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In mathematics, an eigenfunction of a linear operator, , defined on some function space, is any non-zero function in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has : for some scalar, , the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends on any boundary conditions required of . In each case there are only certain eigenvalues that admit a corresponding solution for (with each belonging to the eigenvalue ) when combined with the boundary conditions. Eigenfunctions are used to analyze . For example, is an eigenfunction for the differential operator : for any value of , with corresponding eigenvalue . Similarly, the functions and , have eigenvalue . If a boundary condition is applied to this system (e.g., or ), then even fewer pairs of eigenfunctions and eigenvalues satisfy both the definition of an eigenfunction, and the boundary conditions. Specifically, in the study of signals and systems, the eigenfunction of a system is the signal which when input into the system, produces a response with the complex constant .〔Bernd Girod, Rudolf Rabenstein, Alexander Stenger, ''Signals and systems'', 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 49〕 ==Examples== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Eigenfunction」の詳細全文を読む スポンサード リンク
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