Anonymous at Sat, 15 Mar 2025 09:32:07 UTC No. 16619587

[math] \displaystyle
f(x) = e^{-ix}(\cos x + i \sin x)
\\
f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - i e^{-i x}(\cos x + i \sin x)
\\
f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - e^{-i x}(i \cos x + i^2 \sin x) \equiv 0
\\
f^{\prime}(x) = 0 \;\;\; \forall \; x \in \mathbb{R}\Rightarrow f(x) \text{ is a constant}
\\
f(0) = e^{0}(\cos 0 + i \sin 0) = 1 \cdot(1+0) = 1 \Rightarrow f(x) = 1 \;\;\; \forall \; x \in \mathbb{R}
\\ \\
1 = e^{-ix}(\cos x + i \sin x) \Rightarrow e^{ix}=\cos x + i \sin x \;\;\; \forall \; x \in \mathbb{R}
[/math]

Anonymous at Sat, 15 Mar 2025 11:14:28 UTC No. 16619649

>>16619572
>the most profound thing that I have ever seen
>I have no idea what it means
>pretty sure no one else does
the holy trinity of pseud retard

Anonymous at Sat, 15 Mar 2025 12:11:54 UTC No. 16619690

>>16619572
This one is really cool too.

Anonymous at Sat, 15 Mar 2025 15:26:09 UTC No. 16619771

>>16619572
weak bait

Anonymous at Sat, 15 Mar 2025 17:03:05 UTC No. 16619830

>>16619587
thanks anon. never seen this one before

Anonymous at Sat, 15 Mar 2025 17:12:59 UTC No. 16619845

>>16619572
Is one of those things any smart kid should realize on his own just by knowing Algebra, the Taylor series of e, cos, sin, and the definition of i, otherwise he has no hope to become a great mathematician.

Anonymous at Sat, 15 Mar 2025 17:51:30 UTC No. 16619899

>>16619572
It means that the analytic continuation of exponentiation reproduces circular motion, which you should be able to immediately understand by multiplying two complex numbers on the unit circle.

Anonymous at Sat, 15 Mar 2025 18:01:43 UTC No. 16619911

Just look at the power series interpretations of e^x and sin(x). Both functions are defined by infinite power series. You *never* know exactly the value of e^x and sin(x) because their functions are infinite algorithms.

Let's talk about something else
[eqn] \sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5!} - \frac{x^7}{7!} + \ldots [/eqn]

Do you intuitively understand why this infinite series oscillates between 0 and 1 as x grows bigger and bigger? If not then you are not ready to intuit Euler's equation.

Anonymous at Sun, 16 Mar 2025 05:45:07 UTC No. 16620465

>>16619572
Doesn't mean shit, it's basically just true by the definition of logarithms and trig functions. Pseuds worship it because it's small and easy to remember and it has a lot of symbols that normies don't understand so it makes them look smart when they talk about it. If you really want to be generous you could say it's a (terrible) way to exhibit that logarithms and trig functions are basically the same thing when you consider everything as functions of complex variables.

f(x) = log(x) is by definition the function such that f'(x) = 1/x, f(1) = 0
f(x) = e^x is by definition the inverse of the logarithm

(cos(x) + isin(x))' = i (cos(x) + isin(x))
ix = log(cos(x) + i sin(x))
e^ix = cos(x) + i sin(x)

Anonymous at Sun, 16 Mar 2025 06:36:10 UTC No. 16620506

>>16619911
The larger x gets, another term becomes "relevant" . What I mean is comparable in magnitude to the sum of smaller terms, while the larger terms are still dominated by the factorial. Not op btw

Anonymous at Sun, 16 Mar 2025 06:51:43 UTC No. 16620515

What's special about this? This is literally wikipedia knowledge. As long as you know the Taylor series expansion of sin(x), cos(x) and e^x

Then you simply substitute x = pi.

Anonymous at Sun, 16 Mar 2025 09:11:19 UTC No. 16620568

>rotating a point by 180° in 2D is the most profound thing ever
how many times have I seen this thread before?

Anonymous at Sun, 16 Mar 2025 09:41:19 UTC No. 16620578

>>16619572
Ah sweet, another complex numbers thread on /sci/!

>Yes, we made them up
>No, they don't exist
>Yes, they are useful
>The math just werks

Anonymous at Tue, 18 Mar 2025 23:55:37 UTC No. 16623116

>>16619572

I felt this way after taking calculus in high school. What you wrote is called "Euler's Equation" and it is a pretty nifty equation. It's wild that it contains the 5 of the most important numbers. I still prefer the more descriptive form of the equation:

[eqn] e^{i\pi} = -1 [/eqn]

which sort of means [math] e^i [/math] is just the number 1 in the rotated counter-clockwise around the origin in the complex plane by 1 radian [math] =180/\pi \approx 57.9 [/math] degrees, and if you rotate 1 by [math] \pi [/math] times that angle, you end up rotated by a total of [math] \pi [/math] radians = 180 degrees around the origin to the number -1.

This kind of thinking is easier to think about by using the more general form of Euler's equation, called "Euler's formula" which is just

[eqn] e^{i \theta } = \cos (\theta) + i \sin (\theta) [/eqn]

And if you didn't know that's the exact formula for the complex number you get when you rotate the number 1 counter clockwise around the origin by [math] \theta [/math] radians or [math] \frac {180}{\pi} \theta [/math] degrees. Not only can you get Euler's equation by just plugging in [math] \pi [/math] into Euler's formula. You can also use this equation to compute fourier transforms, do otherwise really complicated integrals pretty simply, and a bunch of other stuff. In fact it's so useful and I saw it so often in my math degree, that I sort of grew numb to how beautiful it is. Thank you for reminding me, OP!

Anonymous at Thu, 20 Mar 2025 18:17:47 UTC No. 16624391

>>16619649
lol true

Anonymous at Thu, 20 Mar 2025 18:19:24 UTC No. 16624392

>>16619572
whenever they right it as +1=0 as opposed to just =-1 you know it's either a LE EPIC SCIENCE youtuber or a retard who's been farmed by the former

Anonymous at Thu, 20 Mar 2025 18:46:46 UTC No. 16624413

e^pi*i = e^-p = 1/e^p = -1
=> e^p = -1

Anonymous at Thu, 20 Mar 2025 20:42:51 UTC No. 16624490

>>16619572
0.9999... != 1

Anonymous at Fri, 21 Mar 2025 11:55:47 UTC No. 16624908

>>16624490
i mean, (0.9...)! equaling 1 is fairly trivial

Anonymous at Fri, 21 Mar 2025 13:16:43 UTC No. 16624946

Should I chat gpt this or ima naturally deep seek it as an American.
0.5^3.14*-1=0
What do you mean I doubt this is correct.

Anonymous at Fri, 21 Mar 2025 15:49:43 UTC No. 16625086

Bro did you know bro that Mexican chicks are real cause she has a belly button. I’m like really absorbing this and I’m confused why I speak Spanish.
I be cat callin niggas so it’s all backwards. You are smart bro. You are like really rewarding I’m just tryna not be racist. And yours is really cool to me. 2pbtid

Anonymous at Fri, 21 Mar 2025 18:38:05 UTC No. 16625236

>>16619587
Why do he be saying the same shit 2ice

Anonymous at Fri, 21 Mar 2025 23:50:08 UTC No. 16625449

>>16624908
>equaling
wrong
they share an equivalence class after you define limits of infinite series