The Greenâs functions are the analytical solutions of boundary value problems when they are excited by an elementary source. This is usually an infinitesimally small vectorial point source. In order for the Greenâs functions to be computationally useful, they must have analytical closed forms. This can be a mathematical expression or a more complex recursive process. It is no surprise that only very few electromagnetic boundary value problems have closed-form Greenâs functions. The total electric ('''E''') field can be expressed in terms of the electric current in the following way:
:<math>\mathbf{E = E^{inc}} + \mathbf{\iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') } d \nu' + \mathbf{\iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') } d \nu'</math>Â Â :<math>\mathbf{H = H^{inc}} + \mathbf{\iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') } d \nu' + \mathbf{\iiint_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') } d \nu'</math><!--[[File:PMOM1(1).png]]-->
where is the dyadic Greenâs functions for electric fields due to electric current sources and '''E<sup>i</sup>''' is the incident or impressed electric field. The incident or impressed field provides the excitation of the structure. It may come from an incident plane wave or a gap source on a line, etc. The simplest background structure is the unbounded free space, which is represented by the following Greenâs function:
:<math>\mathbf{ \overline{\overline{G}}_{EJ}(r|r') = (\overline{\overline{I}} + \nabla\nabla) } G_{\Lambda} (\mathbf{r|r'}), \quadG_{\Lambda} (\mathbf{r|r'}) = \frac{ e^{-jk_0 \mathbf{|r-r'|}} }{ 4\pi \mathbf{|r-r'|} }</math><!--[[File:03_freespace_tn.gif]]-->
where [[File:i_tn.gif]] is the unit dyad, [[File:delta_tn.gif]] is the gradient operator, '''r''' and '''r'''' are the position vectors of the observation and source points, respectively, and k<sub>0</sub> is the free-space propagation constant. This implies that electromagnetic waves propagate in free space in a spherical form away from the source. Note that the Greenâs function has a singularity at the source, i.e. when '''r''' = '''r''''. This singularity must be removed when solving the integral equations.