Derivation under integral

In Mathematical, in analyzes, the derivation under the sign summons , or derivation under integral , is a result established starting from the Théorème of Leibniz.

Statement

Either has and I two intervals of

\ mathbb {R}

. Either F a function of has × I with actual values or complex.

If the following conditions are vérifées:

$\ forall X \ in has; T \ to F (X, T)$ is continuous per pieces and integrable on I ;
$\ forall X \ in has; T \ to \ frac {\ partial F (X, T)} {\ partial X}$ exists and is continuous per pieces;
$\ forall T \ in I; X \ to \ frac {\ partial F (X, T)} {\ partial X}$ is continuous;

And if, moreover, there exists

\ varphi: I \ to \ mathbb {R} ^ {+}

integrable such as:

\ forall X \ in has, \ forall T \ in I, \ left| \ frac {\ partial F (X, T)} {\ partial X} \ right| \ Leq \ varphi (T)

then the function

F: X \ to \ int_I F (X, T) \, dt

is of class

\ mathcal {C} ^1

and

\ forall X \ in has, F' (X) = \ int_I \ frac {\ partial F (X, T)} {\ partial X} \, dt

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