The impossible equation at the heart of Kepler’s laws

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REFERENCES
Colwell, P. (1993). Solving Kepler's Equation Over Three Centuries. United Kingdom: Willmann-Bell.
Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.
Bate, R. R., Mueller, D. D., White, J. E. (1971). Fundamentals of Astrodynamics. Egypt: Dover Publications.
Vallado, D. (2001). Fundamentals of Astrodynamics and Applications. Netherlands: Springer Netherlands.
Borghi R. On the Bessel Solution of Kepler’s Equation. *Mathematics*. 2024; 12(1):154. doi.org/10.339...
Tom Archibald, Craig Fraser, Ivor Grattan-Guinness, The History of Differential Equations, 1670-1950. Oberwolfach Rep. 1 (2004), no. 4, pp. 2729-2794
Francisco G. M. Orlando, C. Farina, Carlos A. D. Zarro, P. Terra; Kepler's equation and some of its pearls. Am. J. Phys. 1 November 2018; 86 (11): 849-858.
Arthur A. Rambaut, M.A. A Simple Method of obtaining an Approximate Solution of Kepler's Equation. *Monthly Notices of the Royal Astronomical Society*, Volume 50, Issue 5, March 1890, Pages 301-302.
Ben Coleman. How to Find the Taylor Series of an Inverse Function. randorithms.co...
17 Recent papers on Kepler’s equation can be found in references 2-16 here: www.mdpi.com/2...

[@WelchLabsVideo]

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[@stopper0203]

Will your book be shipping outside America?

[@Anon54387]

I just factor out the E, set 1-e*sin to zero and call it a day.

[@akaHarvesteR]

Oh man this brings back memories. KSP used three different methods to compute E. The Newton-Raphson method actually turned out to be very unreliable for several cases, and would often blow up. What worked pretty well was a binary partitioning method, where you start with a large correction and half the search space at each iteration. It was slower to converge, but almost always found a decent solution. There was also a dedicated solver for extreme eccentricities, and of course another one for hyperbolic cases.

[@Tuned_Rockets]

Woah it's the man himself!

[@VeganSemihCyprus33]

You don't need to solve it, and that's how it is solved 👉 The Connections (2021) [short documentary] ❤

[@ionre24]

I was wondering how you knew this and then I saw your username. I was working on a 2d orbital space game a few years ago and I wondered how KSP calculated orbits, and I guess now I know :)

[@Bizob2010]

Woah :D. Would you approach it any differently today?

[@jmr5125]

Do you remember under what circumstances Newton's method failed to converge for elliptical orbits? I know that hyperbolic orbits are problematic, especially with high e, but I'm surprised that Newton's method would ever fail for elliptical orbits.

[@revenevan11]

As a space nerd who loves this sort of math, I'm astounded that I never realized how central Kepler's equation was to the historical development of calculus!

[@revenevan11]

And of complex analysis, more broadly I guess.

[@QuantumHistorian]

3:00 That's slightly off: the orbit is a parabola only at exactly e=1. For greater values of e it's a hyperbola. Not that having the right name for the right conic section is all that important here.

[@PhoenixEditz69]

Well coincidentally, our teacher today began teaching about parabola and when he said for e=1, locus is parabola There i got confused because in the video he mentioned something else😅

[@isaacclark9825]

@@PhoenixEditz69 If I recall correctly, a completely different approach is used for hyperbolic orbits. Parabolic orbits, which require e is exactly 1 are as rare as perfect circular orbits. We don't use Kepler's equation for that either, because we don't have to solve for E (eccentric anomaly) to get the true anomaly. We get the true anomaly directly from a much simpler equation.

[@PhoenixEditz69]

@@isaacclark9825 yeah i also think in actual life parabolic trajectories are rare as nothing is perfect and hyperbolas are more relevant I dont know if im correct but this could also be the reason why we study hyperbolic geometry and spherical geometry at higher level I haven't seen many people talk about parabolic geometry specially being a high school student😅 Most of college stuff like multi variable calculus, differential equations, and astrophysics i have learnt through YT videos So finding right stuff sometimes become tricky Like why eccentricity can't be zero But rather it tends to zero🤔 Though visualy it seems correct but mathematically it's counter-intuitive

[@qadirtimerghazin]

While watching the part about Kepler’s numerical approach, I thought “That’s what the Newton(-Raphson) method is for…” Turns out Newton came up with it specifically for this equation!

[@VeganSemihCyprus33]

You don't need to solve it, and that's how it is solved 👉 The Connections (2021) [short documentary] ❤

[@oniondeluxe9942]

I experienced exactly the same thoughts. 😮 It’s so sad when I realize I have learned so many things through life, and still, my life is a failure.

[@TristanPopken]

I remember trying to solve this before knowing it was a famously difficuly/impossible problem, got to the exact form of M=E-esin(E), though with different variable names, and tried to use the taylor series to make an estimate. Worked well for low eccentricities, but lets not even talk about the high ones haha. Great video about this topic, so here is a comment for the algorithm :)

[@maxmn5821]

Similar thing happened to me with x+log(x)=0 For a while I refused to acknowledge the problem - this was just a boring heat exchanger, after all. Until wolfram alpha convinced me

[@andrewzhang8512]

@@maxmn5821 lambert w function moment

[@unflexian]

​@@maxmn5821similar experience, banging my head against a wall trying to invent the Lambert W function until i heard about it in a blackpenredpen video

[@Sluppie]

Same. I was working on a programming project that needed to model orbits. The idea was that you had a "time slider" that you could move back and forth like a scroll bar to make the planets move around the star. I also needed to have a rocket ship that could apply thrust at specific points in time to alter orbits and transfer between different planets and the star. I thought that E would be an easy solve. I thought... I eventually just ended up taking the idea of a binary search tree and applying it to E, so that I could find it out via brute force. It worked pretty fast since I was using a computer, so I didn't worry about it too much beyond that.

[@-_Nuke_-]

I honestly think that this is one of the best channels in all of the internet, not just KZbin. I feel blessed everytime I venture in the minds of the geniuses of our past.

[@VeganSemihCyprus33]

You don't need to solve it, and that's how it is solved 👉 The Connections (2021) [short documentary] ❤

[@marcorademan8433]

Keplers algorithm resembles a control system closed loop in discrete steps, where the error signal is fed back with a proportional gain of 1. Some gains can make the closed loop unstable. This is an early version of a P-controller! You can analyse stability in terms of the z-transform of the open and closed loops.

[@logickedmazimoon6001]

so you can make a block diagram for this equation interesting.

[@paramdandekar562]

yeah lol when i saw that part that's exactly what I thought of too! now this has me wondering if newton's method can be expressed in control system terms too

[@jaspervandenbosch3838]

​@paramdandekar562 It would be a PD method, since it involves both the output value at the time step (P) as well as its derivative (D)

[@paramdandekar562]

@@jaspervandenbosch3838 not exactly though right? PD is when U(s)/E(s) = Kd*s + Kp, whereas Newton's method has the derivative divided by the value?

[@jaspervandenbosch3838]

@paramdandekar562 Yeah, you're right it's not really a PD. I think it should be possible to express in terms of a control scheme though

[@cheesedmacaroni]

I have a beautiful solution to Kepler's equation, but it could not be contained within the margins of this comment section

[@thesmartstickguy1145]

I get that reference

[@sajjadakbar6649]

Good Now i will look for your proof

[@mrfarts5176]

Ou must understand that Einsteins equations but be run assuming a flat earth. Well done...

[@VeganSemihCyprus33]

You don't need to solve it, and that's how it is solved 👉 The Connections (2021) [short documentary] ❤

[@thesmartstickguy1145]

@@VeganSemihCyprus33 Lol. It's still an unsolved problem. Did you not watch the video?

[@mingmiao364]

(Edit: the last paragraph) Superb work!! It always baffles me that Lagrange inversion isn't a standard topic in Calculus courses in universities. Most equations in science and engineering are "impossible" to solve in the sense that no closed-form solutions exist. Yes, we have powerful computers to quickly obtain highly accurate numerical solutions, but an approximate, analytic solution gives a much better qualitative picture. (perhaps it is difficult to justify convergence at elementary level, i.e., without complex analysis, but the inversion technique is just too good to miss out!) And thank you very much for providing reference btw. I'd like to add one more: Theory of Functions of a Complex Variables, by Soviet mathematician Alexei. I. Markushevich, translated by Richard A. Silverman. It has an extensive treatment on the technique of inversion of power series. In particular, the power series solution to Kepler's equation is covered in Example 3, page 99, Volume II. Markushevich's treatise is also extremely thorough (more so than western classic such as Ahlfors) and is the standard reference on complex analysis in the Russian-speaking world. PS. For those who wish to learn about Lagrange inversion but without a background in complex analysis, see pp151-155 Art.125,126 of the book "an Elementary Treatise on the Differential Calculus; Containing the Theory of Plane Curves, with Numerous Examples" by Benjamin Williamson. It is an old book and is available on the Internet Archive.

[@tomerpeled8922]

I solved it actually. Will post proof in a sec

[@platymusPrime]

It’s true, he showed it to me last night

[@DevangPatil-i9f]

It's true my professor told me about him

[@Bombito_]

​@@platymusPrime¿In a dream with a Indian god?

[@ryanschneer]

Ok Fermat

[@maloxi1472]

It's been 676 seconds 😢

[@siddharth-gandhi]

Bro is single handedly bringing back the science in science education channels on YT, nothing but bravos to you! Always excited to see what you teach nexy

[@darrenstensland5301]

Well, with some help from Grant Sanderson and Maxim.

[@Rubrickety]

This Kepler series has been superb. (And let’s all take a moment to thank Grant Sanderson for making Manim open source! Its signature look and power is all over KZbin these days.)

[@janisir4529]

A function not having analytic solution is indeed a good reason why it's hard to solve.

[@carultch]

I stumbled upon this equation when trying to solve the exact same thing, for a recent Walter Lewin problem about the transfer orbit to Mars. I wanted to make a Geogebra animation that showed the path of the rocket as a function of time, with Earth and Mars taking circular orbits at the same time. I derived the equation of this form, and realized there was no analytic solution, and implemented a first order use case of Newton's method to solve it.

[@BernardLowe-v1w]

Oh, man! Believe it or not, I just started taking a course on Numerical methods recently and only 2 days before, I had a lecture on the Newton-Raphson method in class. AND, I just started learning python animations with Manim! That's how you make your videos, right? You chose the perfect time to drop this, lol. I am starting to see just how much code and time you need to devote to get these beautiful animations. Thanks for the great vid!

[@markuswx1322]

Years ago when learning how to compute implied interest rates and future cash flows based on the present value of an investment, I had no idea how closely these were related to fundamental problems in mechanics and other fields. Thanks for a very informative video.

[@chil.6476]

Watching this two weeks into a complex analysis course would add do much motivation

[@swamihuman9395]

- I can't love this enough! :)

[@euromicelli5970]

Not explicitly mentioned I presume for simplicity reasons, the Newton-Raphson method relies on the derivative of the function (see if you can spot it 7:03), which of course Kepler did not have but Newton very much did. Newton-Raphson remains to this day a very practical and useful numerical method (although it’s not the only one and it doesn’t work well for every kind of equation).

[@isaacclark9825]

Wow. I have worked with this equation since I was a high school lad, even up until last year when I implemented a solution (for low eccentricitry orbits) in software last year I expected that the subject would interest only about four KZbin viewers, but you have done much better than that!!

[@liamhagan4434]

Hey man, really excited about the imaginary numbers book. Your series on imaginary numbers in 2016 was extremely impactful to me then and now. I still think about those videos often, and can’t wait to have this book in my possession!!

[@phenixorbitall3917]

Tbh...Cauchy is a real beast! This guy was disrespectacularly smart - no kidding

[@deer001]

After watching your complex number series, I felt heartbroken that you stopped posting.I didn't even know you were back until a short of yours popped up! So glad you are back! Take love❤

[@aljawad]

Ahh, this brings back so many fond memories of my first experience in programming. Some 50 years ago when I started to plot cometary positions based on their orbital elements, I used a programmable calculator to hunt for the value of E, basically using iteration (my first Hewlett Packard calculator had only 49 steps). Eventually when I acquired my first computer (an Apple II), I wrote dedicated programs (initially in Basic, then Pascal) to tabulate an ephemeris of the positions. A fascinating book on the subject is Peter Colwell’s “Solving Kepler’s Equation”, which I highly recommend 👍🏼

[@costbubbles8336]

ive been waiting for this video for so long after seeing the yt short

[@maitland1007]

This was really great. The only thing Id want more of is more explanation of the history/derivation of the equation. Maybe another video?

[@WelchLabsVideo]

I cover some of this here! kzbin.info/www/bejne/hpnWlJ2iZdppiqs

[@maitland1007]

@@WelchLabsVideo Thanks.. just watched the first one.. my reaction is.. WOW!.. so amazingly well done! thanks!

[@kellymoses8566]

Veritasium has a good video about this.

[@Nabla_Squared]

I love so much your videos, your style, explanations and the topics you choose are simply perfect. Excelent work!

[@cparks1000000]

The Lagrange Inversion Formula looks super complicated but it's just Taylors Theorem at f(a) plus "calculating" the (higher) derivative(s) of the inverse function (i.e., prove that $(f^{-1})^{(n)}(f(a))= \lim_{x \to a} (d/dx)^{n-1}[(x-a)/(f(x)-f(a)]$)

[@Calcprof]

Ah, playing with Gil Strang's Introduction to Applied Mathematics. I taught from this book many times in the 1990s.

[@dendarius9906]

Kepler is one of my favorite. His equations are remarkable. Dark matter seem to enforce both his law of equal distance over equal time and newton law.

[@SolarAlmanac]

Ahhh, thank you so much. If only I had known all this when I started writing my astronomy code in high school :) PS. Last year I found Kepler's Equation in a strange place… in a _gardening_ problem. Given a length of chicken wire: Form the chicken wire into a circular arc (less than 2π radians) and place the ends against a straight exterior wall of the house. How large a sector of the circle (call it 2θ) should I use to maximize the enclosed area? … Well, when θ − sin θ appeared in the derivative, I said "No no no! That's Kepler's equation and can't be solved explicitly." By this time I knew better than to try! So I graphed it, and π/2 looked good, so I plugged it in, and yada yada yada the answer is: Semicircle.

[@Altair705]

My favorite method to solve it is the Halley method. It's similar to the Newton-Raphson method but also takes into account the second derivative, which makes the convergence faster. Generally speaking, that method is not very popular because the extra calculations needed to get the second derivative tend to offset the faster convergence, but in this case the second derivative of "E-e.sin(E)" is just "e.sin(E)", so no extra calculation is needed. In the end, the Halley method appears to be more efficient in the case of the Kepler equation, which makes me wonder why it's not more popular.

[@VaraNiN]

16:52 Laplace such a good physicist he still attends lectures 4 years after his death :P

[@tune490]

This is an amazing video! Thank you Stephen :D

[@officialjfendi]

Amazing video! I'm actually taking a grad-level orbital mechanics class right now and had to implement a bunch of these algorithms that you mentioned in the video, this explains it in a really nice and conceptually palatable way so nice job :D. Such a fun topic haha

[@qadirtimerghazin]

8:57 “high curvature causes Newton's method to overshoot the correct answer” I assume this also refers to 6 iterations, as it was for the Kepler’s method? I wonder how many more NR iterations are actually needed to converge? In fact, if it’s done manually without a computer, it seems to me that it would be easier and faster to do a few more NR steps than compute all those really complicated series…

[@jmr5125]

@qadirtimerghazin Yes, Newton's method will generally converge -- sometimes it requires extra iterations. However, based on other comments in this video, there are specific enough values for which Newton's method *won't* converge, no matter how many iterations you use. When / how this occurs I do not know although I strongly suspect it has to do with eccentricity values close to (but below) 1 and a very poor initial guesses for E.

[@qadirtimerghazin]

@@jmr5125 I didn’t have time to investigate this in detail, but our future overlord GPT 4o said that “Newton-Raphson method is widely used for solving Kepler’s equation because it converges very quickly for small and moderate eccentricities. For high eccentricities and problematic initial guesses, however, alternative or hybrid approaches might be necessary to ensure reliable convergence.” As a computational chemist, I’ve dealt with many cases when the NR method frustratingly just couldn’t converge trying to find a critical point on a highly multi-dimensional molecular potential surfaces, but it’s quite interesting that even this seemingly simple function can be challenging for the NR method…

[@exponentmantissa5598]

Well done. I always like your stuff. - Retired Engineer

[@VictorOrji-mz1jw]

Is it me or does the graph at 4:58 like like a cumulative frequency curve

[@witsued]

Edmond Halley, like rally. J.K. Rowling, like bowling.

[@fg786]

5:13 Something is off with your line for M = 77,7°. It intersects the white curve at E >120° but when you zoom in the whole graph is shifting. It looks too low to be the value for E close to 90°. So it's up to you to figure out what's wrong here.

[@ivoryas1696]

10:20 That... was pretty nice! Good point! I've honestly want to someday construct a mechanical integrator for similar purposes. 😅

[@jonquil3015]

Unbelievably good presentation!

[@bobnobrain]

Yay it's finally out!!

[@souldrip2000]

at 16:47... it would have been quite difficult for Laplace to hear Cauchy's 1831 lecture at the French Academy of Sciences. (Laplace died in 1827.)

[@primenumberbuster404]

Cauchy is like that one op anime character that basically does everything with style.

[@thamiordragonheart8682]

He probably got the decade wrong while he was talking or something simple like that.

[@regulus8518]

please do a video like this on the history behind runge kutta methods ode45 etc for differential equation solution ... it was very interesting to learn context behind newton raphson and how and why it was developed, it is used in electrical engineering for something called load flow analysis and allows for things like economic dispatch and maintainence on the grid without swictching the entire thing off

[@Elektrolite111]

It would be nice to see a deep dive into the radius of convergence

[@xanterrx9741]

Great video , thanks for making it

[@michaelpieters1844]

Amazing video! Where did you get all this information? Which sources did you consult or did you go to the original works of Kepler, Newton, Laplace ... themselves?

[@pikkisir4664]

I love your videos a lot

[@amorphant]

The music really adds gravitas.

[@wjalp]

Kepler's method is very smart! 6:00 :))

[@unflexian]

zetamath video about analytic continuation is focused on the subject of the radius of convergence in the complex plane, and it's fantastic!

[@ruperterskin2117]

Cool. Thanks for sharing.

[@alliknowissuperposition2848]

Finally, after a lot of teaser 😅

[@PrimordialOracleOfManyWorlds]

fantastically fascinating.

[@Magicraft13]

Hi! Amazing video, I was wondering if an EU-shipping delivery will be available for your shop in a close future, I really want to buy some items :). Thanks for your astonishing work!

[@-Yousof-]

Great content!

[@cubing7276]

16:25 it should've been > not

[@rb8049]

A Pade expansion including both poles and zeros is more general than a Taylor series expansion.

[@mrhatman675]

It isn t really

[@FadkinsDiet]

Padé approximants are cool but not really more applicable except in special cases.

[@manhhoanguc837]

Just use Bisected Direct Quadratic Regula Falsi. For M in range 0 to π, set the search range [a,b] of E to [0,π] and the start point to M. The same apply for the range [π,2π]. This has the same quadratic convergence rate as the N-R method, but will always converge for e very close to 1.(Even if e is 0.9998 or closer depends on the floating point precision.) If you tweak somewhat, it will also works for the hyperbolic case e>1, and for very large e > 50, and M > 10π or M

[@FadkinsDiet]

That requires many more steps to get the error down to acceptable levels, even if it is theoretically quadratic in the limit

[@manhhoanguc837]

@@FadkinsDiet Nope, each step only requires one additional evaluation of the current function, without having to evaluate any of the derivatives like the harley families. For accuracy up to 12-13 decimal points, this method only requires 5-7 func evaluations.

[@RickyMud]

I swear the algo has been hyping up this equation recently, unless that you guys doing it

[@lorenwilson8128]

Hailey expanded Newton's method to use the second derivative to correct for under or overshoot. It will converge significantly faster.

[@mumujibirb]

4:10 I've seen people use a guess and check approah, i.e. they try a value and see if it's too small or large, then guess again So i think this is really just a natural extension, after all, -since sin(x) = x for small values-

[@Sanchuniathon384]

HOLD UP, where do we find how to use Bessel functions to model light diffracting through DNA strands? I am so down for this.

[@queueeeee9000]

I always thought it was supposed to be pronounced "halle" like Halle Berry? Or am i thinking of something else?

[@darkshoxx]

16:40 I'm sorry I can't see the word "Mecanique" anymore without seeing people dancing to the baseline of "another one bites the dust"

[@WelchLabsVideo]

lol - i should have used that for background music

[@Kaviranghari]

ok this is crazy i was talking about the newton raphson method all day and my teacher was and this video has popped up (by the way i was talking about it because i noticed it converges very quickly )

[@sharpfang]

I wonder, apples to apples, taking computational complexity and not mere number of iterations, how do the methods compare? Take 5 terms of Lagrange's series, it's going to be hundreds, if not thousands of additions, multiplications, exponentiations, sines and so on. The error will be something tiny like 0.001 degree, obviously vastly better than 5 iterations of Kepler's method. But now allocate the same number of operations to Kepler's method, and perform as many iterations as this allotment affords us. Somehow, I have a feeling you'll have something like a hundred iterations and the error will be vastly smaller at the end.

[@bantix9902]

Cauchy was too smart

[@RyanMarshall-lm6uf]

I solved it, E = M/(1 - e sin)

[@trippstreehouse]

god damn this was a good video.

[@kadmii]

get hype, we're doing more astronomy!

[@jemy044]

12:05 you missed a "!" in the Taylor series expansion of sin(x)

[@SolarAlmanac]

Confirmed: the 5-factorial.

[@rgerk]

Maybe the solution will come if you use rational trigonometry, changing angles for spreads and distances for quadrances, avoiding square roots and transcendental functions.

[@plinble]

Use a numerical expansion for sin and a 32 bit or similar computer? It's only a rough model anyway because all the planets and moons interact with gravity, and spacetime (gravity).

[@davidrandell2224]

“The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for proper physics including the CAUSE of gravity, electricity, magnetism, light and well.... everything. Without the cause of gravity these models will remain incomplete. Newton/Einstein etc become obsolete mathematical projections. The Geometric Orbital Equation is: v^2R=K. K is the constant for different planets/sun’s etc.R=radius; v=velocity.

[@electrikhan7190]

I guess that is why the center points of an ellipse are called foci, this is just math to bring something in to focus. Even looks like an eyeball. Maybe a phase diagram , the eccentricity is just the lag time between perfection and a perfect deflection. Relativity in geometry. Should it all be off just a little from procession and using straight lines to cut ovals.

[@matthewjames7513]

3:04 I don't think that's a parabola, I think it's a hyperbola. Typo? EDIT: e < 1 its an ellipse, e = 1 its a parabola, e > 1 it's a hyperbola.

[@sunnythebridger7529]

It's parabola when e is exactly 1, I guess.

[@mingmiao364]

There was a moment that the path is a parabola. The animation shows how a conic section evolves from ellipse to a parabola and then to a hyperbola as eccentricity increases.

[@LupisLight]

When eccentricity = exactly 1, it's a parabola. If eccentricity is greater than ,1 it's a hyperbola.

[@tmst2199]

He doesn't say that the reason for determining the area swept out in the circle is because we don't know the elepticity of the ellipse.

[@LumTheAlien]

I wonder if it is possible to plug many of these equations into a regression algorithm and arrive at a constant value that could be applied to eccentric orbits.

[@spoddie]

Will your book be available on Amazon?

[@white_145]

lmao just use the infamous carybara function: P(E-esinE) = E and there you have it

[@RyanLynch1]

2:13 hey that's my birthday!

[@BPGHchess]

Nice!

[@rodrigoappendino]

I thought I already watched this video before.

[@aleksszukovskis2074]

12:05 theres a mistake in the second denominator "5" -> "5!"

[@screechingtoad2683]

Do any of these take the gravitational pull of the sun and distance to the sun (which affects the gravitational pull)?

[@DavyCDiamondback]

It's always aggravating when classical physics problems are unsolvable, because if classical is unsolvable, what hope is there for relativistic and quantum physics?

[@wetwillyis_1881]

It’s crazy to me how similar the method of Ê is to how we use β and β hat in statistics for Regression analysis.

[@aravindakannank.s.]

i clicked faster than blinking

[@pierret6572]

What is the name of the song at the beginning of the video plz ?

[@stefanocucciati]

16:52 how could Laplace be in the audience if it was held in 1831 and he died in 1827?

[@PhoenixEditz69]

I have one doubt Are e(eccentricity) and e(euler’s number ≈ 2.71) Somehow related? If anyone has any idea, please share😀

[@WelchLabsVideo]

Nope!

[@swamihuman9395]

- Excellent explanation, including graphic illustrations. - In fact, to learn more, I screen captured illustration, then submitted to AI ('Gemini Pro 1.5') for analysis... it did a great job :)

[@mathmachine4266]

Sure I can. Numerically. Start with the initial guess of E+esin(E), then use Newton's or Halley's method.

[@Rope_Adope]

Some things you don’t “solve”. They solve you

[@srilemobitelsrile8809]

What program do you use for animation?

[@mikebon8352]

The Holy Trinity... Mass Energy Speed of Light.

[@whalep]

At 21:02, I don't understand why there is an x in Bessel's solution. Is it supposed to be e?