View source for Eigenfunction

==English==
{{wp}}

===Etymology===
From {{af|en|eigen-|function}}.

===Noun===
{{en-noun}}

# {{lb|en|mathematics}} A [[function]] <math>\phi</math> such that, for a given [[linear]] [[operator]] <math>D</math>, <math>D\phi=\lambda\phi</math> for some [[scalar]] <math>\lambda</math> (called an [[eigenvalue]]). {{C|en|Functions}}
#* {{quote-book|en|year=2000|translator=Hinne Hettema|author=w:John von Neumann; {{w|Eugene Wigner|E. Wigner}}|chapter=On the Behaviour of Eigenvalues in Adiabatic Processes|year_published=1929|editor=Hinne Hettema|title=Quantum Chemistry: Classic Scientific Papers|publisher=World Scientific|pageurl=https://books.google.com.au/books?id=qsidHRJmUoIC&pg=PA25&dq=%22eigenvalue%22%7C%22eigenvalues%22&hl=en&sa=X&ved=0ahUKEwjP97SwuO_XAhUBG5AKHbVzCpAQ6AEI1wIwNg#v=onepage&q=%22eigenvalue%22%7C%22eigenvalues%22&f=false|page=25
|passage=For many quantum-mechanical problems it is important to investigate the change of eigenvalues and '''eigenfunctions''' with the continuous change of one or more parameters. The case in which one knows the eigenvalues and '''eigenfunctions''' for two special values of the parameters, and is interested in the region in between is particularly interesting.}}
#* {{quote-book|en|year=2005|author=Leonid D. Akulenko; Sergei V. Nesterov|title=High-Precision Methods in Eigenvalue Problems and Their Applications|publisher=CRC Press (Chapman & Hall)|url=https://books.google.com.au/books?id=kKjMSwgH-noC&pg=PA1&dq=%22eigenvalue%22%7C%22eigenvalues%22&hl=en&sa=X&ved=0ahUKEwjP97SwuO_XAhUBG5AKHbVzCpAQ6AEIwgEwHA#v=onepage&q=%22eigenvalue%22%7C%22eigenvalues%22&f=false|page=1
|passage=Problems that require an investigation of eigenvalues and '''eigenfunctions''' arise in connection with numerous topics in mechanics, the theory of vibrations and stability, hydrodynamics, elasticity, acoustics, electrodynamics, quantum mechanics, etc.}}

====Synonyms====
* {{l|en|proper function}}, {{l|en|eigenstate}}

====Related terms====
* {{l|en|eigenbasis}}
* {{l|en|eigenbrain}}
* {{l|en|eigendecomposition}}, {{l|en|eigen decomposition}}
* {{l|en|eigenface}}
* {{l|en|eigenhead}}
* {{l|en|eigenmode}}
* {{l|en|eigenpair}}
* {{l|en|eigenspace}}
* {{l|en|eigenspinor}}
* {{l|en|eigenstate}}
* {{l|en|eigensystem}}
* {{l|en|eigentheory}}
* {{l|en|eigenvalue}}
* {{l|en|eigenvector}}

====Translations====
{{trans-top|function}}
* Arabic: {{t-check|ar|دَالَة خَاصَّة|tr=dāla ḵāṣṣa|f}}
* Chinese:
*: Mandarin: {{t|cmn|本徵函數|tr=běnzhēng hánshù}}
* Finnish: {{t|fi|ominaisfunktio}}
* French: {{t+|fr|fonction propre|f}}
* German: {{t+|de|Eigenfunktion|f}}
* Greek: {{t|el|ιδιοσυνάρτηση|f}}
* Icelandic: {{t|is|eiginfall|n}}
* Italian: {{t+|it|autofunzione|f}}
* Japanese: {{t+|ja|固有関数|tr=koyū kansū}}
* Portuguese: {{t|pt|autofunção|f}} {{q|Brazil}}, {{t|pt|função própria|f}} {{qualifier|Portugal}}
* Russian: {{t|ru|со́бственная фу́нкция|f}}
* Spanish: {{t|es|autofunción|f}}, {{t|es|función propia|f}}
* Swedish: {{t+|sv|egenfunktion|c}}
* Welsh: {{t|cy|ffwythiant eigen|m}}
{{trans-bottom}}

===See also===
* {{l|en|hamiltonian}}

===Further reading===
* [http://mathworld.wolfram.com/Eigenfunction.html Mathworld article on eigenfunctions]