Derivative and antiderivative Let f be an infinitely differentiable real valued

Kassandra Ross

Kassandra Ross

Answered question

2022-06-14

Derivative and antiderivative
Let f be an infinitely differentiable real valued function. Can any condition be given under which n-th antiderivative of the n-th derivative of f is equal to f? Any help is appreciated.

Answer & Explanation

humusen6p

humusen6p

Beginner2022-06-15Added 22 answers

Step 1
Let f ( n ) =: g. There is not "the" n t h antiderivative of g, but an n-dimensional affine space G of them. Two functions in G differ by a polynomial of degree n 1, and exactly one element in G is = f.
Step 2
There are cases where f has a special property, e.g., being non-constant and bounded. In such a case it may happen that only one element of G has this property, and this element then has to be the f we started with.

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