Notes
(This page will be continually updated)
The heat kernel K(x,t) (or Fundamental solution or Green's function of the heat equation), which is displayed
in problem 5 on page 200 in the text for one spatial dimension (but it looks the same in any dimension), allowed us to make the following observation
about solutions to the heat equation; (1) signals (or information, or heat energy) travel infinitly fast, and
(2) solutions are smooth (i.e., differentiable) even if the initial data f(x) is not smooth. Both these
properties fail for the wave equation, for example. (To see (1), take f(x) to be 1 for x
in [-1, 1] and 0 elsewhere. Then the formula (7) given on page 198 shows that u(x,t) > 0
for any x and for any t > 0 (take t to be very close to 0 and note that
K(x,t) > 0 for any x and any t > 0 ). To
see (2), just differentiate the formula (7) with respect to x , the derivative goes past f
(because it depends on x' instead of x ) and lands on K(x'-x,t)
in the integral which can be differentiated as often as you like.)
Feb 15 : The significance of the way solutions of the wave equation propagate in even and odd spatial dimensions
(sharply if the spatial dimension is odd, and not sharply if the spatial dimension is even) for the 'suitability'
of life in these dimensions is discussed in the book The Anthropic Cosmological Principle by John
D. Barrow and Frank J. Tipler (Oxford University Press 1986) on pages 266 - 269. A mathematical analysis and
commentary can be found in the classic work Methods of Mathematical Physics Volume II by Richard Courant and David Hilbert
(John Wiley and Sons, 1962) beginning on page 760. These conclusions follow from the structure of the fundamental
solution (Green's function) of the wave equation.
Feb 23 : Harmonic functions are functions u that satisfy Laplace's equation;
Laplacian (u) = 0. Here u(x) is a function of n variables; x = (x1,
x2, . . . , xn) in Rn. The most important property of Harmonic functions is their
Mean-value property (see page 276 in text for the 2-dimensional case); For any r>0, u(x) = mean-value of u
on the surface of the sphere of radius r centred at x. It's easy to deduce the Maximum (and Minimum) Principle
from this (which says that a Harmonic function defined on a bounded region Q has no local maxima or minima
inside Q and achieves its maximum and minimum on the boundary of Q unless u is constant).
Conversely, if u is a function that satisfies the mean-value property (for any r ), then
u is harmonic.
(For a proof of these statements, see the books listed in the
references page, or the book, Maximum Principles in Differential Equations,
by Murray Protter and Hans Weinberger, Prentice-Hall, 1967. A summary of the properties of harmonic functions and
Laplace's equation can be found in Chapter 2 of Elliptic Partial Differential Equations of
Second Order, by David Gilbarg and Neil Trudinger, Springer Verlag.)
April 4 : One way to calculate the inverse Fourier or Laplace transforms is to use the Residue Theorem
of complex analysis. Then, the inverse transform can be expressed in terms of the residues of the transform U(s). The
residues of U(s) are determined at the poles of U(s), i.e., complex numbers so
where U(so) becomes infinite.
If U(s) is a quotient q(s)/p(s) where q(s) is bounded, then the poles will be among the zeros of p(s).
Thus, the "extended Heaviside Formula" described on page 379-381 in the text.
However, to apply the Residue calculus in this context we must have a certain decay rate of U(s) as s tends to
infinity (in the complex plane). Recall that the hyperbolic sine and cosine functions sinh(s), cosh(s) are exponentially
growing at infinity, so quotients that contain these functions in the numerator will have to contain an exponential term
in the denominator (at least) to guarantee sufficient decay. If the extended Heaviside formula is applied to transforms U(s)
that do not satisfy this decay condition, an incorrect answer may result. See, for example, Basic Complex Analysis, by
J. Marsden for a discussion of the complex inversion formula for the Laplace transform.