Question

Explaination wanted -> How can e^(-i*smallomega*t) be equal to cos(-smallomega*t) + i*sin(-smallomega*t) ? A big cookie to whomever explains it so I understand the math behind the conversion. I believe it has something to do with Euler's - however I am unable to piece it together myself :)

Answer 1

You can use taylor series

Answer 2

Ivan have you taken any Calculus?
You can generate the relationship using integration, it's pretty neat :)

Yah Taylor Series is good too.

Answer 3

Yes - I just dont recall much of Taylors though. Working on it.. !

Answer 4

\[e^t = \sum_{n=0}^{\infty} \frac{ t^n }{ n! }, t \in \mathbb{R} \] so you can add an imaginary number \[e^{i t} = \sum_{n=0}^{\infty} \frac{ (i t)^n }{ n! } = \sum_{n=0}^{\infty} \frac{ i^nt^n }{ n! }\] ooh funny thing is if you don't put space in latex between the i and what ever letter, it makes it go away, truly imaginary xD

Answer 5

Yes you will have to note A = a+bi, I think zep can explain it a better way without Taylor

Answer 6

If you have an complex number: \(\large\rm z=a+b i\)
We can write it in terms of trig, yes?

So we'll call the magnitude of this complex number \(\large\rm |z|=r\).
And we can write it in this form: \(\large\rm z=r(\cos\theta+i\sin\theta)\)

Answer 7

Differentiate this with respect to the argument, theta,\[\large\rm \frac{dz}{d \theta}=r(-\sin \theta+i \cos \theta)\]And then I guess this is where the tricky step comes in.
We're going to factor a -1 out of each term:

Answer 8

Woops, I mean factor an \(\large\rm i\) out of each term :)
\[\large\rm \frac{dz}{d \theta}=r i(i\sin \theta+\cos \theta)\]Recall that \(\large\rm -1=i\cdot i\)

Answer 9

So what we've shown is that:\[\large\rm \frac{dz}{d \theta}= i r(\color{orangered}{\cos \theta+i \sin \theta})\]Ooo look we've got our z showing up!\[\large\rm \frac{dz}{d \theta}= i r(\color{orangered}{z})\]

Answer 10

Woops woops, little mistake, my bad.
the r is part of my z

Answer 11

\[\large\rm \frac{dz}{d \theta}= i (\color{orangered}{z})\]

Answer 12

I'm still here - just processing the information and looking everything up and cross referencing in my formula book ... !

Answer 13

Then separation of variables and integrating gives us:\[\large\rm \int\limits \frac{dz}{z}=\int\limits i d\theta\]Leading to,\[\large\rm \ln z=i \theta+c\]

Answer 14

brill astro!!

historically, Euler found this from comparing Taylor series

Answer 15

Exponentiating gives us:\[\large\rm z=e^{i \theta+c}=e^{i \theta}e^c\]
\[\Large\rm z=e^c \color{orangered}{e^{i \theta}}=r(\color{orangered}{\cos \theta+i \sin \theta})\]

Answer 16

Thanks irishboy xD

Nice one zep!

Answer 17

Okay. Working on the trig method. In order to understand (and recall old lessons) I am starting from scratch given from the examples above.

Am I understanding this correctly thus far?

Answer 18

Thank you so much! I finally got it, zapdrix! Thanks to everyone else as well for your input! :)

Answer 19

If anything - the only thing throwing me a bit off is the e^c and r. From the example e^(-iw). Are those just constants we can.... throw away? ooh.. because they are the same we can just "remove" them! Right? No? @zepdrix @Astrophysics

Answer 20

Yeah it's just a constant, you can leave it alone, or just add it to your calculations after

Answer 21

Thanks

Answer 22

Yes, good eye :)
It turns out that the \(\large\rm e^c\) is the same \(\large\rm r\) that we started with.
I was having trouble illustrating that though.

Answer 23

Ah yes, good with the trig.
Sorry I kinda skipped over that at the start >.<