Parameter Integration

  • Norbert Ortner
  • Peter Wagner
Chapter

Abstract

In its simplest form, the method of parameter integration yields a fundamental solution E of a product P1(∂)P2(∂) of differential operators as a simple integral with respect to \(\lambda\) over fundamental solutions \(E_{\lambda }\) of the squared convex sums \({\bigl (\lambda P_{1}(\partial ) + (1-\lambda )P_{2}(\partial )\bigr )}^{2}\). Heuristically, this relies on the representations of E and of \(E_{\lambda }\) as inverse Fourier transforms, i.e.,
$$\displaystyle{\mathcal{F}E = \frac{1} {P_{1}(\text{i}\xi )P_{2}(\text{i}\xi )}\mathop{ =}\limits^{ (\text{F})}\int _{0}^{1} \frac{\text{d}\lambda } {{\bigl (\lambda P_{1}(\xi ) + (1-\lambda )P_{2}(\xi )\bigr )}^{2}} =\int _{ 0}^{1}\mathcal{F}E_{\lambda }\,\text{d}\lambda }$$
where the equation (F) is Feynman’s first formula, see (3.1.1) below (for the name cf. Schwartz [245], Ex. I-8, p. 72). Note that Eq. (3.1.1) boils down to the formula \(a^{-1} - b^{-1} =\int _{ a}^{b}x^{-2}\text{d}x,\) 0 < a < b, from elementary calculus.

Keywords

Fundamental Solution Inverse Fourier Transform Parameter Integration Affine Subspace Feynman Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Norbert Ortner
    • 1
  • Peter Wagner
    • 2
  1. 1.Department of MathematicsUniversity of InnsbruckInnsbruckAustria
  2. 2.Faculty of Engineering ScienceUniversity of InnsbruckInnsbruckAustria

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