MATH 4103 - Introduction to Functions of a Complex Variable, Section 001 - Spring 2012
TR 12:00-1:15 p.m., 222 PHSC

Instructor: Prof. Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Monday 2:30-3:30 p.m., Tuesday 2:45-3:45 p.m., or by appointment.

Prerequisite: The most important prerequisite for the course is a good understanding of Calculus, in particular sequences and series (Calculus III) and calculus of several variables (Calculus IV).

Course description: The class will cover the algebra of complex numbers, analytic functions and their properties, integrals in the complex plane, Cauchy-Goursat theorem, Cauchy integral formula, Laurent series, residues and poles, applications of residues.

Text: J. W. Brown, R. V. Churchill. Complex Variables and Applications, 8th edition, 2009, McGraw Hill, ISBN-13: 9780073051949.
Optional: M. R. Spiegel, S. Lipschutz, J. D. Schiller, D. Spellman. Schaum's Outline of Complex Variables, 2nd edition, 2009, McGraw-Hill, ISBN-13: 978-0071615693.

Check out the OU Math Blog! It is REALLY interesting!

Check out the Problem of the Month !

If you are interested in solving Math problems, check out the web-page of our Putnam problem-solving seminar .

Homework:

• Homework 1, due Thu, Jan 26.
• Homework 2, due Thu, Feb 2.
• Homework 3, due Thu, Feb 9.
• Homework 4, due Tue, Feb 21. Please note the unusual date!
• Homework 5, due Thu, Mar 1.
• Homework 6, due Thu, Mar 8.
• Homework 7, due Thu, Mar 15.
• Homework 8, due Thu, Mar 29.
• Homework 9, due Thu, Apr 5.
• Homework 10, due Thu, Apr 12.
• Homework 11, due Thu, Apr 19.
• Homework 12, due Thu, May 3.

Content of the lectures:

• Lecture 1 (Tue, Jan 17): Sums and products: complex numbers, the complex plane, sums and products of complex numbers (Sec. 1).
  Basic algebraic properties: commutative and associative laws of addition and multiplication, additive inverse of a complex number, multiplicative inverse of a non-zero complex number (Sec. 2).
• Lecture 2 (Thu, Jan 19): Further properties: subtraction and division of complex numbers, binomial formula (Sec. 3).
  Vectors and moduli: analogy between adding vectors in the plane and adding complex numbers, modulus of a complex number, the distance between two points z₁ and z₂ in the complex plane is equal to |z₁-z₂|, inequalities involving |z|, triangle inequality and its consequences and generalizations (Sec. 4).
  Complex conjugates: definition of a complex conjugate, complex conjugages of sums, products, differences, and quotients of two complex numbers, more properties (Sec. 5).
• Lecture 3 (Tue, Jan 24): Exponential form: exponential form of a complex number, arg(z) and Arg(z), Euler's formula, examples (Sec. 6).
  Products and powers in exponential form: derivation of a formula for multiplication, division, and integer powers, of complex numbers in exponential forms, examples (Sec. 7).
  Arguments of products and quotients: a recap of all properties derived earlier (Sec. 8).
  Roots of complex numbers: derivation of a formula for the nth root of a complex number (Sec. 9).
• Lecture 4 (Thu, Jan 26): Regions in the complex plane C: an ε-neighborhood of z₀∈C, a deleted ε-neighborhood of z₀; interiour, exterior, and boundary points of a set S⊆C; boundary ∂C of a set S⊆C; open and closet sets, the closure of a set; connected set, domain, region; bounded and unbounded sets; accumulation point of a set. (Sec. 11).
  Functions of a complex variable: domain and range of a function f of a complex variable, examples, polynomials and rational functions of a complex variable; multiple-valued functions (Sec. 12).
• Lecture 5 (Tue, Jan 31): Mappings: geometric methods of visualising functions of complex variables; example of w=z²: constant-u and constant-v lines, mapping regions, the mapping in exponential form (Sec. 13).
  Mappings by the exponential function: definition of e^z; examples of mapping straight lines, rectangles, and horizontal strips in the (x,y) plane (Sec. 14).
  Limits: definition of a limit, uniqueness of the limit, generalization to the case of boundary points (Sec. 15).
• Lecture 6 (Thu, Feb 2): Limits (cont.): examples of computing limits or demonstrating that a limit does not exist (Sec. 15).
  Theorems on limits: relating the existence of a limit of a complex-valued function f(z) with the existence of the limits of its real part u(x,y) and imaginary part v(x,y) (Theorem 1), limit of a sum, product, and quotient of two functions (Theorem 2), existence of limits of polynomials and rational functions (Sec. 16).
  Limits involving points at infinity: Riemann sphere, stereographic projection (Sec. 17).
• Lecture 7 (Tue, Feb 7): Limits involving points at infinity (cont.): ε-neighborhood of ∞, generalization of the definition of a limit of a function of a complex variable to the cases when either the argument tends to ∞ or the value of the function tends to ∞ or both the argument and the value of the function tend to ∞, a Theorem equivalent conditions for limits involivng ∞ with proofs of parts (a) and (b) done in class; please do the proof of part (c) yourself (Sec. 17).
• Lecture 8 (Thu, Feb 9): Continuity: definition of continuity; continuity of a composition of two continuous functions (Theorem 1); if a continuous function is nonzero at a point, then it is nonzero in a neighborhood of this point (Theorem 2) (Sec. 18).
  Derivatives: definition of derivative, examples of finding derivatives (derivatives of f(z)=z, f(z)=z³), examples of functions that have no derivatives (f(z)=the complex conjugate of z, f(z)=Re(z)), an example of a function that has derivative at one point only (f(z)=|z|²) (Sec. 19).
  Differentiation formulas: derivative of a constant, derivatives of f(z)=zⁿ; derivatives of sums, products and quotients of two functions (derivation of the formula for derivative of a product) (Sec. 20).
• Lecture 9 (Tue, Feb 14): Differentiation formulas (cont.): chain rule for the derivative of composition of functions (math majors should read the proof!) (Sec. 20).
  Cauchy-Riemann equations: Cauchy-Riemann equations for the real and imaginary parts of a differentiable function (read the proof!), examples (Sec. 21).
  Suffiient conditions for differentiability: the Cauchy-Riemann equations as sufficient conditions for differentiability of a function (skip the proof), examples (Sec. 22).
• Lecture 10 (Thu, Feb 16): Polar coordinates: changing from Cartesian coordinates (x,y) to polar coordinates (r,θ): x=X(r,θ)=rcosθ, y=Y(r,θ)=rsinθ; expressing the real and imaginary parts of a function f(z)=u(x,y)+iv(x,y) using Cartesian coordinates z=x+iy; expressing the real and imaginary parts of a function f(z)=U(r,θ)+iV(r,θ) using polar coordinates z=re^iθ; expressing U(r,θ) and V(r,θ) through u(x,y) and v(x,y): U(r,θ)=u(X(r,θ),Y(r,θ)), V(r,θ)=v(X(r,θ),Y(r,θ)); chain rule: U_r(r,θ)=u_x(X(r,θ),Y(r,θ))X_r(r,θ)+u_y(X(r,θ),Y(r,θ))Y_r(r,θ)=u_x(X(r,θ),Y(r,θ))cosθ+u_y(X(r,θ),Y(r,θ))sinθ, U_θ(r,θ)=u_x(X(r,θ),Y(r,θ))X_θ(r,θ)+u_y(X(r,θ),Y(r,θ))Y_θ(r,θ)=-ru_x(X(r,θ),Y(r,θ))sinθ+ru_y(X(r,θ),Y(r,θ))cosθ, and similarly for V_r(r,θ) and V_θ(r,θ); Cauchy-Riemann equations in polar coordinates: rU_r=V_θ, rV_r=-U_θ; examples of application (Sec. 23).
  Analytic functions: definitions of a function analytic at a point, in an open set, in a closed set; entire function; singular point (singularity) of a function (page 73 of Sec. 24).
• Lecture 11 (Tue, Feb 21): Analytic functions (cont.): a proof that the sum and the product of two analytic functions is an analytic function, a proof that the composition of two analytic functions is an analytic function, a proof that the ratio of two analytic function is an analytic function, a proof that if f'(z)=0 everywhere in a domain, then f(z) is constant throughout the domain (Sec. 24).
  Examples: definition and some properties of the hyperbolic functions (page 75 of Sec. 25).
• Lecture 12 (Thu, Feb 23): Exam 1.
• Lecture 13 (Tue, Feb 28): Examples (cont.): singlar points of a rational function; proof that if both f(z) and its complex conjugate are analytic in a domain, then f(z) is constant throughout the domain; if |f(z)| is constant in a domain, then f(z) is constant throughout the domain - read the proof in the book (Sec. 25).
  Harmonic functions: heat equation T_t(t,x,y,z)=ΔT(t,x,y,z), where Δ=∂_xx+∂_yy+∂_zz is the Laplace's operator; stationary heat equation in two spatial dimensions: T_xx+T_yy=0; definition of a harmonic function, the real and the imaginary parts of an analytic function are harmonic functions (Theorem 1), a harmonic conjugate of a harmonic function, an example that "harmonic conjugate" is not a symmetric relation between two functions (Sec. 26).
• Lecture 14 (Thu, Mar 1): Harmonic functions (cont.): a function is analytic if and only if its imaginary part is a harmonic conjugate of its real part (Theorem 2), finding the harmonic conjugate of a harmonic function (Sec. 26).
  The exponential function: definition and basic properties of e^z, derivative of e^z, periodicity of e^z (with purely imaginary period 2πi) (Sec. 29).
  The logarithmic function: definition of log(z), log(z) as a multivalued function, definition of Log(z), examples (Sec. 30).
• Lecture 15 (Tue, Mar 6): Branches and derivatives of logarithms: multiple-valued versus single-valued functions, branch points and branch cuts, principal branch of log, idea of Riemann surfaces, cautionary examples with Log (Sec. 31).
  Some identities involving logarithms: arg vs. Arg, log vs. Log, identities for log of a product and a ratio of complex numbers, defining the nth root of a complex number through log and comparison with the "old" definition (Sec. 32).
  Complex exponents: definition of z^c for complex z and c, the principal value of z^c, examples; cautionary examples of violation of some identities like z^cw^c=(zw)^c (Sec. 33).
• Lecture 16 (Thu, Mar 8): Some scattered remarks: log(z) is not exactly the inverse function of e^z, derivatives of log(z) and Log(z), derivatives of z^c and c^z for complex z and c (based on material from Sections 30-33).
  Trigonometric and hyperbolic functions: definitions and basic properties of the trigonometric and the hyperbolic functions (material from Sections 34 and 35).
  Inverse trigonometric and hyperbolic functions: definition of these functions and discussion of some of their basic properties (in particular, they are multiple-valued), derivation of an expression for arccos(z) in terms of log(z) (Sec. 36).
• Lecture 17 (Tue, Mar 13): Derivatives of w(t): definition of a function w:R→C as w(t)=u(t)+iv(t), where u:R→R, v:R→R; definition of the derivative: w'(t):=u'(t)+iv'(t), basic rules for differentiation: [z₀w(t)]'=z₀w'(t), (e^z₀t)'=z₀e^z₀t (Sec. 37).
  Definite integrals of w(t): definition of integral of w(t)=u(t)+iv(t) from a to b as a sum of integrals of u(t) and iv(t) from a to b, examples, generalization to improper interals; properties of the integral: linearity, additivity with respect to the domain of integratoin; the Fundamental Theorem of Calculus holds for w(t); review of the Mean Value Theorem for real-valued functions (in terms of derivatives and in terms of integrals), the Mean Value Theorem of Calculus fails for complex-valued functions w(t) (Sec. 38).
  Contours: definitions of an arc in C, a simple arc (Jordan arc), a simple closed curve, a positively (negatively) oriented simple closed curve; examples; changing of the parameterization of a curve in C by t=φ(τ), where φ is a strictly monotonically increasing real-valued function of a real variable; linear reparameterization taking τ∈[α,β] into t∈[a,b]; a differentiable arc; length of a differentiable arch; the length of a differentiable arch does not depend on the parameterization (think how you can prove this!) (Sec. 39).
• Lecture 18 (Thu, Mar 15): Contours (cont.): proof that the length of a differentiable arch does not depend on the parameterization; unit tangent vector to a curve, a smooth arc, a contour (a piecewise smooth arc), a simple closed contour, Jordan curve theorem (Sec. 39).
  Contour integrals: definition of a contour integral, invariance of the value of a contour integral with respect to change of parameterization of the contour, linearity of the contour integral, opposite contour (-C) of a contour C, definition of the contours C₁+C₂ (if the start of C₂ concides with the end of C₁) and C₁-C₂ (if the ends of C₁ and C₂ concide), the contour integral over C₁+C₂ is equal to the sum of the contour integrals over C₁ and C₂ (Sec. 40).
  Some examples: Example 1 (Sec. 41).
• Lecture 19 (Tue, Mar 27): Some examples (cont.): more examples of contour integrals (Sec. 41).
  Examples with branch cuts: examples of contour integrals in the presence of a branch cut (Sec. 42).
• Lecture 20 (Thu, Mar 29): Upper bounds for moduli of contour integrals: Lemma (the modulus of an integral of a function w:[a,b]→C is no greater than the integral of the modulus of the function); Theorem (an upper bound of the modulus of the contour integral of a piecewise continuous complex-valued function defined at the points of the contour); examples of application of the Theorem (Sec. 43).
• Lecture 21 (Tue, Apr 3): Antiderivatives: definition of an antiderivative of a function defined on a domain; elementary observations (antiderivatives are analytic functions, antiderivatives are defined up to an additive constant); Theorem on the equivalence on (i) the existence of antiderivative, (ii) path-independence of contour integrals, and (iii) vanishing of the contour integrals for any closed contour (with a complete proof); an example of application (Example 1); analogies with the case from multivariable Calculus (potential forces and potential energy); please read Examples 2-4 of Sec. 44 (Sec. 44, Sec. 45).
• Lecture 22 (Thu, Apr 5): Cauchy-Goursat theorem: statement and proof of the theorem (assuming that the derivative of f is continuous), an example of application (Sec. 46).
  Simply connected domains: definition of a simply connected domain, Theorem stating that an integral of a function analytic in a simply connected domain D over a closed contour in D is zero, an example of application, Corollary that a function analytic in a simply connected domain D has an antiderivative everywhere in D, entire functions have antiderivatives everywhere in C (Sec. 49).
  Multiply connected domains: Theorem claiming the vanishing of the sum of the integrals of a function f on a big contour (in positive direction) and small contours (in negative direction) inside the big one (Sec. 49).
• Lecture 23 (Tue, Apr 10): Multiply connected domains (cont.): Corollary about equality of integrals over identically oriented contours between which the integrand is analytic, examples (Sec. 49).
  Cauchy integral formula: statement and proof of Cauchy integral formula, an example of application (Sec. 50).
• Lecture 24 (Thu, Apr 12): An extension of the Cauchy integral formula: an extension of the Cauchy integral formula for the f⁽ⁿ⁾(z), a formal "proof" (math majors should read the complete proof) (Sec. 51).
  Some consequences of the extension: Theorem 1 (if f(z) is analytic at z₀, then all derivatives of f(z) are analytic at z₀ as well), Corollary (the real and imaginary part of a function analytic at z₀ have continuous partial derivatives of all orders at z₀), Morrera's Theorem, Theorem 3 (Cauchy inequality) (Sec. 52).
  Liouville's theorem and the fundamental theorem of algebra: Liouville's theorem, an idea of the fundamental theorem of algebra (without the proof) (Sec. 53).
  Convergence of sequences: definition and elementary properties of convergent sequences (z_n)_n∈N (Sec. 55).
  Convergence of series: partial sums of a series, definition of a convergent series, sum of a series, relation between convergence and absolute convergence, remainder of a series, a series converges iff the sequence of its remainders tends to 0 (Sec. 56).
  Taylor series: statement of Taylor's theorem (Sec. 57).
• Lecture 25 (Tue, Apr 17): Proof of Taylor's theorem: a complete proof of the theorem (Sec. 58).
  Examples: examples of applications of Taylor's theorem: series for exp(z), sin(z), cos(z), sinh(z), cosh(z), 1/(1-z) for |z)|<1, using the formula for the geometric series to find the Taylor expansion of rational functions, determining the radius of convergence of the Taylor series by calculation and by the geometry of the problem (from the location of the nearest function where the function f is non-analytic with respect to the point z₀ around which f is expanded) (Sec. 59).
  Laurent series: statement of the Laurent's theorem, obtaining the Taylor's theorem as a particular case of Laurent's theorem (Sec. 60).
  Proof of Laurent theorem: the beginning of the proof (Sec. 61).
• Lecture 26 (Thu, Apr 19): Proof of Laurent theorem: finishing the proof of the theorem (Sec. 61). Examples: examples of application of Laurent series (Sec. 62).
• Lecture 27 (Tue, Apr 24): Exam 2.
• Lecture 28 (Thu, Apr 26): Continuity of sums of power series: Theorem on the continuity of the sum of a power series in its disk of convergence (without proof) (Sec. 64).
  Integration and differentiation of power series: theorems on integration and differentiation of power series and on analyticity of the sum of a power series in its disk of convergence (without proofs), examples (Sec. 65).
  Isolated singular points: singular points, isolated singular points, examples (Sec. 68).
• Lecture 29 (Tue, May 1): Residues: a definition of a residue, expression for the residue of a function as a contour integral, examples of applications of residues to computations of contour integrals (Sec. 69).
  Cauchy Residue Theorem: statement and proof of the theorem, an example of application (please read the example from the book) (Sec. 70).
  The three types of isolated singular points: simple poles, removable singular points, essential singular points, examples of the three kinds of isolated sinuglar points (Sec. 72).
• Lecture 30 (Thu, May 3): Residues at poles: Theorem giving a necessary and sufficient condition for an isolated singular point of a function to be a pole of order m, and an explicit expression for the residue at a pole of order m (with sketch of proof), examples (Sec. 73, 74).
  Evaluation of improper integrals: definition of an improper integral, an example of computing a contour integral by using contour integration (Sec. 78, 79).

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent). You should come to class on time; if you miss a quiz because you came late, you won't be able to make up for it.

Grading: Your grade will be determined by your performance on the following coursework:

Homework: It is absolutely essential to solve the assigned homework problems! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. Your homework should have your name clearly written on it, and should be stapled. No late homework will be accepted!

You are encouraged to discuss the homework problems with other students. However, you have to write your solutions clearly and in your own words - this is the only way to achieve real understanding! It is advisable that you first write a draft of the solutions and then copy them neatly. Please write the problems in the same order in which they are given in the assignment.

Quizzes: Short pop-quizzes will be given in class at random times; your lowest quiz grade will be dropped. Often the quizzes will use material that has been covered very recently (even in the previous lecture), so you have to make every effort to keep up with the material and to study the corresponding sections from the book right after they have been covered in class.

Exams: There will be two in-class midterm exams and a comprehensive in-class final. All tests must be taken at the scheduled times, except in extraordinary circumstances. Please do not arrange travel plans that will prevent you from taking any of the exams at the scheduled time.
Tentative date for the midterm exams: February 23 (Thursday), April 20 (Thursday).
The final is scheduled for May 7 (Monday), 1:30-3:30 p.m.

Academic calendar for Spring 2012.

Course schedule for Spring 2012.

Policy on W/I Grades : Through February 24 (Friday), you can withdraw from the course with an automatic "W". In addition, from February 27 (Monday) to May 4 (Friday), you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after April 2 (Monday) requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the Academic Calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

• The Greek_alphabet.
• Some useful notations.