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Second-order Hadamard lemma

For a smooth function

f (x, y)

,

f (x, y) = f (x, 0) + y D_{2} f (x, 0) + y^{2} h (x, y),

where the smooth remainder is

h (x, y) = \int_{0}^{1} (1 - t) D_{2}^{2} f (x, t y) d t .

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