Inserting this expansion into the FRGE and expanding the trace on its right-hand side in order to extract the beta-functions, one obtains the exact RG equation in component form: k ∂ ∂ k g α α ( k ) = β β α α ( g 1 , g 2 , ⋯ ⋯ ) . Together with the corresponding initial conditions these equations fix the evolution of the running couplings g α α ( k ) , and thus determine Γ Γ k completely. As one can see, the FRGE gives rise to a system of infinitely many coupled differential equations since there are infinitely many couplings, and the β β -functions can depend on all of them. This makes it very hard to solve the system in general.