Difference of solutions for the inversion problem of ultra-elliptic integrals

Let V be a hyperelliptic curve of genus 2 defined by Y²=f(X), where $f(X)$ is a polynomial of degree 5. The sigma function associated with V is a holomorphic function on C². For a point P on V, we consider the problem to express the X-coordinate of P in terms of the image of P under the Abel-Jacobi map. Two meromorphic functions f₂ and g₂ on C² which give solutions of this problem are known. Since f₂ and g₂ coincide on the zero set of the sigma function, it is expected that f₂-g₂ can be divided by the sigma function. In this paper, we decompose f₂-g₂ into a product of the sigma function and a meromorphic function explicitly.