Need help with Impulse function and unit step function at singularity

[phoenixy]

I have some trouble wrapping my head around singularity

One of assignment question is to show that the unit function is not defined at 0. To do that, I need to show

$$\lim_{\Delta\to0}[u_{\Delta}(t)\delta(t)]=0$$
$$\lim_{\Delta\to0}[u_{\Delta}(t)\delta_{\Delta}(t)]=\frac{1}{2}\delta(t)$$


Also, I need to show that the following is identical to u(t)
$$g(t)=\int u(t)\delta(t-\tau)d\tau$$
integrating from negative infinity to positive infinity


One more question, what's the derivative of the impulse function?

PS: what's the tex code for integration from infinity to infinity? I tried \int_-\infty^+\infty, but the tex output is messed up

 

[dextercioby]

Of course.Click on this

$$\int_{-\infty}^{+\infty}$$

and use {} wherever necessary.

Daniel.

 

[phoenixy]

thanks dex,


still need help with the signularity functions ...

 

[circuits83]

Hello very body
i got confused understanding how we can get the derivative of V(t) equal to Hv(t) :


V= (1/R) e^(-t/RC) U(t) its derivative V' = Hv(t) = -(1/R²C)e^(-T/RC) U(t) +(1/R)&(t)
where : U(t) is the unite step fonction and &(t) is the delta fonction
please some help
thanks for your time

 

[tanujkush]

thanks dex,


still need help with the signularity functions ...

Second part:

The Kronecker delta is non zero only when the parameter in its brackets is equal to zero. So for the integral, its only when tao is equal to t that delta(t) is non zero. Set tau=t, and you have the result.

The derivative of the step function is the Kronecker delta function. check this out
http://mathworld.wolfram.com/HeavisideStepFunction.html

 

[trambolin]

It is Dirac Delta function that is mentioned here. Kronecker Delta is different.

 

[tanujkush]

Second part:

The Kronecker delta is non zero only when the parameter in its brackets is equal to zero. So for the integral, its only when tao is equal to t that delta(t) is non zero. Set tau=t, and you have the result.

The derivative of the step function is the Kronecker delta function. check this out
http://mathworld.wolfram.com/HeavisideStepFunction.html

Oh sorry, I meant the Dirac delta function, not the Kronecker, my bad.