Errors and inaccuracies in Haberman's Applied PDE, 5th ed

1. Formula 7.10.24 on page 335 is wrong: one needs to add a factor (2/π)^{1/2} (that's the square root of 2/π). Compare section 2.4 of my notes on Bessel and Legendre functions, or section 10.47 and 10.4 of NIST digital library of mathematical functions
   
   
2. Page 370, the line after (8.6.14), is a serious and inexplicable error. The context is as follows. You are ask to solve a Poisson equation Δ u = Q on a bounded domain D in the plane, with a non-homogeneous boundary condition u=α on the boundary of D, where Q is a given function on D and α The eigenvalue problem for Laplacian with Dirichlet boundary condition is supposed to have been solved. In other words you have a orthogonal family of eigenfunctions φ_i, which form a complete orthogonal system of functions on D for the square norm. In (8.6.14), Haberman wants to represent the unknown function u as an infinite sum of the form u=Σ b_i φ_i, where the coefficients b_i are to be determined.
   
   On the next line, Haberman says that "it is NO LONGER true that Δ u = Σ b_i Δ φ_i, since u does not satisfy the homogeneous boundary condition". This is nonsense--I have no idea what he was thinking. The Laplacian is a linear operator, so if u=Σ b_i φ_i, then obviously we will have Δ u = Σ b_i Δ φ_i, assuming that the coefficients b_i die off fast enough as n goes to infinity so that the infinite sums Σ b_i φ_i and Σ b_i Δ φ_i both converge in suitable spaces of functions.
   
   The real trouble is that each of the eigenfunctions φ_i vanishes at the boundary of D, therefore every infinite linear combination Σ b_i Δ φ_i of them also vanish at the boundary, assuming that the coefficients b_i goes to zero rapid enough so that the infinite series converges in a strong sense. Thus if the given boundary value α is non-zero, we won't get a solution of the form Σ b_i Δ φ_i.

Inaccuracies

1. page 479, line -4: The sentence "Since u(x,y) -> 0 as y-> ∞, its Fourier transform (in the variable x) also vanishes as y->∞" is problematic, because the reasoning given is flawed. This problem traces back to the formulation of the boundary conditions (10.6.42)-(20.6.44) on the same page. These boundary conditions requires that (a) for every fixed y we have u(x,y) -> ∞ as x->∞ and also as x->∞, and (b) for every fixed x, we have u(x,y) -> ∞ as y->∞. These conditions do not imply the asserted vanishing of Fourier transform at infinity.
   
   There are various ways to formulate the "boundary conditions at infinity" more precisely that will make the reasoning on line -4 valid. For instance we can require that the integral of the squre of the absolute value of u(x,y) over all x goes to 0 as y -> ∞. Then the integral of the squre of the absolute value of its Fourier transform will also go to 0 as y->∞.
   
   
2. The treatment of the Dirac delta function leaves the reader confused, to say the least. There was not even a sentence saying that "Dirac delta function" is not actually a function in any sense when the Dirac delta is first introduced in 9.3. No explanation was given about the meaning of formula (9.3.32) on the derivative of the Heaviside function, nor the meaning of the "scaling formula (9.3.34).
   
   Exercise 10.3.18 gives many formulas involving Dirac delta and Fourier transforms. I have no idea what the author expected of his readers. For instance part (e) ask the reader to show that the Fourier transform of exp(-i x \omega) is the delta function. But the Fourier transform is defined as an integral in the book, and the integral of exp(-i x\omega) over the real line diverges. What are the readers supposed to do?
   
   On page 452, it is asserted that the limit of the fundamental solution of the heat kernel goes to a delta function as t goes to 0, and that this is "shown in Exercise 10.3.18". But statement is not "shown" in Exercise 10.3.18. Moreover the author did not make any attempt to explain what it means to say that a sequence of functions converges to a delta function.

Passage in Haberman which I strongly disagree

1. page 321: The "explanation" on why the eigenvalue \lambda has the asserted sign is unsatisfactory and misleading. No one can be reasonably expected to "see" from Figure 7.9.1 that "the oscillations in z should be expected for the u_3-problam but not for the u_1- and u_2-problems".
   
   The author created more confusion, saying that "perhaps \lambda <0 for the u_3-problem but not for the u_1- and u_2- problems. Thus we do not specify \lambda at this time". (It is unclear what the \lambda's refer to.) But he never came around to explain why the eigenvalue \lambda in (7.9.5) is negative.
   
   The real explantion is that if the eigenvalue \lambda for (7.9.5) is positive, then the equation (7.9.9) for f(r) will not have non-zero solution satisfying f(a)=0.
   
   
2. page 324, the third line after (7.9.30): The author asserted that "We have oscillations in z and \theta. The r-dependent solutions should not be oscillatory" without explanation, perhaps because no reasonable explanation can be offered to justify the statement that "the r-dependent solutions should not be oscillatory". This statement is misleading, to put it mildly, and misses an important point.
   
   A better explanation why the eigenvalue \lambda in the differential equation (7.9.26) for h(z) is negative is that for \lambda positive, the only solution of (7.9.26) d^2 h/dz^2 =\lambda which satisfies the boundary conditions (7.9.27) and (7.9.28) that h(0)=h(H)=0 is the trivial solution h(z)=0.
   
   
3. lines 4-8 of page 333: the result that the eigenvalues of the Laplace operator on a three-dimensional unit sphere can only be twice a triangular number is characterized as being ``mysterious but elegant'', as if the two adjectives provide enough explanation. Such treatment is intellectually dishonest and conveys a wrong impression on the nature of mathematics.
   
   There is nothing mysterious about this fact, which can be proved in a several ways. The proof often found in books on "equations of mathematical physics" is that the linear span of the eigenfunctions found by the method of separation of variables are seen directly to be dense in the space of continuous function on the unit sphere, for the sup norm.
   
   Such a statement does the readers a disservice. Give a precise pointer to one of the books in the biliography will be much better.