A Formula for the Local Solution of the Self-Dual Yang-Mills Equations

Twistor techniques are used to give all local solutions to the self-dual Yang-Mills equations for any matrix gauge group. The field is expressed in terms of an infinite sum, which is a contour integral analogue of a path-ordered exponential integral, over the twistor datum. The key step is the solution of the matrix-valued Cousin-Hilbert-Riemann problem on the sphere, that is, of trivializing a holomorphic vector bundle given by a transition function over an annulus.