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The principal value of argz is the unique choice of

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Unformatted text preview: z= ø= since the a rctan function takes values between -π/2 and π/2. The principal value of arg(z) is the unique choice of ø such that -π < ø ≤ π. We write this as Arg(z) -π < Arg(z) ≤ π 3 -π ≤ Arg( Examples (a) Express z = 1+i in polar form, using the principal value (b) Same for 3 + 3 3 i (c) 6 = 6(cos 0 + i sin 0) 6. Multiplication in Polar Coordinates If z1 = r1(cosø1 + i sinø1) and z2 = r2(cosø2 + i sinø2), then z1z2 = r1r2(cosø1 + i sinø1)(cosø2 + i sinø2) = r1r2[(cosø1cosø2 - sinø1sinø2) + i (sinø1cosø2 + cosø1sinø2). z1z2 = r1r2[[ cos(ø1+ø2) + i sin(ø1+ø2)] Thus That is, we multiply the magnitudes and add the arguments. Examples In class. 7. Multiplicative Inverses in Polar Coordinates Once we know how to do multiplication, division follows formally: Let z = r(cosø + isinø) be given. We want to find z -1. So let z -1 = s(cos˙ + isin˙). Then, since zz -1 = 1, we have rs(cosø + isinø)(cos˙ + isin˙) = 1 ie., rs(cos(ø+˙) + isin(ø+˙)) = 1 = 1(cos0 + isin0). Thus, we can take s = 1/r and ˙ = -ø. In other words, z -1 = --r -1(cos(-ø) + i sin(-ø)) Examples In class. 8. Divisio...
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