MAT334H1 Chapter : arguments.pdf
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For z c, we de ne the arguments of z arg(z) =(cid:8) r(cid:12)(cid:12)z = |z|ei (cid:9) . If z (cid:54)= 0, we de ne the principal argument of z, denoted. Arg(z), to be the unique ( , ] such that z = |z|ei . Suppose 1 = ei = cos( ) + i sin( ). For z1, z2 c, de ne arg(z1) + arg(z2) = { 1 + 2| 1 arg(z1) and 2 arg(z2)} . : suppose 1 arg(z1) and 2 arg(z2) so that 1 + 2 arg(z1) + arg(z2). We must show that arg(z1) + arg(z2). Case 1: suppose that z1 or z2 is zero. Without loss of generality, suppose z1 = 0 (otherwise, switch them). Then z1z2 = 0, and arg(z1z2) = arg(0) = r. but also arg(z1) + arg(z2) = { 1 + 2| 1 arg(z1) and 2 arg(z2)} = { 1 + 2| 1 r and 2 arg(z2)}
