Math 401: Complex Analysis

Spring 2020

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Jan. 28	Review of metric spaces: Metric or distance function, sequences, convergent and Cauchy sequences, complete metric spaces, open and closed subsets, interior, boundary, and closure of subsets, continuous functions mapping a metric space to another; Various mathematical structures associated with set of complex numbers: Set structure, metric space structure, real vector-space structure, complex structures on R^2and their existence and uniqueness	-
2	Jan. 30	The field structure on the set of complex numbers, definition of multiplication of complex numbers, calculation of the inverse of nonzero complex numbers, real and imaginary parts, modulus, and conjugate of a complex numbers and some of their properties, Argand diagram and complex plane, Euler’s formula and polar representation of complex numbers. The argument and principal argument of a complex number.	Pages 19-30 of textbook (Howie’s Complex Analysis)
3	Feb. 04	De Moivre’s formulas, roots of complex numbers, construction of the two-sheeted Riemann surface associated with the square root function, its branches, and the branch cut; Spherical representation of the complex numbers and the extended complex plane, rational function of a complex variable, the exponential function f(z):=e^z and its basic properties, trigonometric functions of a complex variable	Pages 5-20 & 29 of Gamelin’s book
4	Feb.06	Non-uniqueness of the choice of branch cuts, branch cut for f(z)=[z(1-z)]^1/2, contours and associated phase shifts for multi-valued functions, unification of triangular and hyperbolic function, logarithm and complex powers of complex numbers; Theorem: C together with the Euclidean metric forms a complete metric space; open discs, open and closed subsets of C.	Pages 21- 27, 30-31 of Gamelin’s book & Pages 35-38 of textbook
5	Feb. 11	Limit and continuity of complex-valued functions of a single complex variable; derivative of such a function	Pages 41-45 & 51 of textbook
Quiz 1	Feb 12
6	Feb. 13	Chain rule for complex-valued functions, Cauchy-Riemann relations as necessary conditions for the differentiability of a complex-valued function, holomorphic and entire functions	Pages 44-45 of Gamelin’s book & Pages 52-53 of textbook
7	Feb. 18	Theorem: If real and imaginary parts of a function f: C → C have continuous partial derivatives in an open disc with center c and satisfy the Cauchy-Riemann conditions at c, f is differentiable at c. Furthermore, if f satisfies Cauchy-Riemann conditions in this open disc, it is holomorphic in the disc; Corollary: The preceding theorem holds if we replace the disc by an open subset containing c. Theorem: If f: C → C is holomorphic in an open disc and its derivative vanishes in this disc, it is constant in the disc.	Pages 53-58 of textbook
8	Feb. 20	Theorem: If f: C → C is holomorphic in an open disc and |f| is constant on the boundary of this disc, then f is constant in the whole disc; Series of complex numbers, their convergence and absolute convergence; If a series converges, the sequence of its terms converges to zero, bounded sequences, monotonically increasing and decreasing real sequences, Monotone Convergence Theorem (without proof), Theorem: Every absolutely convergent series of complex numbers converges; Power series; Theorem: If a power series centered ‘a’ converges at a point ‘z★’ other than ‘a’ , then it converges absolutely in the open disc with center ‘a’ and radius |z★-a|.	Pages 60-62 of textbook
9	Feb. 25	Theorem: Given a power series with variable ‘z’ and center ‘a’ one of the following statements holds: 1) The series converges absolutely in the whole complex plane, 2) It converges only at ‘a’, 3) There is a positive real number R such that the series converges absolutely for |z-a|<R and does not converge for |z-a|>R. Radius of convergence, examples of series with infinite, zero, and finite radius of convergence.	Pages 62-63 of textbook
10	Feb. 27	Theorem: Consider a power series with coefficients cn and a finite or infinite radius of convergence R. If one of the following limits exists it is equal to R: 1) limn→∞|cn/ cn+1|, 2) limn→∞|cn|^-1/n. Theorem the power series ∑ cn(z-a)^n and ∑ ncn(z-a)^n-1 have the same radius of convergence.	Pages 63-64 of textbook
11	Mar. 03	Theorem: Consider a power series ∑ cn(z-a)^n with a nonzero radius of convergence R and let f(z) be its sum for z ϵ NR(a). Then f is holomorphic in NR(a) and the f’(z)= ∑ n cn(z-a)^n-1. Corollary: Let f be as in the preceding theorem. Then it has derivatives of all order and its m-derivative is given by f^(m)(z)= ∑n [(m+n)!/n!]cm+n (z-a)^n. The power series ∑ z^n/n! and the equivalence of its sum to e^z, power series for sin(z), cos(z), sinh(z), and cosh(z).	Pages 64-69 of textbook
Quiz 2	Mar. 04
12	Mar. 05	Open covering and subcovering of subsets of complex plane C, compact subsets of C; Theorem: Every compact subset of C is closed and bounded. Theorem (Heine-Borel): Every closed and bounded subset of C is compact. Corollary: Let f: C → C be a function that is continuous in a closed and bounded subset S of C. Then f is bounded on S and it attains its maximum value in S.	Pages 79-82 of textbook
13	Mar. 10	Parameterized curves complex plane, their graph, and image; reparameterization of a parameterized curve; closed, simple, piecewise continuous and piecewise continuously differentiable (p.c.d) curves, Jordon Curve Theorem, contours, rectifiable curves and the their length, integral of a complex-valued function along a p.c.d curve	Pages 83-95 of textbook
14	Mar. 12	Integral of the derivative of a complex-valued function along a p.c.d curve, bounds on the modulus of the integral of a bounded function along a p.c.d curve, convex curves; Theorem: Let C be a convex contour and f: C → C be a function that is continuous in C and its interior I(C). If the integrals of f along triangular contours lying in I(C) vanish, there is a function F that is holomorphic in I(C) such that f(z)=F’(z) for all z ϵ I(C) and the integral of f along C vanishes. Uniform convergence: Piecewise convergence of sequences of functions, sup-norm, and uniformly convergent sequences of functions	Pages 95-103 of textbook
15	Mar. 31	Uniform convergence: Piecewise convergence of sequences of functions, sup-norm, and uniformly convergent sequences of functions, Theorem: Limit of a uniformly convergent sequence of continuous functions is continuous, uniformly convergent series of functions, Weierstrass M-test, Uniform convergence of power series, Theorem: Given a uniformly convergence series of functions fn, the series whose n-th term is the integral of fn along a piecewise differentiable curve C converges and its sum equals the integral of the sum of the original series of functions along C.	Pages 103-106 of textbook
16	Apr. 02	Statement of Cauchy’s theorem (the general case); Goursat’s lemma, Proof of Cauchy’s theorem for triangular contours: Corollaries: Cauchy’s theorem for polygonal and convex contours; Lemma: If a curves lies in an open subset O of C, there is a positive real number r such that for every z on the curve the open disc Nr(z) is a subset of O.	Pages 107-110 of textbook & Page 51 of H.A.Priestly’s “Introduction to Complex Analysis.”
17	Apr. 07	Two lemmas on covering of a curve by open discs, Lemma: Let C be a contour and f: C → C be a function that is holomorphic in C and its interior I(C). Then there are open subsets O and O' and a compact set S such that f is holomorphic in O', the closure of the interior of C is a subset of O, O is a subset of S, and S a subset of O'. Proof of Cauchy theorem (general case).	-
18	Apr. 09	Deformation theorem for contour integrals, Cauchy’s integral formula, Cauchy’s integral formula for the derivative of a holomorphic function.	Pages 115-122 of textbook
19	Apr. 14	Existence of derivatives of arbitrary order for a holomorphic function, Morera’s theorem, Liouville’s theorem, existence of roots of a complex polynomial, Proof of fundamental theorem of algebra.	Pages 123-128 of textbook
20	Apr. 16	Theorem: Holomorphic functions are derivatives of other holomorphic functions, Lemma: Every pair of points in the interior of a contour can be connect via a piecewise differentiable curve. Deformation theorem for integrals over open curves. Theorem: Let C be a contour and f: C → C be a function that is holomorphic in C and its interior I(C). Then there is a holomorphic function F: I(C) → C such that F’(z)=f(z) for all z ϵ I(C). Application for defining holomorphic functions: Principal logarithm and error function; Theorem: Holomorphic functions are analytic.	Pages 128-134 of textbook
21	Apr. 21	Zeros of holomorphic functions, Laurent series, its existence and uniqueness theorems	Pages 137-143 of textbook
22	Apr. 28	Isolated, removable, and essential singularities of a function, poles of a function, residue of an isolated singularity, Riemann’s theorem on removable singularities, characterization theorems for poles, Casorati-Weierstrass Theorem on essential singularities and its application for nonpolynomial entire functions	Pages 144-145 of textbook
23	Apr. 30	Residue theorem, finding residue of a function at a pole, using Residue Theorem to perform angular integrals	Pages 146-151 & 158-160 of textbook
24	May 05	Using Residue Theorem to perform improper integrals, Jordan’s Lemma	Pages 152-157 & 161-164 of textbook
25	May 07	Real integrals with singularity of the integrand on the real line, use of special contours	Pages 164-174 of textbook
26	May 12	Contour integral of f'/f over a contour C for a function that is meromorphic in I(C) and holomorphic in C, with finitely many zeros and poles in I(C), finiteness of the set of zeros of a function belonging to a compact set where it is holomorphic, Rouche's theorem.	Pages 183-185 of textbook
27	May 14	Argument principle, the Open Mapping and Maximum Modulus theorems	Pages 186-190 of textbook
28	May 21	Inverse function theorem, characterization of 1-to-1 holomorphic functions	Pages 190-192 of textbook

Note: The pages from the textbook listed above may not include some of the material covered in the lectures.