 Hadamard's lemma

On a star-shaped neighbourhood of $a$, a smooth function satisfiesThe fundamental theorem of calculus along the segment proves the identity, and differentiation under the integral sign proves smoothness of the coefficients.

Hadamard's lemma is a result in the field of differential calculus that relates to the expansion of a function in terms of its derivatives. Specifically, it provides a formula for expressing the value of a function at a point in terms of its Taylor series expansion around another point.