Complex analysis notes

Class notes:

• Synopsis of complex numbers [.pdf]
• Convergence in the plane [.pdf]
• Topology; complex functions; complex infinity [.pdf]
• Complex differentiation; holomorphy [.pdf]
• Derivatives and integrals over curves; complex line integrals [.pdf]
• Complex elementary functions; Cauchy's Theorem and its consequences [.pdf]
• The Integral Theorems of complex analysis [.pdf]
• Uniqueness consequences; root functions; harmonic functions [.pdf]
• Power functions; circles in the plane [.pdf]
• Complex series; convergence tests [.pdf]
• Series of functions; power series and analyticity [.pdf]
• Uniqueness theorems; Laurent series [.pdf]
• Laurent expansion; singularities, poles etc. [.pdf]
• Residues; the Argument Principle [.pdf]

Exams with full solutions:

• Midterm, Part 1 [.pdf]
• Midterm, Part 2 [.pdf] A different approach to problem 5 can be found here [.pdf]
• Final exam [.pdf]

Important proofs:

• Equivalence of differentiability and CR equations [.pdf]
• Goursat's Theorem [.pdf]
• Existence of antiderivatives [.pdf]
• Cauchy Integral Formula [.pdf]
• Cauchy Estimates and Liouville's Theorem [.pdf]
• Convergence of the Taylor Series for holomorphic functions [.pdf]