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This introduction to Green's functions is based on their role as kernels ofdifferential equations. The procedures to construct solutions to a differentialequation with an external source or with an inhomogeneity term are put togetherto derive the Dyson equation for the Green's function of the inhomogeneoussystem. Very different areas of physics such as, for example, electrodynamicsand quantum transport, can profit from this Green's function formalism. Thefundamental homogeneous-medium Green's tensor of electrodynamics is deducedfrom the field of a dipole. Based upon that a numerical procedure is presentedto solve the wave-equation for the near-field in a scattering setup forarbitrary material distributions. The full inhomogeneous system's Green'sfunction is not explicitly needed to get the fields, although it can beobtained by a very similar calculation and in optics can be interpreted as adensity of states. It is demonstrated how the transport problem for two openfree-electron gas reservoirs with arbitrary coupling can be solved by findingthe system's Green's function. In this sense the article is an introduction onGreen's functions for treating interaction. A very detailed discussion of thecurrent formula is given on an elementary basis.

Green's functions for solving differential equations, in non-boundary  value problems in near-field optics and in quantum transport through point  contacts