Automorphic function

Japanese: 保型関数 - ほけいかんすう（英語表記）automorphic function

There is a group G whose elements are linear fractional transformations that leave the planar domain D invariant, and a meromorphic function f whose domain is D and whose value does not change when a transformation belonging to G is applied to the independent variable, in other words, if the following holds, is called an automorphic function. It is a generalization of periodic functions such as trigonometric functions and doubly periodic functions such as elliptic functions. The concept of automorphic functions was introduced by HJ Poincaré, and has been applied to the theory of algebraic functions and differential equations with great success. For f to be non-constant, G must have the following properties:

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

平面領域Dを不変にする一次分数変換を元とする群Gがあり，Dを定義域とする有理型関数fで，Gに属する変換を独立変数にほどこしても値の変わらないもの，すなわち，が成り立つようなものを保型関数という。三角関数のような周期関数，楕円関数のような二重周期関数の一般化である。保型関数の概念はH.J.ポアンカレが導入し，代数関数論や微分方程式論に応用して大きな成果をあげた。　fが非定数であるためには，Gはつぎの性質をもっていなければならない。

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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