Euler's Formula through Picard–Lindelöf (Mathvengers)

Something I covered on the channel already, but here's my clip from Mathvengers: Eulergame. We prove Euler's formula by using the uniqueness of solutions of differential equations.

(as well as previews, behind-the-scenes, and more content).

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Music by Sakis Strigas:
(Eulogy originally by Tool)

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[EpicMathTime]

Many uploads to come. This video is representative of what future videos will be like. 👍

[nikiangelov]

Bruh that version of Eulogy SLAPS my man! Very nice!

[AndresFirte]

This style of editing is so cool! I love that the tiny boxes give some formal extra information, it allows you to keep the relaxed pace of the video without giving up the formal aspects.

Also I like your “cool” attitude, it looks natural and not cringy.

I found you thanks to the mathvangers video, it’s great to see that there are so many skilled math youtubers!

[thesecretguy]

We actually got taught this method instead of the Taylor series one when we learnt it, although a slightly different way.

Clearly any complex number z can be written in terms of the angle and modulus as well as coordinates, so we can say z=cos(a)+isin(a) (because there's no theta on my keyboard). Now we ask, what happens if we differentiate this? Does it even mean anything?

And that then gets you to this result, because the differential equation can be solved to give e^(ia).

[alkankondo]

What a huge jump in video production value!! Great work!

[clivenazareth]

Hey, I couldn't watch every video, the editing quality has gone up over the past couple months. Great job and happy holidays

[Assault_Butter_Knife]

Easily the best segment of Mathvengers. The card at the end is just the cherry on top.
Merry Christmas to you my man!!

[frozenmoon]

The oily-vengers video was amazing and this one was a great part of it.

[SakisStrigas]

Congratulations man! Amazing work! The sky is the limit!

[fermibubbles]

the way u transmogrify equations <3 <3 bet u hit 1 mil before my dog dotson hits 500k

[oo]

Another way to do this is taking the second derivative & getting y”= (i^2) exp(ix) = -exp(x), which makes the connection to cosines obvious

[ivanjorromedina]

Is complex differentiation, the same as real differentation? And did you check the sufficient conditions of the Picard-Lindëlof theorem?
Pd: Great video!

[AndrewKln]

I'm estimating in 1 year I should be able to understand calculus videos. Looking forward towards the day when I solve my first derivative.

[goodhuman]

how do you write stuff facing front, shouldnt it be inverted for you

[ventrue]

"named after some dead guys" HAHAHAHAHA

[alexandersanchez]

Nice! I still think integrating factors from the differential equation induced by f(x) = cos(x) + isin(x) is the more natural derivation. But, the differential equation approach is certainly the most natural, and it's always fun to fire bazookas at insects!

(Assume (e^ix)' = i*e^ix)
f(x) = cos(x) + isin(x)
f'(x) = i*f(x)
(e^(-ix)f(x))' = 0
f(x) = Ce^ix
C = f(0) = cos(0) + isin(0) = 1
Thus, cos(x)+isin(x) = e^ix. QED
(Just to demonstrate that no heavy artillery is required, except perhaps that a function's derivative is identically zero iff it's constant.)

[headlibrarian]

For completeness, shouldn't the continuity requirements of Picard-Lindelof be demonstrated?

[Kolinnor]

Why would the chain rule be in the definition of the complex exponential ? It doesn't feel that natural to me
Otherwise great argument, clearly more understandable than Taylor for high school students !

[MrCmon]

When you pull out your functional analysis for your son's homework.

[yogeshwagh]

Hey Epic math time, do you really write backwards?