Mathematical Tools for Physics - Department of Physics - University

17—Densities and Distributions 414 Green’s functions In the discussion of the Green’s function solution to a differential equati ...

17—Densities and Distributions 415 Usek=mω 02 , and this is zero. Now go back to the extra term inθ. Thekxterms doesn’t have it, ...

17—Densities and Distributions 416 The idea of a generalized function is that you can manipulate itas if it were an ordinary fun ...

17—Densities and Distributions 417 At this point I can introduce a notation: “ ∫∞ −∞ dxδ(x)φ(x)” MEANS nlim→∞ ∫∞ −∞ dxδn(x)φ(x) ...

17—Densities and Distributions 418 This turns out to have all the properties that you need, though again, you don’t have to invo ...

17—Densities and Distributions 419 Now I have to makegsatisfy Eq. (17.30)atx=y. Computedg/dx. But wait. This is impossible unles ...

17—Densities and Distributions 420 Now to solve the original differential equation, Eq. (17.29). Substitute into this equation t ...

17—Densities and Distributions 421 The right side is designed to be easy. For the left side, do some partial integrations. I’m l ...

17—Densities and Distributions 422 as long as the position~r 0 is inside the volume of integration. For an example that uses thi ...

17—Densities and Distributions 423 There are two terms; start with the first one.r > 0 , soeiqr→ 0 asq→+i∞. The contour is al ...

17—Densities and Distributions 424 The expression−∂(1/r)/∂zis such a simple way to compute this, the potential of an electric di ...

17—Densities and Distributions 425 This is the potential from the charge densityρ= +Q∂^2 δ(~r)/∂z^2 , whereQ=qa^2. [What about ∂ ...

17—Densities and Distributions 426 Problems 17.1 Calculate the mean, the variance, the skewness, and the kurtosis excess for a G ...

17—Densities and Distributions 427 17.16Starting with the formulation in Eq. (17.23), what is the result ofδ′and ofδ′′on a test ...

17—Densities and Distributions 428 students? That is, given this function for all students, what is the resulting distribution o ...

Index . Abramowitz and Stegun, 7 , 51, 341, 365 absolute convergence, 29 acceleration, 63 Adams methods, 279 stable 281 air mass ...

Index 430 combinatorial factor, 31, 46 commutative law, 124 commutator, 176 commute, 155 comparison test, 26–28 complete, 135 co ...

Index 431 singular point 73 trigonometry 77, 95 vector space 129 differential operator, 220 differentiation, 349, 375, 385 diffr ...

Index 432 functional, 296 –309, 385, 411–415 derivative 412 integral 412 3-linear 302 bilinear 297, 298, 300, 302 derivative 387 ...

Index 433 infinite-dimensional, 134 infinitesimal, 183, 184, 312 inflation, 45 instability, 269, 281, 288 integral, 9 –14 contou ...