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In mathematics, the Cauchy–Kovalevskaya theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by , and the full result by . ==First order Cauchy–Kovalevski theorem== This theorem is about the existence of solutions to a system of ''m'' differential equations in ''n'' dimensions when the coefficients are analytic functions. The theorem and its proof are valid for analytic functions of either real or complex variables. Let ''K'' denote either the fields of real or complex numbers, and let ''V'' = ''K''''m'' and ''W'' = ''K''''n''. Let ''A''1, ..., ''A''''n''−1 be analytic functions defined on some neighbourhood of (0, 0) in ''V'' × ''W'' and taking values in the ''m'' × ''m'' matrices, and let ''b'' be an analytic function with values in ''V'' defined on the same neighbourhood. Then there is a neighbourhood of 0 in ''W'' on which the quasilinear Cauchy problem : with initial condition : on the hypersurface : has a unique analytic solution ''ƒ'' : ''W'' → ''V'' near 0. Lewy's example shows that the theorem is not valid for all smooth functions. The theorem can also be stated in abstract (real or complex) vector spaces. Let ''V'' and ''W'' be finite-dimensional real or complex vector spaces, with ''n'' = dim ''W''. Let ''A''1, ..., ''A''''n''−1 be analytic functions with values in End (''V'') and ''b'' an analytic function with values in ''V'', defined on some neighbourhood of (0, 0) in ''V'' × ''W''. In this case, the same result holds. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cauchy–Kowalevski theorem」の詳細全文を読む スポンサード リンク
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