greetings . it's known that if [itex] g(x), f(x)[/itex] are two functions ,and [itex]f(x)[/itex] is sufficiently differentiable , then by repeated integration by parts one gets :(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int f(x)g(x)dx=f(x)\int g(x)dx -f^{'}(x)\int\int g(x)dx^{2}+f^{''}(x)\int \int \int g(x)dx^{3} - .... +(-1)^{n+1}f^{(n)}(x)\underbrace{\int.....\int}g(x)dx^{n+1}+(-1)^{n}\int\left[ \underbrace{\int.....\int}g(x)dx^{n+1}\right ]f^{n+1}(x)dx[/tex]

now, if [itex]f(x) [/itex] is a smooth function , one would expect the formula above to be repeatable infinitely man times . therefore :

[tex] \lim_{n \to \infty }\int\left[ \underbrace{\int.....\int}g(x)dx^{n+1}\right ] f^{n+1}(x)dx=0[/tex]

is a necessary but not sufficient condition for the summation to converge . my question is , what are the conditions needed to extend the scope of the formula !?!?

also, are there any theorems on the multiple integrals - the ones containing [itex]g(x)[/itex] - besides cauchy formula for repeated integration

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# On the integration by parts infinitely many times

**Physics Forums | Science Articles, Homework Help, Discussion**