Answer to Question #161952 in Complex Analysis for Sai

Question

Given: 𝑧 = 6 + 7𝑖

Calculate the following:

1) 𝑧^-4

2) 𝑧^1/3, graph the roots.

3) ln 𝑧 , for 𝑛 = 1, 3, 5 & 7

4) 𝑓(𝑧) = 𝑧𝑧̅− 𝑧̅𝑖 + 5𝑖, and check your answer using 𝑧 = 𝑥 + 𝑖y

Answer 1

1) $z=6+7i\Rightarrow z^{-1}=\frac{1}{6+7i}=\frac{6-7i}{85}.$ Hence $z^{-4}=\frac{(6-7i)^4}{85^4}=\frac{-6827+i2184}{85^4}.$

2) $z=\sqrt{85}(\frac{6}{\sqrt{85}}+i\frac{7}{\sqrt{85}}).$ Hence, $z=re^{i\theta}$ where $r=\sqrt{85}, cos \ \theta=\frac{6}{\sqrt{85}}, sin \ \theta =\frac{7}{\sqrt{85}}.$ Hence $z^{1/3}=r^{1/3}e^{i\theta/3}, r^{1/3}e^{i(\theta/3+2\pi/3)}, r^{1/3}e^{i(\theta/3+4\pi/3)}.$ In other words, $z^{1/3}= \xi\ or\ \xi\omega \ or\ \xi \omega^2$ where $\xi=r^{1/3}e^{i\theta/3} .$ (Diagram is attached below) Here $\omega$ denotes complex cube root of unity.

3) $ln (z)= ln |z|+i\theta+i2\pi n= ln \sqrt{85} +i\theta+i2\pi n.$ Now we put $n=1,3,5,7.$ to get, $ln (z)= ln \sqrt{85} +i\theta+i2\pi , ln \sqrt{85} +i\theta+i6\pi n, ln \sqrt{85} +i\theta+i10\pi n, ln \sqrt{85} +i\theta+i14\pi n.$4) $f(z)=(6+7i)(6-7i)-(6-7i)i+5i=78-i.$

Verification: $z\overline{z}=(x+iy)(x-iy)=x^2+y^2.$ Also $i\overline{z}=ix-y.$ Hence, $f(z)=x^2+y^2-ix-y+5i=x^2+y^2+y+i(5-x).$ Hence $f(6+7i)=6^2+7^2-7+i(5-6)=$ $78-i.$