  The Euler-Gauss linear transformation formula for the hypergeometric function
was extended by Goursat for the case of logarithmic singularities. By replacing
the perturbed Bessel differential equation by a monodromic functional equation,
and studying this equation separately from the differential equation by an
appropriate Laplace-Borel technique, we associate with the latter equation
another monodromic relation in the dual complex plane. This enables us to prove
a duality theorem and to extend Goursat's formula to much larger classes of
functions.
