Question

find the invariant points of the transformation f(z)=1/z-2i

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Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{f(z)=\dfrac{1}{z-2i}}$

$\underline{\textbf{To find:}}$

$\textsf{Invariant points of f(z)}$

$\underline{\textbf{Solution:}}$

$\textsf{The invariant points are obtained by solving}$

$\textsf{the equation f(z)=z}$

$\mathsf{Consider,}$

$\mathsf{f(z)=z}$

$\mathsf{\dfrac{1}{z-2i}=z}$

$\mathsf{1=z(z-2i)}$

$\mathsf{1=z^2-2i\,z}$

$\textsf{Rearranging terms, we get}$

$\mathsf{z^2-2i\,z-1=0}$

$\mathsf{z^2-2i\,z+i^2=0}$

$\mathsf{(z-i)^2=0}$

$\implies\mathsf{z=i}$

$\therefore\textbf{The invariant point is i}$

Answer 2

Solution:

The invariant points are obtained by solving

the equation f(z)=z

Consider,

f(z) = z

1 Z N 21 Z

1 = z(z = 2i)

1 = z² - 2iz

Rearranging terms, we get

z² - 2iz-1=0