#161

This video is about how the most beautiful formula in mathematics works: the Euler formula e ^ (iπ) + 1 = 0.
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Hello! In this video, within the school curriculum we will try to figure out what is the decomposition of a function into a Taylor series (Maclaurin series) on the example of an exponent, look at the graphic connection of functions and power series. Well, in the final stage we shall deal with the well-known Euler identity, which many mathematicians recognize as the most beautiful of all.
In the course of the video, many different theorems from the course of mathematical analysis are mentioned; if you have a desire to deal with the strict proof of the statements used, you can refer to the book by V.А. Zorich on mathematical analysis. If you like math - be sure to subscribe to this channel: there is something to watch!
Hoping to see more viewers who understand the content of the video, I summarize and retell it with the text.
ESSENTIAL IN BRIEF.
We are trying to understand how the Taylor formula (its special case is the Maclaurin formula) works on the example of the function f (x) = e ^ x: the sense is that many functions, the exponent in particular, can be represented in another, more convenient in some problems , form - using a power series. Further, working in this convenient form, we make several simple transformations and prove the validity of the equality e ^ (iπ) + 1 = 0.
SPECIFIC STEPS.
1. We were personally convinced of the existence of such polynomials, the graphs of functions of which can be as much as the graphs of the functions e ^ x, sinx and cosx [0:01].
2. Saw formulas that allow you to get such magic polynomials [1:24].
3. We try to deal with these formulas using the exponential example: we restricted ourselves to finding the first five derivatives for f (x) = e ^ x and for g (x) = a + bx + cx² + .... Differentiate f (x) - times , then the resulting function again, then another, more and more ..., the same thing with g (x) - we successively find the derivatives [2:37].
4. Found the values ​​of all these derivatives and the functions themselves at the point x = 0: substituted instead of "X" a zero in the function f (x) and g (x) [3:00].
5. Equated the found values ​​(3rd and 5th columns), thereby finding the values ​​of the unknown coefficients a, b, c, etc. [3:17].
6. Summarizing the whole thing, we got the expansion of e ^ x in a series, which is called the Maclaurin series. You can even put emphasis on "e", I’m not offended, the main thing is to realize the promise: if functions, to put it simply, are the same, then they cannot have different values ​​of derivatives - they should also be the same [4:27].
7. With the help of the same Maclaurin formula, one can obtain decompositions for sinx and cosx - I propose to do this as an exercise. I show the result at the time of [4:49].
8. All three presented expansions of the functions e ^ x, sinx, cosx are true for complex arguments [5:09]. Why - this is a separate story, but about complex numbers something told here: • Video
9. Instead of z, we took iy for the function e ^ z: since iy is also some complex argument, the formulas (or rather definitions) for our functions still work [5:18].
10. Grouped the terms, and it turned out that the series for the exponent of the argument iy contains decompositions for sine and cosines - got the identity e ^ (iy) = cozy + isiny [5:40]. There are small slips in the frame - the squares of the players have been missed, fixed it here: vk.cc/8mH0xI
11. They took y = π, they remembered that cosπ = -1, sinπ = 0. Hence, e ^ (iπ) + 1 = 0, pm, etc. [5:54].
Then there were jokes about an empty wallet and other things. Happy end!
Mathematics # Matan # Euler