18.112 | Fall 2008 | Undergraduate

Functions of a Complex Variable

Lecture Notes

The readings from this course are assigned from the text and supplemented by original notes by Prof. Helgason. The lecture notes were prepared by Zuoqin Wang under the guidance of Prof. Helgason.


Ahlfors, Lars V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd ed. New York, NY: McGraw-Hill, 1979. ISBN: 9780070006577.

L1 The algebra of complex numbers: the geometry of the complex plane, the spherical representation Ahlfors, pp. 1-11 and 19-20 (PDF)
L2 Exponential function and logarithm for a complex argument: the complex exponential and trigonometric functions, dealing with the complex logarithm   (PDF)
L3 Analytic functions; rational functions: the role of the Cauchy-Riemann equations Ahlfors, pp. 21-32 (PDF)
L4 Power series: complex power series, uniform convergence Ahlfors, pp. 33-42 (PDF)
L5 Exponentials and trigonometric functions Ahlfors, pp. 42-47 (PDF)
L6 Conformal maps; linear transformations: analytic functions and elementary geometric properties, conformality and scalar invariance Ahlfors, pp. 69-80 (PDF)
L7 Linear transformations (cont.): cross ratio, symmetry, role of circles Ahlfors, pp. 80-89 (PDF)
L8 Line integrals: path independence and its equivalence to the existence of a primitive Ahlfors, pp. 101-108 (PDF)
L9 Cauchy-Goursat theorem Ahlfors, pp. 109-115 (PDF)
L10 The special cauchy formula and applications: removable singularities, the complex taylor’s theorem with remainder Ahlfors, pp. 118-126 (PDF)
L11 Isolated singularities Ahlfors, pp. 126-130 (PDF)
L12 The local mapping; Schwarz’s lemma and non-Euclidean interpretation: topological features, the maximum modulus theorem Ahlfors, pp. 130-136 (PDF)
L13 The general Cauchy theorem   (PDF)
L14 The residue theorem and applications: calculation of residues, argument principle and Rouché’s theorem   (PDF)
L15 Contour integration and applications: evaluation of definite integrals, careful handling of the logarithm Ahlfors, pp. 154-161 (PDF)
L16 Harmonic functions: harmonic functions and holomorphic functions, Poisson’s formula, Schwarz’s theorem   (PDF)
L17 Mittag-Leffer’s theorem: Laurent series, partial fractions expansions Ahlfors, pp. 187-190 (PDF)
L18 Infinite products: Weierstrass’ canonical products, the gamma function Ahlfors, pp. 191-200 (PDF)
L19 Normal families: equiboundedness for holomorphic functions, Arzela’s theorem   (PDF)
L20 The Riemann mapping theorem Ahlfors, pp. 229-231 (PDF)
L21-L22 The prime number theorem: the history of the theorem and the proof, the details of the proof   (PDF)
L23 The extension of the zeta function to C, the functional equation

Ahlfors, pp. 214-217

For the original proof, see p. 146 of Weber, Heinrich, ed. The Collected Works of Bernhard Riemann. Reprint ed. New York, NY: Dover Publications, 1953. ASIN: B000KGER80. (With the assistance of Richard Dedekind.)


Course Info

As Taught In
Fall 2008
Learning Resource Types
Exams with Solutions
Lecture Notes
Problem Sets with Solutions