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Express each function in the form

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 3×2ðiyÞ þ 3xðiyÞ2 þ ðiyÞ3 ¼ x3 þ 3ix2y 3xy2 iy2¼ x3 3xy2 þ ið3x2y y3ÞThen uðx; yÞ ¼ x3 3xy2; vðx; yÞ ¼ 3x2y y3.ðbÞ w ¼11 z ¼11 ðx þ iyÞ ¼11 x iy 1 x þ iy1 x þ iy ¼1 x þ iyð1 xÞ2 þ y2Then uðx; yÞ ¼1 xð1 xÞ2 þ y2; vðx; yÞ ¼yð1 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 3yðdÞ ln z ¼ lnðeiÞ ¼ ln þ i ¼ lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2qþ i tan1 y=x

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