Proof using deravatives.

markrvr · #1 $Old$ July 28th, 2009, 07:25 AM

Hi,
Wasn't too sure where to post this so feel free to move.
If
$\int x^MK(x)dx\not=0$
and
$\int x^mK(x)dx=0$
where m = 1,2,3......M-1
Show that the formula
$K_{[M+2]}(x)=\frac{3}{2}K_{[M]}(x)+\frac{1}{2}xK'_{[M]}(x)$ .
Produces K of order M+2
and now
$\int x^{M+2}K(x)dx\not=0$

Any help appreciated cos I don't have a clue where to begin.

HallsofIvy · #2 $Old$ July 28th, 2009, 09:43 AM

Quote:

Originally Posted by markrvr $View Post$

Hi,
Wasn't too sure where to post this so feel free to move.
If
$k_M=\int t^MKtdt\not=0$
and
$\int t^mK(t)dt=0$
where m = 1,2,3......M-1
Show that the formula
$K_{[M+2]}(x)=\frac{3}{2}K_{[M]}(x)+\frac{1}{2}xK'_{[M]}(x)$ .
Produces K of order M+2
Any help appreciated cos I don't have a clue where to begin.

Your notation is a bit confusing. Is the " $K_{[M]}$ " the same as the " $k_M$ " in your first equation?

markrvr · #3 $Old$ July 28th, 2009, 10:21 AM

Sorry, no they are different. That notation was from a different part I forgot to delete it.