- #1
Choragos
- 20
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Hello all. In short, I am wondering what the second derivative of the Heaviside function (let's say H[(0)]) would be. I'm presuming that it's undefined (or more accurately, zero everywhere but at x=0), but I would like to know if that is correct.
Essentially, I am attempting to extend a proof which uses f(x) where f''(x) is defined. I would like to extend this proof to a piecewise-constant f(x). However, one of the requirements is that f'(x) >> f''(x) >> f(n)(x). If H''[(0)] is undefined, than this approach will not work.
My attempt at a solution is as follows. Given a general, differentiable function f(x), then
[itex]\int[/itex]f(x)δ(n)dx [itex]\equiv[/itex] -[itex]\int[/itex][itex]\frac{∂f}{∂x}[/itex]δ(n-1)(x)dx
I am, however, unclear as to the meaning of δ(n-1), especially when n=1.
Thanks for your help.
Essentially, I am attempting to extend a proof which uses f(x) where f''(x) is defined. I would like to extend this proof to a piecewise-constant f(x). However, one of the requirements is that f'(x) >> f''(x) >> f(n)(x). If H''[(0)] is undefined, than this approach will not work.
My attempt at a solution is as follows. Given a general, differentiable function f(x), then
[itex]\int[/itex]f(x)δ(n)dx [itex]\equiv[/itex] -[itex]\int[/itex][itex]\frac{∂f}{∂x}[/itex]δ(n-1)(x)dx
I am, however, unclear as to the meaning of δ(n-1), especially when n=1.
Thanks for your help.