleclair663 I did say “modern”, did I not ...