The Kernel of (Laplacian Plus Exponential)

Abstract

Among the ordinary differential equations of the form

Appendix: Vertices in the Theory of Fuchsian Functions

Let $\mathbf{R}$ be the real line, and let a, b be complex numbers, with complex conjugates $a'$ and $b'$, respectively. Following Poincaré [113], we define

$$\begin{aligned} (a, b) \equiv {(a-a') \over (a-b')} {(b-b') \over (b-a')}. \end{aligned}$$

(26.a.1)

Let us consider two arcs of circle, ab and cd, centred on $\mathbf{R}$. If one has

$$\begin{aligned} (a, b)=(c, d), \end{aligned}$$

(26.a.2)

this means that there exists a linear substitution with real coefficients that changes ab into cd. This is said to be the substitution

$$\begin{aligned} (a, b; c, d). \end{aligned}$$

(26.a.3)

Consider now a curvilinear polygon located entirely below $\mathbf{R}$, and whose sides are of two kinds: those of the first kind are arcs of circles centred on $\mathbf{R}$; those of the second kind are segments of the real axis.

There exist 2n sides of the first kind; two consecutive sides of the first kind are separated in one of the following three ways:

• (i) By a vertex located below $\mathbf{R}$, called vertex of the first category (or species or kind).

• (ii) By a vertex located on the real line, which is therefore a vertex of second category .

• (iii) By a side of second kind, which is said to be a vertex of third category (or, again, species or kind).

By virtue of this convention, it is clear that one meets, by following the perimeter of the polygon, alternatively a side of first kind and a vertex of first, second or third category. The side that one meets after a given vertex is the subsequent side; the vertex that one meets afterwards is the subsequent vertex, and so on.

One assumes that the sides of first kind are distributed in pairs in arbitrary fashion, and a side is said to be conjugate of the side belonging to the same pair. The vertices are taken to be distributed into cycles in the following way. One starts from an arbitrary vertex; one considers the subsequent side, then its conjugate, then the subsequent vertex, then the following side, thereafter its conjugate, and afterwards the subsequent vertex, and so on, until one reverts to the original vertex. All vertices met in this way are said to belong to the same cycle .

Poincaré made the following assumptions:

(1) All vertices of a same cycle are of the same category.

(2) If all vertices of a cycle are of the first category, the sum of the corresponding angles of the curvilinear polygon is a portion of $2 \pi$.

(3) if $a_{i}b_{i}$ and $a_{i}'b_{i}'$ are two sides of a same pair, one has, with the notation of Eq. (26.a.1), the equality

$$\begin{aligned} (a_{i}, b_{i})=(a_{i}', b_{i}'). \end{aligned}$$

(26.a.4)

Under these three conditions, the group of substitutions

$$ (a_{i}b_{i}; a_{i}'b_{i}') $$

turns out to be a Fuchsian group, and one obtains in this way all Fuchsian groups.