COMPLEX ANALYSIS (MA 214)

lecturer Boris Kruglikov

Books:             ST:      I. Stewart & D. Tall "Complex Analysis" (basic),           L:        S. Lang "Complex Analysis" (optional).

Lect. 1.            Basics.             ST Ch. 0-1 or L Ch. I, §1-3.

Sem. 1.            See the list (also in ps).

Lect. 2.            Topology.         ST Ch. 2 or L Ch. I, §4.

Sem. 2.            ST Ex.2: 3(i-iv),4(i,iii,v),6(i,ii,iv),8,9(i-iii),10(i-iv); L Ex.ch.I§3: 1,2,3.

Lect. 3.            Power Series.   ST Ch. 3 or L Ch. II, §2 (first part).

Sem. 3.            ST Ex.3: 1(i-v),2(i-v),6,7,8(i-iv),11,13; L Ex.ch.I§4: 1,2,3,4,ch.II§2: 1,2,6.

Lect. 4.            Differentiation.  ST Ch. 4 or L Ch. I, §5-6, Ch. II, §5.

Sem. 4.            ST Ex.4: 1(i-iii),2,3(i),5,6(i,ii),7(i,iii),9,14; L Ex.ch.II§5: 1,3.

Lect. 5.            Exponential.      ST Ch. 5, L Ch. II, §2 (second part).

Sem. 5.            ST Ex.5: 1(i-iii),2(i-iii),3(i-ii),7(i,ii),10,11(i-iv),15(i,ii); L Ex.ch.II§5: 6(i).

Lect. 6.            Integration.       ST Ch. 6 or L Ch. III, §2-4.

Sem. 6.            ST Ex.6: 1,2,4,5(i-iv),7,11,12,14(i),16(i-v); L Ex.ch.III§2: 3,4,5.

Lect. 7.            The winding number.     ST Ch. 7 or L Ch. III §6, Ch. IV §1.

Sem. 7.            ST Ex.7: 1,3,6,7,8,10,15,16,19-22; L Ex. ch.III §6: 1,2,5, ch. IV §2: 3.

Lect. 8.            Cauchy’s theorem.        ST Ch. 8 or L Ch. III §1-4.

Sem. 8.            ST Ex.8: 1-6; L Ex.ch.III§2: 4-6,8, ch.III§7: 1.

Lect. 9.            Homotopy and Integral.            ST Ch. 9 or L Ch. III §5, Ch. IV §2.

Sem. 9.            ST Ex.9: 1,2,4,5,7(i,iii,iv); L Ex.ch.III§5: 1,4.

Lect. 10.          Taylor expansion.                     ST Ch. 10 or L Ch. V §1.

Sem. 10.          ST Ex.10: 1,3(i-iv),5,8,9(i),13,15,18; L Ex.ch.V§1: 3,4,9.

Lect. 11.          Laurent expansion.        ST Ch. 11, L Ch. V §2-3.

Sem. 11.          ST Ex.11: 1(i-iii,v-vi),2(i-ii,iv-v),3(i-v),5,8(i-iii),9(i,iii,v)10,14(ii-iv),17, 18(iv-vi),22; L Ex.ch.V§2: 4-6,9,10,12, ch.V§3: 1,3.

Lect. 12.          Integration via Residues.           ST Ch. 12 or L Ch. VI §1, 2(first part).

Sem. 12.          ST Ex.12: 1(i,ii,iv,v),2(i-iii),4,5(i-iv),6,8(i),10(i,ii),16(i),21,23,24,27; L Ex.ch.VI§1: 1-10,15,16,19,20(a,b),26(a-d), ch.VI§2: 1,2,3,5,14(a).

 

Remark.           Additional chapters for lectures from L are optional, but for exercises would be better to do both ST and L.