Indian Institute of Space Science and Technology
Complex Analysis
TUTORIAL - IV
1. Let C0 denote the circle |z − z0 | = R, taken counterclockwise. Use the parametric representation
z = z0 + Reiθ (−π ≤ θ ≤ π) for C0 to derive the following integration formulas:
dz
= 2πi;
C0 z − z0
Z
(a)
Z
(b)
(z − z0 )n−1 dz = 0
(n = ±1, ±2, · · ·).
C0
2. Apply the Cauchy-Goursat theorem to show that
Z
f (z) dz = 0
C
when the contour C is the circle |z| = 1, in either direction, and when
z2
1
; (b) f (z) = ze−z ; (c) f (z) = 2
;
z−3
z + 2z + 2
(d) f (z) = sech z; (e) f (z) = tan z; (f) f (z) = log(z + 2).
(a) f (z) =
3. Let C denote the positively oriented boundary of the half disk 0 ≤ r ≤ 1, 0 ≤ θ ≤ π, and let f (z)
be a continuous function defined on that half disk by writing f (0) = 0 and using the branch
f (z) =
√
reiθ/2
r > 0,
−π
3π
<θ<
2
2
of the multiple-valued function z 1/2 . Show that
Z
f (z)dz = 0
C
by evaluating separately the integrals of f (z) over the semicircle and the two radii which make up
C. Why does the Cauchy-Goursat theorem not apply here?
4. Let C denote the positively oriented boundary of the square whose sides lie along the lines x = ±2
and y = ±2. Evaluate each of these integrals:
e−z dz
;
C z − (πi/2)
Z
cos z
(b)
dz;
2 + 8)
z(z
C
Z
z dz
(c)
;
C 2z + 1
Z
cosh z
(d)
dz;
z4
C
Z
tan(z/2)
(e)
dz (−2 < x0 < 2).
2
C (z − x0 )
Z
(a)
5. Let C be the circle |z| = 3, described in the positive sense. Show that if
2z 2 − z − 2
dz
z−w
C
Z
g(w) =
(|w| =
6 3),
then g(2) = 8πi. What is the value of g(w) when |w| > 3?
1
6. Show that if f is analytic within and on a simple closed contour C and z0 is not on C, then
f 0 (z) dz
=
C z − z0
f (z) dz
.
2
C (z − z0 )
Z
Z
7. Let f denote a function that is continuous on a simple closed contour C. Prove that the function
g(z) =
f (s) ds
C s−z
1
2πi
Z
is analytic at each point z interior to C and that
g 0 (z) =
1
2πi
f (s) ds
(s
− z)2
C
Z
at such a point.
8. Obtain the Maclaurin series representation
z cosh(z 2 ) =
∞
X
z 4n+1
n=0
9. Obtain the Taylor series
ez = e
(2n)!
∞
X
(z − 1)n
n=0
n!
(|z| < ∞).
(|z − 1| < ∞)
for the function f (z) = ez by
(a) using f (n) (1) (n = 0, 1, 2, · · ·);
(b) writing ez = ez−1 e.
10. Show that when 0 < |z| < 4,
∞
X
1
1
zn
=
+
.
4z − z 2
4z n=0 4n+2
11. Give two Laurent series expansions in powers of z for the function
f (z) =
1
z 2 (1 − z)
,
and specify the regions in which those expansions are valid.
12. Show that when 0 < |z − 1| < 2,
∞
X
z
(z − 1)n
1
= −3
−
.
n+2
(z − 1)(z − 3)
2
2(z − 1)
n=0
13. Write the two Laurent series in powers of z that represent the function
f (z) =
1
z(1 + z 2 )
in certain domains, and specify those domains.
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