Proof of the completeness relation

Let us now expand a complex function / (r) in terms of some orthonormal set of functions (pi (r) [cf. Appendix B.5.1] according to 

Comparing the right side of this expression with Eq. (B.82), we conclude that 

Finally, take as a specific example the eigenfunctions of the momentum operator (see Section 2.5) as a basis that is 

Replacing in this expression the suimnation by an integration according to 

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