We now proceed term-by-term in (I.2). The zeroth-order
is

(I.15) |

(I.16) |

(I.17) |

The Helmholtz equation and the BCs together imply , a relation used both above and below. In general our task is to reduce the number of -derivatives; it is always possible using such manipulations to leave only terms containing a single -derivative. This will be desirable for manipulation of boundary integrals by parts later.

The third-order result (included below) requires use of the following. simplifies to when the BCs are applied. This required the normal derivative of curvature since now needs to be regarded as a scalar field with -dependence. The tangential derivative is given the name . Also simplifies to when BCs are applied.

Combining everything and finally substituting for gives
the expansion of a Dirichlet scaling eigenfunction at location on :

The expression is believed to be correct to order . The complexity increases greatly with each power of , and higher derivatives of the curvature enter. However, in the case of the circle billiard (where is constant) it is easy to verify the expansion. One general pattern is that the -order terms all contain powers of , and for each term the number of spatial derivatives of must equal the power of minus the power of . It is important to notice that the term differs by the presence of a factor of from the term given by VS.