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22.2 The Residue Theorem 739

SECTION 22.2 PROBLEMS

In each of Problems 1 through 16, use the residue theorem Hint: Use Theorem 22.6. Begin by writing h(z) =
to evaluate the integral. (z − z 0 ) ϕ(z) where ϕ(z 0 ) = 0.
2
1 + z 2 18. Suppose f is differentiable at points on a closed path

1. dz with γ the circle of radius 7 γ and at all points in the region G enclosed by γ ,
γ (z − 1) (z + 2i)
2
about −i. except possibly at a finite number of poles of f in G.
Let Z be the number of zeros of f in G and P be the
2z

2. dz with γ the circle of radius 3 about 1. number of poles of f in G with each zero and pole
γ (z − i) 2
counted as many times as its multiplicity. Show that
e z
3. dz with γ the circle of radius 2 about −3i. 1 f (z)
γ z dz = Z − P.
2πi γ f (z)
cos(z)
4. dz with γ the square of side length 3 and This formula is known as the argument principle.
γ 4 + z 2 Hint:If f has a zero of order k at z 0 , show by looking
sides parallel to the axes centered at −2i.
at the Taylor expansion of f (z) about z 0 that
z + i

f (z) k g (z)
5. dz with γ the square of side length 8 and
γ z + 6 = +
2
sides parallel to the axes centered at the origin. f (z) z − z 0 g(z)
where g is differentiable at z 0 and g(z 0 ) = 0. Use this
z − i

6. dz with γ the circle of radius 1 about the to evaluate Res( f /f, z 0 ).
γ
2z + 1 If f has a pole of order m at z 1 ,showby
origin.
examining the Laurent expansion of f (z) about z 1 that
z

7. dz with γ the circle of radius 1 about 1/2. f (z) m h (z)

γ 2
sinh (z) =− +
f (z) z − z 1 h(z)
cos(z)
8. dz with γ the circle of radius 1/2 about i/8. for some h(z) that is differentiable and nonzero at z 1 .
γ ze z
Use these facts and the residue theorem to derive
iz
the argument principle.
9. dz with γ the circle of radius 2
γ (z + 9)(z − i)
2
about −3i. 19. Evaluate
z
2/z 2 dz
10. e dz with γ the square with sides parallel to the γ 2 + z 2
γ
axes and of length 3 centered at −i.
with γ as the circle |z|= 2 first by using the residue
8z − 4i + 1
11. dz with γ the circle of radius 2 about theorem and then by using the argument principle.
γ
z + 4i
−i. 20. Evaluate γ tan(z)dz with γ the circle |z|= π first
by using the residue theorem and then by using the
z 2

12. dz with γ the square of side length 4 and argument principle.
γ
z − 1 + 2i
sides parallel to the axes centered at 1 − 2i. 21. Evaluate
z + 1

13. coth(z)dz with γ the circle of radius 2 about i. dz
γ 2
γ z + 2z + 4
(1 − z) 2 with γ the circle |z|= 2, first by using the residue
14. dz with γ the circle of radius 2 about 2.
γ z − 8 theorem and then by using the argument principle.
3
e 2z
22. Let
15. dz with γ any closed path enclosing 0
γ z(z − 4i)
and 4i. p(z) = (z − z 1 )(z − z 2 )···(z − z n )
2 with z 1 ,··· , z n distinct complex numbers. Let γ be

z

16. dz with γ any closed path enclosing 1. a positively oriented closed path enclosing each z j .
γ z − 1
Evaluate
17. Let h and g be differentiable at z 0 and g(z 0 ) = 0. p (z)

Suppose h has a zero of order 2 at z 0 . Show that dz
γ p(z)
(3)
2g (z 0 ) 2 g(z 0 )h (z 0 ) first by using the residue theorem and then by using

Res(g(z)/h(z), z 0 ) = − . the argument principle.
h (z 0 ) 3 (h (z 0 )) 2

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October 14, 2010 15:37 THM/NEIL Page-739 27410_22_ch22_p729-750