Page 882 - Advanced_Engineering_Mathematics o'neil

862 Answers to Selected Problems

Using this, obtain
∞ ∞ n
1 2n 1 2π z ze iθ iθ
z = e e dθ
(n!) 2 2π n!e i(n+1)θ
n=0 0 n=0
1 2π 1 2π
= e z(e iθ +e iθ ) ,dθ = e 2z cos(θ) dθ.
2π 0 2π 0
19. 4 21. 3 23. 2
25. Show that
f (n) (z 0 )
a n = = b n for n = 0,1,2,··· .
n!

Section 21.2 The Laurent Expansion
1 1 ∞ (−1) n
1. + n=0 (z − i) for 0 < |z − i| < 2
n
z − i 2i (2i) n
4
∞ (−1) n+1 n

3. z 2n−2 ,0 < |z| < ∞
n=1
(2n)!
1
5. − − 2 − (z − 1),0 < |z − 1| < ∞
z − 1
∞ 1

7. n=1 z 2n−2 ,0 < |z| < ∞
n!
2i
9. 1 + ,0 < |z − i| < ∞
z − i
CHAPTER TWENTY TWO SINGULARITIES AND THE RESIDUE THEOREM
Section 22.1 Singularities
1. Pole of order 2 at 0 3. Essential singularity at 0
5. Simple poles at i,−i and a double pole at 1
7. Removable singularity at i, simple pole at −i
9. Simple poles at 1,−1,i,−i
11. Simple poles at (2n + 1)π/2for n any integer

Section 22.2 The Residue Theorem
1. 2πi 3. 0 5. 2πi 7. 2πi
9. −πi/4 11. 0 13. 2πi
15. π[cos(8) − 1 + i sin(8)]/2 19. 2πi 21. 2πi

section 22.3 Evaluation of Real Integrals
√
1. 2π/ 3
3. π/3
√ √
5. πe −2 2 sin 2 2 /4
−4
7. π(1 + 5e )/32
9. π/4
iαz
−α
2
11. e /(z + 1) has one singularity in the upper half-plane and a simple pole at i. Compute the residue at i to be πe .
iθ
13. With z = e , substitute for cos(θ),sin(θ) and dz to obtain
1
2π
dθ
2
2
2
2
0 α cos (θ) + β sin (θ)
4 z
= dz.
2
2
2
2
2
i |z|=1 (α − β )z + 2(α + β )z + (α − β )
2
4
2
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October 14, 2010 17:50 THM/NEIL Page-862 27410_25_Ans_p801-866