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Express each function in the form ux; y ivx; y, where u and v are real: (a) z3; b 1=1 z; c e3z; d ln z.
Express each function in the form uðx; yÞ þ ivðx; yÞ, where u and v are real:
(a) z3; ðbÞ 1=ð1 zÞ; ðcÞ e3z; ðdÞ ln z.
ðaÞ w ¼ z3 ¼ ðx þ iyÞ3 ¼ x3 þ 3x2ðiyÞ þ 3xðiyÞ2 þ ðiyÞ3 ¼ x3 þ 3ix2y 3xy2 iy2
¼ x3 3xy2 þ ið3x2y y3Þ
Then uðx; yÞ ¼ x3 3xy2; vðx; yÞ ¼ 3x2y y3.
ðbÞ w ¼
1
1 z ¼
1
1 ðx þ iyÞ ¼
1
1 x iy
1 x þ iy
1 x þ iy ¼
1 x þ iy
ð1 xÞ2 þ y2
Then uðx; yÞ ¼
1 x
ð1 xÞ2 þ y2
; vðx; yÞ ¼
y
ð1 xÞ2 þ y2
:
ðcÞ e3z ¼ e3ðxþiyÞ ¼ e3x e3iy ¼ e3xðcos 3y þ i sin 3yÞ and u ¼ e3x cos 3y; v ¼ e3x sin 3y
ðdÞ ln z ¼ lnðeiÞ ¼ ln þ i ¼ ln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2 þ y2
q
þ i tan1 y=x