Thread: Theorem involving delta function: proof by induction

Hi Folks,

I am working through a theorem involving the delta function. I have some queries in the early stages of derivation. See attached.

Thanks

1. There is k -th derivative of delta function and differencial of k-th derivative
gives k+1 derivative.
2. The k-th derivative of delta function at =0.
You may think delta function being proportional to .

Originally Posted by zzzoak

1. There is k -th derivative of delta function and differencial of k-th derivative
gives k+1 derivative.
2. The k-th derivative of delta function at =0.
You may think delta function being proportional to .

I am confused now because in the first line of attachment
but then for the delta function you state that


Are both treated differently? Thanks

Yes, they are treated differently.

Please look here
Delta Function -- from Wolfram MathWorld
(17) equation.

For k-th derivative we write


and k-th degree

Originally Posted by zzzoak

Yes, they are treated differently.

Please look here
Delta Function -- from Wolfram MathWorld
(17) equation.

For k-th derivative we write


and k-th degree

Thanks,

I have written out the problem fully, just have 2 queries highlighted in red. Hopefully it can be read!!

1) At the Start, integration by parts query
2) how we got from (k+1)(-1)(-1)^kK!=(-1)^(k+1)!

I am not sure you have some problems with this line

Originally Posted by zzzoak

I am not sure you have some problems with this line

Yes, Im wondering how the LHS equates to the RHS? Ie the k -th derivative goes to k+1 etc

Thanks

(k+1)!=(k+1)k! and (-1)(-1)^k=(-1)^(k+1)

Cheers