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Unformatted text preview: lVcV> <V AW " © V Mz V v •. ^ Xtr C i r * r f o ^ <s > o ^< \^ r "  '~ J <^>l v •? yP u; lf ? t>\ o " £ Sk c icr x (/> •T \4 H • r M M C \ — O — O U O]D u r ^ i \ 2 T .5  ^ O T ^I *\ *r* \ T _ = t ^ C= Q  ^ V ^ ^K h ' C D d >i\^ tM, PC,^_J P VY\ V t :.>vu\ OV U U 3 42, • V M I f ^ H \ UTV J J GA •2.V <>o Y\ & %l v> & T L.OV M ^ fi'A £ _— __— . * —— — ^ _3 o *KA , ex Cn "I T 3 i H T '' VvX^\ S4>n^jp APPE&DIX ALPHA: Determining the n th root of J Finding the n th root of a negative number requires that we recall how to represent it as a complex number in both polar and exponential form. What I mean is that a complex number given in rectangular form bi a + can be represented in either polar or exponential form via the following: [ ] θ θ θ i re i r bi a = + = + sin cos where 2 2 b a r + = . (1) We can think of J − as a complex number using Eq. (1) by nothing that J a − = and = b , which upon substitution into Eq. (1) yields: [ ] θ θ θ i Je i J J = + = − sin cos . (2) The most obvious way for the first equality to be true is that the polar angle should equal to π, leading to [ ] π π π i...
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 Spring '10
 L.ARCHER
 Complex number, ax sin bx, ax cos bx, R.C. Elementary Differential

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