Linear fractional transformation

Japanese: 一次分数変換 - いちじぶんすうへんかん（英語表記）linear fractional transformation

$Linear fractional transformation$

It is also called the Möbius transformation. When the constants a , b , c , and d are complex numbers and ad - bc ≠ 0, the function is called a linear fractional function, and the mapping determined by this is called a linear fractional transformation. It is defined on the complex plane C other than -d / c when c ≠ 0, and on the complex plane C when c = 0. Since it represents the same function if a : b : c : d is constant, there is no loss of generality in assuming ad - bc = 1. When c ≠ 0, this linear fractional function gives a one-to-one mapping from C -(- d / c ) to C -( a / c ), and the inverse mapping is given by.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

メービウス変換Möbius transformationともいう。定数a,b,c,dが複素数で，ad－bc≠0であるとき関数を一次分数関数といい，これより定まる写像を一次分数変換という。これはc≠0のとき－d/c以外の複素平面C上で定義され，c＝0のときは複素平面C上で定義される。a：b：c：dが一定であれば同じ関数を表すから，ad－bc＝1と仮定しても一般性を失わない。c≠0のときはこの一次分数関数はC－(－d/c)からC－(a/c)への1対1上への写像を与え，逆写像はで与えられる。

出典　株式会社平凡社世界大百科事典第２版について　情報

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