  Five dimensional supersymmetric gauge theory compactified on a circle defines
an effective N=2 supersymmetric theory for massless fields in four dimensions.
Based on the relativistic Toda chain Hamiltonian proposed by Nekrasov, we
derive the Picard-Fuchs equation on the moduli space of the Coulomb branch of
SU(2) gauge theory. Our Picard-Fuchs equation agrees with those from other
approaches; the spectral curve of XXZ spin chain and supersymmetric cycle in
compactified M theory. By making use of a relation to the Picard-Fuchs equation
of SU(2) Seiberg-Witten theory, we obtain the prepotential and the effective
coupling constant that incorporate both a perturbative effect of Kaluza-Klein
modes and a non-perturbative one of four dimensional instantons. In the weak
coupling regime we check that the prepotential exhibits a consistent behavior
in large and small radius limits of the circle.
