Abstract
The previous chapter gathered together some general properties of the GFs and their companion, the Dirac delta function. This chapter considers the Green’s functions for elliptic, parabolic, and hyperbolic equations that satisfy the BCs appropriate for each type of PDE.
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Notes
- 1.
Actually, to be general, we must add an arbitrary function f(r″) to this. However, as the reader can easily verify, the following argument will show that f(r″)=0. Besides, we are only interested in a solution, not the most general one. All simplifying assumptions that follow are made for the same reason.
- 2.
The heat equation turns into the Schrödinger equation if t is changed to \(\sqrt{-1}\,t\); thus, the following discussion incorporates the Schrödinger equation as well.
- 3.
This will determine how to (semi)circle around the poles.
- 4.
The inner product is defined as a double integral over the rectangle.
References
Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series, and Products. Academic Press, New York (1965)
Weinberg, S.: The Quantum Theory of Fields (2 volumes). Cambridge University Press, Cambridge (1995)
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Hassani, S. (2013). Multidimensional Green’s Functions: Applications. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_22
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DOI: https://doi.org/10.1007/978-3-319-01195-0_22
Publisher Name: Springer, Cham
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