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Math 3364, Section 12181
Spring 2025

Introduction to Complex Analysis

General info                 Assignments (and notes, stats)                 (bottom) - latest HW                

Notes

Graph of the complex logarithm (from Wikipedia)

Assignments

 ASSIGNMENTS (FUTURE ONES MIGHT CHANGE)  
     Section    TO DO   Optional  
HW 1
  1.1
page 4		 8, 10, 16, 22, 24 7, 11, 13, 14, 15
1.2
page 12		 10, 16 (hint: one approach is to write \(z= a+ i b\) and compute, using that \(|z|=1\)) 3, 4, 7
1.3
page 22		 6 (a, b), 12 5, 7, 9, 10, 13, 15
HW 2
1.4
page 31		 2, 7, 10, 22 1, 3, 12, 13, 17, 20*
1.5
page 37		 4, 5 (a,e), 10 5, 16, 19
1.6
page 43		 20 [Hint: use Theorem 1 to obtain uniqueness] 1, 19, 24
1.7
page 50		 none 2(b), 5
HW 3
2.1
page 56		 1(c), 4(c), 6(b) [Hint: \(|z|=1 \iff 1/z=\overline{z}\)] 3, 5, 7-10
2.2
page 63		 read this section (definitions same as for real variables)
20 		3, 4, 5, 7, 15, 18
2.3
page 70		 4(a), 10, 12 (OK to prove this without induction), 15, 16 8, 9, 11, 14
2.4
page 77		 3, 8 [enough to prove the case "\(\operatorname{Re} f(z)\) constant"; the "\(\operatorname{Im} f(z)\) constant" case reduces to this by looking at \(i f\)] 1(b), 7, 9, 16*
2.5
page 84		 6, 8, 9, 12
1, 2, 3, 13, 14, 18
HW 4
3.1
page 108		 7, 12, 14, 19 (can use problems 17 and 18 without proof) 3, 5, 9, 10, 16, 17, 18, 21
3.2
page 115		 2, 14(b), 20 1, 3, 5, 7, 9, 11, 22*
3.3
page 123		 5, 14, 16 1, 3, 8, 15, 17
3.5
page 136		 1 (a, b, c), 8, 11 1, 4
4.1
page 159		 8, 9 1, 5, 11
4.2
page 170		 6, 12, 14(d), 16, 17 3, 5, 8, 18 (on a closed contour, can start the parametrization at any point)
HW 5
4.3
page 178		 2, 4 1, 3, 5, 6 and 7
4.4
page 199		 11, 13, 15, 17 1, 2, 9, 10, 15, 18-20
HW 6
4.5
page 212		 4, 6 (use Thm 19), 7, 10 1, 2, 3, 11
if you have more time, look also at 9, 13*, 14*, 15*, 16*, 17*
4.6
page 219		 3, 5, 6, 8, 10, 16, 18 4, 7*,11, 13, 14-15, 19
HW 7
5.1
page 239		 7, 11, 21 1, 2, 5, 8, 9, 10, 19*, 20
5.2
page 249		 4, 11(a, b, c) [Hint for (c): write \(\cos z = 1 - u\) for some expression \(u=u(z)\), and then use the power series of \(1/(1-u)\)], 13 1, 2, 3, 6, 8, 11, 12, 18
5.3
page 258		 2, 3 (b, c), 4, 5, 8 [Hint: use the information about the derivatives to compute the first few terms of the power series of \(f\) and note that one can now factor out \(z^2\)] 1, 3, 6, 7, 10, 11
5.4*
page 266		 none for now
5.5
page 276		 1(d), 4, 9, 11 [Do only \(n=1, 2, 3\). Hint: compute \(n=1\), then differentiate.] 1, 5, 7(a,b), 13 [added 5, 7(b) later]
HW 8
5.6
page 285		 2, 5 (see more details), 7, 19(a) 1, 3, 6, 14* (Theorem of Casorati and Weierstrass, a weaker version of Picard's Theorem, about behavior near an essential singularity), 17*-18* (Schwarz's Lemma)
5.7
page 290		 none for now 1, 2, 5, 6, 7* (consider \(f(z)=1/z\)), 13
6.1
page 313		 3(a, b, g), 4, 6, 7 1, 2, 3, 5
HW 9
6.2
page 317		 2, 5, 9 (do for \(n=3\) only) 1-8
6.3
page 325		 1, 4 2, 6 [Hint: \(\int_0^\infty = \frac{1}{2}\int_{-\infty}^\infty\) in this case], 11, 14*, 15
HW 10
6.4
page 336		 1, 2, 6 3, 10, 11, 12*; can try the others too
6.5
page 344		 1(c), 3, 4, 10 1, 2, 11*, 12; can try the others too
HW 11 (to come)