Mobius transformation

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A Möbius transformation is a function on the complex numbers which is only defined on the upper half plane (i.e. all z with imaginary part greater than 0). These transformations are all of the form , where a,b,c, and d are real values with ad-bc>0. These transformations are named in honor of the German mathematician August Ferdinand Möbius, because if you topologically identify the upper half plane with the Möbius strip, then each Möbius transformation (to varying degrees) "flips" the Möbius strip inside out. Möbius transformations are the basis of modular forms.