Answer to Question #91339 in Complex Analysis for Ra

Question #91339
Describe the geometric, polar and exponential representations of:
(\bar -5i+2)⁻¹
1
Expert's answer
2019-07-05T08:54:14-0400

Let's find a conjugate to -5i+2


"\\overline{-5i+2}=2+5i"

let's find the rectangular representation of this number:


"z=( 2+5i )^{-1}= \\frac{1}{2+5i}= \\frac{2-5i}{(2+5i)(2-5i)}= \\frac{2-5i}{29}"

Real part


"Re(z)=\\frac{2}{29}"

Imaginary part


"Im(z)=-\\frac{5}{29}"

Modulus of z


"|z|=\\sqrt{\\frac{4}{29^2}+\\frac{25}{29^2}}=\\frac{\\sqrt{29}}{29}"

Re(z)>0 and Im(z)<0 hence


"arg(z)=\\theta=2\\pi-arctan \\frac{|Im(z)|}{Re(z)}\n\u200b\t =2\\pi-arctan \\frac{\\frac{5}{29}}{\\frac{2}{29}}=\n2\\pi-arctan\\frac{5}{2}"

Therefore the polar representation of z will be


"z=\\frac{\\sqrt{29}}{29}\\left(cos\\left(2\\pi-arctan\\frac{5}{2}\\right )+isin\\left(2\\pi-arctan\\frac{5}{2}\\right )\\right)"

Exponential representation of z


"z=\\frac{\\sqrt{29}}{29}e^{i\\left(2\\pi-arctan\\frac{5}{2}\\right )}"

Geometric representation





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