Integration by Parts: Formula & Real/Non-Integer n

[zetafunction]

is the following formula of integration by parts

\int_{-\infty}^{\infty}dxf(x)D^{n}g(x) = (-1)^{n} \int_{-\infty}^{\infty}dxg(x)D^{n}f (x)

valid for real or non-integer n? the problem i see here is the term (-1)^{n} , which may be not so well defined for non-integer 'n'

 

[mathman]

Integration by parts takes place in integer steps, so the formula you presented requires n to be an integer. If n is not an integer, you need to use h(x)=Dⁿg(x) and carry out the parts integration in integer steps. You will never get the right hand side.