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QUESTION

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

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