Page 770 - Advanced_Engineering_Mathematics o'neil

750 CHAPTER 22 Singularities and the Residue Theorem

Now use the facts that

I (z) = iJ (iz)

0 0
and

J (z) =−J 1 (z) = J 1 (−z)
0
to obtain
Res(g(z)/h(z), j n )

−2R J 0 ( j n r/R) 2 2
= e − j n t/R .
J 1 ( j n )
j n
The solution is therefore

∞
2
−2R J 0 ( j n r/R) − j n t/R 2
u(r,t) = T 0 1 − 2 e .
j n J 1 ( j n )
n=1
SECTION 22.4 PROBLEMS

1
In each of Problems 1 through 10, use Theorem 22.7 to 5.
find the inverse Laplace transform of the function. (z + 5) 3
1
6.
z + 8
3
z 1
1. 7.
4
2
z + 9 z + 1
1 1
2. 8. z
(z + 3) 2 e (z − 1)
1 z 2
3. 9.
(z − 2) (z + 4) (z − 2) 3
2
1 z + 3
4. 10.
3
2
(z + 9)(z − 2) 2 (z − 1)(z + 2)

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October 14, 2010 15:37 THM/NEIL Page-750 27410_22_ch22_p729-750