Homework assignments for M361
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Nr Section : Problems
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[1] 1.2 : 2,4,6,8,10,18,24 Sep 1 → Sep 8
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[2] 1.3 : 2,4,8,20,26,28 Sep 8 → Sep 15
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[3] 1.4 : 7,8,11,14,15,18,21 Sep 15 → Sep 22
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[4] 1.5 : 18a-c,19,23,27a-b
1.6 : 2,3,10 Sep 22 → Oct 6
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[5] 2.1 : 1,7a,9,12
2.2 : 1,5,9 (see R1 below) Oct 6 → Oct 13
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[6] 2.3 : 1,2,7,10 Oct 13 → Oct 20
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[7] 2.4 : 1,5,13,17
... : A1 (see below) Oct 20 → Oct 27
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[8] 2.5 : 2,3,5,10
3.1 : 3,4,6 Oct 27 → Nov 3
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[9] 3.2 : 1,3,5b,8,13,18 Nov 3 → Nov 10
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[10] 3.3 : 1a,1d,4,9,11 Nov 10 → Nov 17
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[11] 4.1 : 2a,8
4.2 : 3,5,10 Nov 17 → Nov 24
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[12] 4.3 : 3,8,20 extra problems
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REMARKS R1 ..
Denote by C the set of all complex numbers.
Definition.
A bounded subset of C is said to be simply connected
if both the set and its complement are connected.
Theorem. (Cauchy's theorem)
Let A be a simply connected bounded open subset of C, and let f: A→C be analytic.
If γ is a closed piecewise C1 curve in A then ∫γf(z)dz=0.
R1. For the problems in Section 2.2 you can use Cauchy's theorem above.
You may take for granted that disks and rectangles are simply connected.
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PROBLEMS A1 ..
A1. Let 0<a<1. Compute the integral I=∫(1+a2cos(t))-1dt where t ranges from 0 to 2π.
Hint: Rewrite I as an integral ∫γf(z)dz along the unit circle γ(t)=eit,
using that cos(t)=(z+1/z)/2 for z=eit.