Integral Transforms Generated by Green’s Functions

  • B. Davies
Part of the Applied Mathematical Sciences book series (AMS, volume 25)


In this section we will investigate (in a purely formal manner) some properties of the self-adjoint differential operator [see (10.15)]
$$ L\left[ u \right] = {\left[ {p\left( x \right)u'\left( x \right)} \right]^\prime } + q\left( x \right)u\left( x \right), $$


Eigenfunction Expansion Singular Problem Alternative Formula Fourier Sine Fourier Sine Series 
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  1. 1.
    For example, STAKGOLD (1968).Google Scholar
  2. 2.
    STAKGOLD (1968), Ch. 4.Google Scholar
  3. 3.
    TITCHMARSH (1953), Ch. 6.Google Scholar
  4. 4.
    If any one of these conditions is not satisfied, we have a singular problem.Google Scholar
  5. 5.
    These manipulations involve assumptions about the solution which can only be verified a posteriori. Alternatively, we could work with a suitable set of generalized functions from the outset.Google Scholar
  6. 6.
    R. D. Turner, Q. Appl. Math. (1956), 14, 63.zbMATHGoogle Scholar
  7. 7.
    MORSE & FESHBACH (1953), p. 842.Google Scholar
  8. 8.
    The evaluation of the right-hand side (58) is discussed at length in Turner’s paper.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • B. Davies
    • 1
  1. 1.The Australian National UniversityCanberraAustralia

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