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→Peano–Baker series: wrap long formula |
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==Peano–Baker series==
The most general transition matrix is given by a [[product integral]], refered to as the '''Peano–Baker series'''
:<math>\begin{align}
:<math> \mathbf{\Phi}(t,\tau) = \mathbf{I} + \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\int_\tau^{\sigma_2}\mathbf{A}(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 + ...</math>▼
\mathbf{\Phi}(t,\tau) = \mathbf{I} &+ \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 \\
&+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 \\
▲
&+ \cdots
\end{align}</math>
where <math>\mathbf{I}</math> is the [[identity matrix]]. This matrix converges uniformly and absolutely to a solution that exists and is unique.<ref name=rugh />
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