Euler's formula

>>9159987
You're discounting the imaginary value.
cos(2*pi)+i*sin(2*pi) =/= 1.

>>9159987
Oh man haha great thread. I mean, obviously no complex analysis textbook has a chapter on the complex logarithm and how it actually works. Impossible. It just isn't known! AT ALL! Please post your address so that we can mail you the nobel prize and the fields medal because you are actually the first human being in the universe to realize that the complex logarithm doesn't work like the real logarithm WOW. You must feel real good about yourself :)

https://en.wikipedia.org/wiki/Complex_logarithm#Problems_with_inverting_the_complex_exponential_function

>>9160004
Yes it does

>>9160004
No, I'm not.
[math] \sin 2 \pi = 0 [/math]
Therefore: [math] \sin 2 \pi i = 0 [/math]
Or is that wrong?

>>9159987
the "natural logarithm" function is not injective in the complex field, it's actually periodic with the period [math]2\pi i[/math]. So whenever you have
[eqn]e^{y} = e^{x}[/eqn]with [math]x,y \in \mathbb{C}[/math], you can only be sure that
[eqn]y = x + 2k\pi i, k \in \mathbb{Z}[/eqn]

>>9160016
"natural logarithm" function should be exponential function. my bad

>>9159987
WHICH BRANCH OF THE LOGARITHM OP

>>9160016
Not sure I get it, could you phrase it in a way such that someone with no background in real or complex analysis could understand?

>>9160027
In the real numbers log is one to one dog, in the complex number log is periodic. you acting like the sin of 4 Pi equaling the sin of 2 Pi is evidence that 4 equal 2.

>>9160027
Let me prove that [math]\pi = 0[/math] without using complex numbers. Let [math]x = \pi[/math]. We know that [math]\sin x = \sin \pi = 0[/math]. But it's also true that [math]\sin 0 = 0[/math]. We obtain the following equation:
[eqn]\sin x = \sin 0[/eqn]Taking [math]\arcsin[/math] of both sides, we conclude [math]x = 0[/math] and so [math]\pi = 0[/math].

Do you know where I cheated ?

>>9160037
Bloody hell, I think I get it. Thanks, mate.

>>9160012

[math] \displaystyle
sin( 2 \pi) i \neq sin(2 \pi i)
[/math]

>>9160006
Jesus Christ I thought /lit/ was toxic. You fuckers are sad.

>>9160063
FedEx just called us and they said that "Jesus Christ I thought /lit/ was toxic. You fuckers are sad." isn't a valid address so they are sending the fields medal we sent you back. Please answer with a real address because shipping through FedEx costs a bit so if you give us a fake address a second time I am afraid you will have to come and pick your fields medal in person.

>>9160037
>Ackhchually

pi = -0

cute proof of euler's formula

[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]

>>9160253
holy shit this good