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 Exercise 5.7 
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Joined: 26 Mar 2008, 17:06
Posts: 2
Post Exercise 5.7
I am puzzled by exercise 5.7. I can do the exercise easily enough, but only if I forget about the "i" in the exponent. I mean, why am I allowed to expand out e^(a+b) = e^a x e^b in real and imaginary terms (using the cos(theta) and i*sin(theta)) when there aren't any "i"s in the exponents?


31 Mar 2008, 18:14
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Joined: 19 Mar 2008, 14:09
Posts: 36
Post Re: Exercise 5.7
Yes that should probably be:

e^{i(a+b)} = e^{ia}e^{ib}


31 Mar 2008, 22:20
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Joined: 19 Mar 2008, 14:09
Posts: 36
Post Re: Exercise 5.7
Ok, so some sort of answer to this would be.

e^{i\theta}=\cos \theta+i\sin \theta

So we have:

e^{i(a+b)}=e^{ia}e^{ib}

\cos(a+b)+i\sin(a+b) = [\cos a+i \sin a][\cos b+i \sin b]

\cos(a+b)+i\sin(a+b)=\cos a  \cos b +i\cos a  \sin b +i\sin a  \cos b -\sin a  \sin b

Both real an imaginary parts of both sides must be equal and so:

\cos(a+b)=\cos a \cos b-\sin a \sin b
\sin(a+b)=\sin a \cos b+\cos a \sin b


09 Apr 2008, 23:39
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