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30 Nov 2021
@_discord_336567624982986752:t2bot.iomilia oh boy 01:01:04
@_discord_336567624982986752:t2bot.iomilia from one awkward moment to the next 01:01:11
@_discord_559418497734803470:t2bot.ioISO 4683-1#3216 uh-oh 01:01:11
@_discord_559418497734803470:t2bot.ioISO 4683-1#3216 lol 01:01:14
@_discord_336567624982986752:t2bot.iomilia sry 😐 01:01:14
@_discord_559418497734803470:t2bot.ioISO 4683-1#3216 you're fine 01:01:18
@_discord_336567624982986752:t2bot.iomilia 🙂 01:01:23
@_discord_336567624982986752:t2bot.iomilia so, this is due to E. Landau 01:01:29
@_discord_336567624982986752:t2bot.iomilia 1 = cos(0) = cos(x-x) = cos(x)*cos(x) + sin(x)*sin(x) 01:01:51
@_discord_613847106993782835:t2bot.ioGarklein#9297 you can escape *s with \ 01:02:05
@_discord_336567624982986752:t2bot.iomilia * 1 = cos(0) = cos(x-x) = cos(x)*cos(x) + sin(x)*sin(x) 01:02:20
@_discord_336567624982986752:t2bot.iomilia tx 🙂 01:02:24
@_discord_559418497734803470:t2bot.ioISO 4683-1#3216 neat 01:02:57
@_discord_336567624982986752:t2bot.iomilia 🙂 01:03:03
@_discord_613847106993782835:t2bot.ioGarklein#9297 wait how do you get that last part 01:03:15
@_discord_336567624982986752:t2bot.iomilia you'll find it in his integral + differential calculus book 01:03:21
@_discord_559418497734803470:t2bot.ioISO 4683-1#3216 cos(a-b) = cos(a)*cos(b) + sin(b)*sin(a) 01:03:40
@_discord_336567624982986752:t2bot.iomilia it's the trig identity cos(a-b) = cosa * cosb + sina * sinb 01:03:42
@_discord_336567624982986752:t2bot.iomilia yeh 01:03:47
@_discord_613847106993782835:t2bot.ioGarklein#9297 oh 01:03:55
@_discord_336567624982986752:t2bot.iomilia what blew my mind was the cos(0) = cos(x-x) part 01:04:19
@_discord_336567624982986752:t2bot.iomilia pretty creative moment 01:04:26
@_discord_336567624982986752:t2bot.iomilia creating a solution out of zero 01:04:33
@_discord_559418497734803470:t2bot.ioISO 4683-1#3216 mhm 01:04:51
@mlochbaum:matrix.orgMarshallFollows pretty naturally from the complex unit circle: conjugate of exp(ix) is exp(-ix), so magnitude-squared of exp(ix) is exp(ix)exp(-ix) or exp(i(x-x)).01:10:31

Huh, the typical way you'd prove exp(ix) has magnitude 1 for real x is to take the derivative of exp(ix)exp(-ix), knowing that exp(z) is its own derivative by definition (you also need exp(0)=1). Same thing works for cos(x)cos(x) + sin(x)sin(x), since the cos term gives you the negative of the sin one.

@_discord_336567624982986752:t2bot.iomilia oh yes, I assume you've used the euler formula at some point ? 01:52:11
@mlochbaum:matrix.orgMarshallI take that to be the definition of sine and cosine, so yes.02:07:45
@_discord_336567624982986752:t2bot.iomilia Nice. I think in the case of Landau he took the taylor expansions as the definitions of the sine and cosine functions. 02:37:13

Makes sense. I think proving existence of a solution to the exp(z) differential equation might have been complicated to do rigorously at the time? Then again I don't remember how to prove that at all.


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