The Cauchy–Kowalevski Theorem

Given a Cauchy problem, the Cauchy–Kowalevski theorem states:
If all the functions $F_{i}$ are analytic in some neighborhood of the point

(t^{0}, x_{1}^{0}, x_{2}^{0}, \dots, ϕ_{j, k_{0}, k_{1}, \dots, k_{n}}^{0}, \dots),

and if all the functions $ϕ_{j}^{(k)}$ are analytic in some neighborhood of the point

(x_{1}^{0}, x_{2}^{0}, \dots, x_{n}^{0}),

then the Cauchy problem has a unique analytic solution in some neighborhood of the point

(t^{0}, x_{1}^{0}, x_{2}^{0}, \dots, x_{n}^{0}) .