belleamie

Hi there, I was given these proofs to do for my quantum class.
proofs are the worst for me, I know it work and i have and idea how it starts which i wrote in the image but I can't seem to figure out the inbetweens. I've attach the images, if anyone can help me that would be great! thanks

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The first one you do use Taylor expansion and make an observation about regrouping the series with the $$\imath$$ stuff. The other ones follow from this first proof.

belleamie
MalleusScientiarum said:
The first one you do use Taylor expansion and make an observation about regrouping the series with the $$\imath$$ stuff. The other ones follow from this first proof.

I'm not so sure how to regroup the i's? do you mean figure out what sin x identity is and cos x idenity and plus it in, in the 2! group and the the other for the 3! group?

beautiful1
You are on the right track with Part 1.
The result is known as Euler's formula.
Look here for more help with the proof
http://en.wikipedia.org/wiki/Euler's_formula

You are on the right track with Part 2a.
You should next use Euler's formula to write
$$e^{ia} e^{ib} = (\cos(a) + i \sin(a)) (\cos(b) + i \sin(b))$$

Part 2b) is same approach as part 2a)

I'm sorry, I don't understand part 3.

Good luck

amcavoy
You could prove #2 very easily geometrically.

belleamie
Thanks a bunch i understand #1, 2a&b now, can anyone help me on the 3rd question?

Staff Emeritus
Did you copy #3 exactly as it is written in the book? I am thinking that you did not, because the question as written implies that $z_1*z_2=z_1+z_2$, which is certainly not true.
Ir's easy enough to find out what you do get when you multiply $z_1$ and $z_2$ together. Just let $z_1=|z_1|e^{i\phi_1}$ and $z_2=|z_2|e^{i\phi_2}$ and multiply them together using the rules for multiplying exponential functions that you learned in precalculus.