Indian Institute of Space Science and Technology
Complex Analysis
TUTORIAL - IV
1. Let C0denote the circle |z−z0|=R, taken counterclockwise. Use the parametric representation
z=z0+Reiθ(−π≤θ≤π) for C0to derive the following integration formulas:
(a) ZC0
dz
z−z0
= 2πi;
(b) ZC0
(z−z0)n−1dz = 0 (n=±1,±2,···).
2. Apply the Cauchy-Goursat theorem to show that
ZC
f(z)dz = 0
when the contour Cis the circle |z|= 1, in either direction, and when
(a) f(z) = z2
z−3; (b) f(z) = ze−z; (c) f(z) = 1
z2+ 2z+ 2;
(d) f(z) = sech z; (e) f(z) = tan z; (f) f(z) = log(z+ 2).
3. Let Cdenote 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/2r > 0,−π
2< θ < 3π
2
of the multiple-valued function z1/2. Show that
ZC
f(z)dz = 0
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 Cdenote the positively oriented boundary of the square whose sides lie along the lines x=±2
and y=±2. Evaluate each of these integrals:
(a) ZC
e−zdz
z−(πi/2) ;
(b) ZC
cos z
z(z2+ 8)dz;
(c) ZC
z dz
2z+ 1;
(d) ZC
cosh z
z4dz;
(e) ZC
tan(z/2)
(z−x0)2dz (−2< x0<2).
5. Let Cbe the circle |z|= 3, described in the positive sense. Show that if
g(w) = ZC
2z2−z−2
z−wdz (|w| 6= 3),
then g(2) = 8πi. What is the value of g(w) when |w|>3?
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