Editing Green's function (section)

===Eigenvalue expansions===
If a [[differential operator]] {{math|''L''}} admits a set of [[eigenvectors]] {{math|Ψ<sub>''n''</sub>(''x'')}} (i.e., a set of functions {{math|Ψ<sub>''n''</sub>}} and scalars {{math|''λ''<sub>''n''</sub>}} such that {{math|1=''L''Ψ<sub>''n''</sub> = ''λ''<sub>''n''</sub> Ψ<sub>''n''</sub>}}&thinsp;) that is complete, then it is possible to construct a Green's function from these eigenvectors and [[eigenvalues]].

"Complete" means that the set of functions {{math|{{mset|Ψ<sub>''n''</sub>}}}} satisfies the following [[completeness relation]],
<math display="block">\delta(x-x') = \sum_{n=0}^\infty \Psi_n^\dagger(x') \Psi_n(x).</math>

Then the following holds, 
{{Equation box 1
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|equation = <math>G(x, x') = \sum_{n=0}^\infty \dfrac{\Psi_n^\dagger(x') \Psi_n(x)}{\lambda_n},</math>
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where <math>\dagger</math> represents complex conjugation.

Applying the operator {{math|''L''}} to each side of this equation results in the completeness relation, which was assumed.

The general study of Green's function written in the above form, and its relationship to the [[function space]]s formed by the eigenvectors, is known as [[Fredholm theory]].

There are several other methods for finding Green's functions, including the [[method of images]], [[separation of variables]], and [[Laplace transform]]s.<ref>{{cite book |first1=K.D. |last1=Cole |first2=J.V. |last2=Beck |first3=A. |last3=Haji-Sheikh |first4=B. |last4=Litkouhi |chapter=Methods for obtaining Green's functions |title=Heat Conduction Using Green's Functions |publisher=Taylor and Francis |year=2011 |pages=101–148 |isbn=978-1-4398-1354-6}}</ref>