- #1
Jaynte
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Homework Statement
I want to know how I could extract T from Tz in matrix form so I can
get [tex]T^{-1}[/tex] to get the inverse of w(z).
w(z)=Tz=[tex]\frac{az+b}{cz+d}[/tex]
The Inverse of Möbius transform, also known as the inverse Möbius transformation or inverse Cayley transformation, is a mathematical operation that undoes the effect of a Möbius transformation. It maps a point in the complex plane back to its original location before the transformation was applied.
The Inverse of Möbius transform is calculated using the same formula as the Möbius transform, but with the coefficients in reverse order. For example, if the Möbius transform is w(z) = (az + b)/(cz + d), then the Inverse of Möbius transform is z(w) = (dw - b)/(a - cw).
The Inverse of Möbius transform has many applications in mathematics and physics. It is used to solve problems involving conformal mapping, complex dynamics, and the study of Riemann surfaces. It is also useful in geometric transformations and in the study of symmetry in various systems.
Yes, the Inverse of Möbius transform can have multiple solutions or be undefined for certain values of the coefficients a, b, c, and d. This is because the Möbius transform is not bijective, meaning that multiple different points in the complex plane can map to the same point after the transformation. Therefore, the inverse operation may not be unique.
The Inverse of Möbius transform is closely related to other mathematical concepts such as complex analysis, group theory, and projective geometry. It is also connected to the theory of fractional linear transformations and has applications in calculus, differential equations, and quantum field theory.