Mobius transformation :
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and gives thatMATH Consequently, $z_{1} = z_{2}$. Therefore, the function $T$ is a one-to-one function in MATH.

Let $w$ be an arbitrary complex number such that $w \neq a / c$. Then we want to find a complex number $z$ such that $T(z) = w$. If MATH then it gives that MATH. Consequently, the function $T$ is onto. Therefore, the Mobius transformation $T(z)$ is a one-to-one function from MATH onto MATH.
Suppose that $c \neq 0$ in the expression of MATH. Observe that the Mobius transformation $T$ has a simple pole at $-d / c$ and MATH. Let us define $T(-d/c) = \infty$ and $T(\infty) = a /c$, if $c \neq 0$. If $c =0$ then define $T(\infty) = \infty$. Then, $T$ is a one-to-one function from the extended complex plane MATH onto the extended complex plane MATH and hence invertible. The inverse transformation $T^{-1}$ of the Mobius transformation MATH is given by MATH which is also a Mobius transformation.

Note:In the above, we have shown that the Mobius transformation is a one-to-one mapping of the extended complex plane onto itself. Conversely, a one-to-one (analytic/meromorphic) mapping of the extended complex plane onto itself is the Mobius transformation.

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