Page 432 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 423

1 1 1 1 1 z z 2 z 3
Ans: þ þ þ 1 þ þ
z 5 z 4 z 3 z 2 z 2 4 8
ð
1
16.94. Let e st FðtÞ dt ¼ f ðsÞ where f ðsÞ is a given rational function with numerator of degree less than that of the
0
denominator. If C is a simple closed curve enclosing all the poles of f ðsÞ,we can show that
1 þ
zt
zt
e f ðzÞ dz ¼ sum of residues of e f ðzÞ at its poles
FðtÞ¼
2 i C
2
s 1 s þ 1 1
Use this result to find FðtÞ if f ðsÞ is (a) ; ðbÞ ; ðcÞ ; ðdÞ and
2
2
s þ 1 s þ 2s þ 5 2 2 2
sðs 1Þ ðs þ 1Þ
check results in each case.
[Note that f ðsÞ is the Laplace transform of FðtÞ, and FðtÞ is the inverse Laplace transform of f ðsÞ (see Chapter
12). Extensions to other functions f ðxÞ are possible.]
2t
3 2t
5
Ans. (a)cos t; 1 t 1 þ te þ e ; 1
e sin 2t;
2 ðcÞ 4 2 4 ðdÞ 2 ðsin t t cos tÞ
ðbÞ