The Subtle Reason Taylor Series Work | Smooth vs. Analytic Functions

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• MO

  @morphocular · 2 years ago · pinned

  Hey all. Just a few clarifications I&apos;d like to make in response to comments I&apos;ve seen. It seems I&apos;ve had to do this a lot lately, huh? Nothing gets past you guys :)<br><br><a href="https://www.youtube.com/watch?v=0HaBNdmUWXY&amp;t=424">7:04</a> - Many have taken issue with the piecewise function g(x) being a satisfying example of a function that fails to equal its Taylor series because it&apos;s a &quot;gluing&quot; of two completely different functions, so it&apos;s natural to expect its Taylor series to behave incorrectly at the join. But the critical trait with this particular piecewise function that makes it different from most others is that the join is truly &quot;seamless&quot;: despite being a &quot;gluing&quot; of two &quot;different&quot; functions, it&apos;s perfectly smooth (i.e. has derivatives of all orders) at the join point, which is not usually true of most piecewise functions you could construct. Because certainly if a function fails to be smooth at a point, its Taylor series will break there.<br><br><a href="https://www.youtube.com/watch?v=0HaBNdmUWXY&amp;t=5">0:05</a> - I was aware that many calculators do not actually employ Taylor series directly to compute sin, cos, and e^x. My intent there was just to make it as clear as possible that I&apos;m talking about computing these functions at a truly arbitrary input (like a calculator can). In my defense, that&apos;s why I used the word &quot;might&quot; in that line, but I was probably asking that word to do too much work, so if I could go back, I&apos;d rewrite that line. Apologies for any confusion!

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• MA

  @maxwellhunt3732 · 2 years ago

  I love Taylor&apos;s Theorem. &nbsp;It&apos;s one of those results that is so incredibly important, but is not at all obvious at first sight.

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• OW

  @OwlonH.Christ · 2 years ago

  It&apos;s honestly wild how well he can explain these things using the visuals.

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• BI

  @billcook4768 · 2 years ago

  The crazy thing about analytic functions is that if you know everything that is going on in a “small” region around a point, you understand the entire function.

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• JA

  @jacob_90s · 2 years ago

  I know this wasn&apos;t the primary point of the video, but I just wanted to note this because it&apos;s something I was very interested in when I first started programming, and I had a hard time learning this because every first year calc student would just copy and paste the same damn explanation about taylor series in every online forum.. &nbsp;Most programming math libraries DO NOT use infinite series of continued fractions to calculate elementary functions (the exceptions generally being arbitrary precision libraries) &nbsp;The issue with them is that they are in general too slow, and often times require the intermediate calculations to be computed at a greater precision than the final results needs to be. &nbsp;<br><br>Instead, when writing the math library, the developers will curve fit either a polynomial or rational function, which can compute the function within a certain range to the required level of precision. &nbsp;Additionally, identities are often used to reduce the input to a smaller range so that you don&apos;t have to try and compute the values for all possible floating point values; the trig functions are probably the best example of this; sin and cos are defined for all x values from -infinity to +infinity, but since it just repeats, you can reduce the input value into the range -2pi to +2pi (depending upon the library sometimes it will be reduced even further). &nbsp;Similar tricks can be used for exponential and logarithmic functions using the layout of floating point numbers.<br><br>For anyone who wants to read up more on this, I would suggest <br>* Approximations for Digital Computers by Hastings (1955)<br>* Computer Approximations by Hart (1968)

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• AN

  @angelofdeth94 · 2 years ago

  One interesting thing about analytic functions is they behave more like &quot;infinite degree polynomials&quot; than a general smooth function. Polynomials are very rigid. If you know the value at n+1 points of a degree-n polynomial, then you know the whole polynomial. So even though it might seem like you could express a lot of different shapes with a degree-4 polynomial, it only takes 5 points to completely pin it down. There&apos;s a theorem in complex analysis that says if you know the value of an analytic function at a sequence of points and at a limit point of the sequence, then &nbsp;you know the analytic function everywhere. For example, if you know the values at 1/n for every natural number n, and at 0, then you uniquely determine the analytic function. In retrospect, it&apos;s kind of &quot;obvious&quot; that analytic functions would act like infinite-degree polynomials, because that&apos;s basically what a power series is.

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• MI

  @MisterTutor2010 · 2 years ago

  If anyone tells you that math is boring, just shake it off.

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• AS

  @asgfudr4954 · 1 year ago

  <a href="https://www.youtube.com/watch?v=0HaBNdmUWXY&amp;t=167">2:47</a> the Taylor series is slapping the x axis

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• KI

  @kingbeauregard · 2 years ago

  Taylor Expansions are great. For concept, I recommend this: a given function f(x) is actually built out of a bunch of polynomial terms (ax, bx^2, cx^3, etc) but it does not readily admit to what the coefficients a, b, c, etc are for the various terms. So we need to torture the function into confessing each coefficient. The method of torture that works is taking the derivative the appropriate number of times for a given polynomial term, and then setting x equal to zero. It&apos;s brutal and harrowing work, but it&apos;s also brutally efficient.

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• CA

  @calmkat9032 · 2 years ago

  This is my #1 favorite subject! All of calculus feels like a narrative, but none moreso than Taylor series. The way it starts with something plain with approximating functions, to turning irrational, even transcendental, functions into these weird work-around ratios, it&apos;s just such a cool story! &nbsp;<br><br>It even ends what I consider a years-long story arc in math. &nbsp;Since algebra 1, we learned about functions. &nbsp;Then we steered into the seemingly unrelated geometry. &nbsp;Then we alternate with algebra 2 and trigonometry. &nbsp;And it all comes together here at the end of calculus 2, when you turn sin(x) and cos(x) into plain algebra, and vice versa. &nbsp;<br><br>And as a bonus, you learn that the taylor series of cos(x) + i*sin(x) is the same as e^x. &nbsp;Meaning trigonometry, algebra, and calculus all meet here. &nbsp;Add the cornerstone of geometry, pi, by making x=(i*pi), and boom. &nbsp;The one and only e^(i*pi)=-1

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• FR

  @FredericoKlein · 2 years ago

  i remember getting really spooked about this in college thinking that if you could know every derivative of a path, that you could calculate it in the future and how future information would be somehow hidden in higher order derivatives, I kinda forgot about this, but i think it has to do with my incomplete understanding of the taylor theorem and the limitations of taylor explansions

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• TO

  @tomkerruish2982 · 2 years ago

  I&apos;d speculate that the reason analytic functions show up so much is because we&apos;re using them to solve (relatively simple) differential equations.

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