Plotting the approximation of the Dirac delta function

Hi,

I am not sure if I have solved the following task correctly



I did the plotting in mathematica and got the following


Would the plots be correct? What is meant by check for normalization, is the following meant?

For the approximation for ##\epsilon > 0##, does it mean that for the area of ## 0 < x < \epsilon## the area must be 1, so for the case ##\epsilon=1## the total area would be 1, the half for ## 0 < x < \epsilon## would then be ##\frac{1}{2}##, so the normalization constant should be 2?

To sketch usually means to draw by hand. Is this behavior that you expected? Look at the formula. Between minus epsilon and +epsilon, what is the function doing (does it change with x)?

Normalization refers to the area under the "curve" equal to 1 over the entire spectrum. Do these graphs satisfy that?

Lambda96 said:

Would the plots be correct?

Yes, those plots look correct.

Lambda96 said:

What is meant by check for normalization, is the following meant?

It means to check that the integral is 1.

Lambda96 said:

For the approximation for ##\epsilon > 0##, does it mean that for the area of ## 0 < x < \epsilon## the area must be 1,

No. It means that the area of ##- \epsilon < x < \epsilon ## must be 1.

Lambda96 said:

so for the case ##\epsilon=1## the total area would be 1, the half for ## 0 < x < \epsilon## would then be ##\frac{1}{2}##, so the normalization constant should be 2?

No. No further normalization is needed. The area between each graph and the x axis is already 1.

Thank you scottdave and FactChecker for your help