17—Densities and Distributions 425This is the potential from the charge densityρ= +Q∂^2 δ(~r)/∂z^2 , whereQ=qa^2. [What about
∂^2 /∂x∂y?]
Exercises1 What is the analog of Eq. (17.4) for the linear mass densityλ(x) =C(a constant) for 0 < x < L
and zero otherwise?
2 Take the preceding mass density and add a point massm 0 atx=L/ 2. What is the distribution
m([x 1 ,x 2 ])now?
3 Use theλfrom the first exercise and define the functionalF[φ] =
∫∞
−∞dxλ(x)φ(x). What is the
total mass,F[1] =M? What is the mean position of the mass,F[x]/M?
4 As in the preceding exercise, what are the variance, the skewness, and the kurtosis excess?
5 What is
∫ 1
0 dxδ(x−x^0 )?
6 Pick any two of Eq. (17.20) and show that they are valid delta sequences.
7 What is
∫x−∞dtδ(t)?
8 In Eq. (17.12) the functionφn(x)appeared. Sketch a graph of it and of−φ′n(x), which is needed
in Eq. (17.11).