- #1

mmzaj

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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 :

[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

[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

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