Answer to Question #187973 in Complex Analysis for Muzala Seleji

Question #187973

Solvetheequationsbelowandhence,showthesolutionsoftheseequationsontheArganddiagram:(i)z4−i=0(ii)z3=−√3−i(iii)z5=−18i(iv)z=16i1+i18(v)z=1+i√3+i16


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Expert's answer
2021-05-07T10:09:43-0400

(i)z4i=0z4-i=0

4(x+iy)i=04x+(4y1)i=0x=0 and y=144(x+iy)-i=0\Rightarrow 4x+(4y-1)i=0\Rightarrow x=0 \text{ and }y=\dfrac{1}{4}


(ii) z3=3iz3=-\sqrt{3}-i

3(x+iy)+3+i=0(3x+3)+i(y+1)=0x=13,y=13(x+iy)+\sqrt{3}+i=0\Rightarrow (3x+\sqrt{3})+i(y+1)=0\Rightarrow x=\dfrac{-1}{\sqrt{3}}, y=-1


(iii)z5=18iz5=-18i

5(x+iy)+18i=05x+i(y+18)=0x=0,y=185(x+iy)+18i=0\Rightarrow 5x+i(y+18)=0\Rightarrow x=0,y=-18


(iv) z=16i+18iz=16i+18i

x+iy=34ix=0,y=34x+iy=34i\Rightarrow x=0,y=34


(v) z=1+i3+16iz=1+i\sqrt{3}+16i

x+iy=1+i(3+16)x=1,y=3+16x+iy=1+i(\sqrt{3}+16)\Rightarrow x=1,y=\sqrt{3}+16


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