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MAT334H1
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# arguments.pdf

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University of Toronto St. George

Mathematics

MAT334H1

Jaimal Thind

Winter

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Arguments
De▯nition 1. For z 2 C, we de▯ne the arguments of z
▯ ▯
arg(z) = ▯ 2 R z = jzje i▯ :
Note that arg(0) = R. If z 6= 0, we de▯ne the principal argument of z, denoted
i▯
Arg(z), to be the unique ▯ 2 (▯▯;▯] such that z = jzje .
Lemma 2. arg(1) = f2▯kjk 2 Zg.
Proof. Suppose 1 = e = cos(▯) + isin(▯). Then ▯ is a multiple of 2▯.
Theorem 3. For z ;z 2 C1 de2ne
arg(z 1 + arg(z ) 2 f▯ + ▯1j▯ 2 2rg1z ) and ▯ 1 arg(z )g2 2
Then arg(z z )1 2arg(z ) + a1g(z ). 2
Proof. We need to show two containments.
▯: Suppose ▯ 2 1rg(z ) and1▯ 2 arg(2 ) so that2▯ + ▯ 2 arg(z1) + 2rg(z ). Th1n 2
i▯1 i▯2 i(1 +▯2)
z1 2= (jz j1 )(jz2je ) = jz 1 2e ;
so ▯ + ▯ 2 arg(z z ).
1 2 1 2
▯: Suppose that ▯ 2 arg(z z ). 1 2must show that ▯ 2 arg(z ) + arg(z ). 1 2
Case 1: Suppose that z or z 1s zero2 Without loss of generality, suppose z = 0 1
(otherwise, switch them). Then z z = 0, 1 2 arg(z z ) = arg(0)1 2R. But a

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