Elasticity_ Barber

390 24 Spherical harmonics
χ 0 0 = 1 ; χ 0 1 = z ; χ 0 2 = z2
2 − ζ ¯ζ
4 ; χ0 3 = z3 zζ ¯ζ
−
6 4 ,
χ 1 0 = ζ ; χ 1 1 = zζ ; χ 1 2 = z2 ζ
2 − ζ2 ¯ζ
8 ; χ1 3 = z3 ζ
6 − zζ2 ¯ζ
8
χ 2 0 = ζ 2 ; χ 2 1 = zζ 2 ; χ 2 2 = z2 ζ 2
24.8.2 Singular cylindrical harmonics
2
, (24.72)
− ζ3 ¯ζ
12 ; χ2 3 = z3 ζ 2
− zζ3 ¯ζ
6 12 .
A similar procedure can be used to develop complex versions of the singular
harmonics of equations (24.53, 24.55), starting with the function
f 0 (ζ, ¯ζ) = ζ −m ; m ≥ 1
= ln(ζ ¯ζ) ; m = 0 . (24.73)
Notice that in the complex variable formulation of two-dimensional problems,
ln(ζ) is generally excluded as being multivalued and hence non-holomorphic.
However,
ln(ζ ¯ζ) = ln(ζ) + ln(¯ζ) = 2 ln(r)
is clearly a real single-valued harmonic function (being of the form (24.61).
Singular potentials with m > 0 will eventually integrate up to include logarithmic
terms when following the procedure (24.59). We see this for example
in the real stress function versions (24.55) for m=1.
PROBLEMS
1. Use equation (24.22) to evaluate the function P 2 1 (x) and verify that it
satisfies Legendre’s equation (24.9) with m = 2 and n = 1. Construct the
appropriate harmonic potential function from equation (24.24) and verify that
it is singular on the z-axis.
2. The functions (23.23) are singular only on the negative z-axis (r =0, z <0).
Similar functions singular only on the positive z-axis are given in equations
(23.25). Develop the first three of (24.42) by superposition, noting that ln(R+
z)+ln(R−z)=2 ln(r).
3. Use the methodology of §5.1 to construct the most general harmonic polynomial
function of degree 3 in Cartesian coördinates x, y, z. Decompose the
resulting polynomial into a set of spherical harmonics.
4. Express the recurrence relation of equations (24.43–24.46) in spherical polar
coördinates R, θ, β. Check your result by deriving the first three functions
starting from the source solution φ 0 = 1/R and compare the results with the
expressions derived in the Maple or Mathematica file ‘spn’.