The constants A, B, and C may be determined by use of the boundary and compatibility con- 
ditions. Finally 
I, (pr) 
Py (r, s) et Hone 
Kk, SA(s) 1, (pd) [23] 
(7, 8) : bD,, (qb,9r) a D,, (qb 
F938) aoe 9 = ? 
Po x, 8A(s) q 10 629997) + Ky oo ‘2 a”)| 
where G is written for 
_ KP 4 | (pd) [24] 
I, (p 5) 
and the determinant A(s) is 
G 
A(s) = qa [a Dy, (92,94) + — Do (9,90) 
297) [25] 
P G 
+ i 76D, (96,94) + sar Doo (9,00) 
2 2 
The functions Diy, (x, y) are the modified cylinder functions, defined by Jaeger* as 
git 
Diy, (es y) = | Up (@) Kg (y) - Kg (2) Ig (y)] [26] 
oz dy 
Some of the properties of these functions and the modified Bessel functions are listed in the 
Appendix. 
The inverse of the Laplace transform is 
Figure 2 - Contour of Integration in the ) Plane 
T +i 0 
AGis—— 
Qt iPofcs 
b(r,r) eA? dd [27] 
The integrand has simple poles at A = 0 and: at 
an infinite number of points A, on the negative 
real axis where A (r,,) = 0. The integration is 
taken along a line, parallel to the imaginary 
axis, which lies to the right of all the singu- 
larities, see Figure 2. The integral from A to 
B plus the integral over a portion C of a circle 
which does not pass through any of the poles 
is equal to 272 times the sum of the residues 
within the contour. As the radius tends to in- 
finity the integral over C tends to zero. Thus 
¢ (r,t) becomes the sum of the residues at the 
poles of the integrand. 
