MATH 4103 - Introduction to Functions of a Complex Variable, Section 001 - Spring 2016
TR 10:30-11:45 a.m., 117 PHSC

Office Hours: M 2:30-3:30 p.m., T 9:30-10:30 p.m., or by appointment, in 802 PHSC.

First day handout

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, 9th edition, 2014, McGraw Hill, ISBN: 978-0-07-338317-0.
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.

Homework

• Homework 1 (problems assigned on January 19 and 21), due January 26 (Tuesday)
• Homework 2 (problems assigned on January 26 and 28), due February 2 (Tuesday)
• Homework 3 (problems assigned on February 2 and 4), due February 9 (Tuesday)
• Homework 4 (problems assigned on February 9 and 11), due February 16 (Tuesday)
• Homework 5 (problems assigned on February 16 and 18), due February 23 (Tuesday)
• Homework 6 (problems assigned on February 23 and 25), due March 1 (Tuesday)
• Homework 7 (problems assigned on February March 1, 8, 10), due March 22 (Tuesday)
• Homework 8 (problems assigned on March 22, 24), due March 29 (Tuesday)
• Homework 9 (problems assigned on March 29, 31), due April 5 (Tuesday)
• Homework 10 (problems assigned on April 5, 7), due April 12 (Tuesday)
• Homework 11 (problems assigned on April 12, 14), due April 19 (Tuesday)
• Homework 12 (problems assigned on April 19, 21), due April 26 (Tuesday)

Content of the lectures

• Lecture 1 (Tue, Jan 19): A brief tour of the history of the concept of a number: counting objects - introducing the natural numbers N; what does 3−5 mean? - introducing the integers Z; what is 3 divided by 5? - introducing the rational numbers Q; the roots of the quadratic equation x²−2=0 are ±2^1/2 which are not rational numbers (with a proof) - introducing the real numbers R; what are the roots of the quadratic equation x²+1=0? - defining the number i:=(−1)^1/2; introducing the complex numbers C - all numbers of the form x+iy with x,y∈R; every quadratic equation has exactly two roots in C (if counted with their multiplicities).
  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]
  Homework: Sec. 2 / 1(a,b), 9, 11 (pages 4, 5).
• Lecture 2 (Thu, Jan 21): Basic algebraic properties (cont.): explicit forms of the additive inverse of z∈C and the multiplicative inverse of a nonzer z∈C; proof of the uniqueness of the additive and multiplicative inverses; [Sec. 2]
  Further algebraic properties: if z₁z₂=0, then at least one of the factors z₁ and z₂ is equal to 0; subtraction and division of complex numbers; examples; binomial coefficients, Newton binomial formula [Sec. 3]
  Vectors and moduli: analogy between adding vectors in the plane and adding complex numbers; modulus of a complex number (analogous of the length of a vector); the distance between two points z₁ and z₂ in the complex plane is equal to |z₁-z₂|; inequalities involving Re(z), Im(z), and |z|; describing circles and ellipses in the complex plane by equations [Sec. 4]
  Triangle inequality: statement and geometric meaning of the triangle equality, corollaries of the triangle inequality; examples of usage of the triangle inequality and its corollaries to find bounds on distances between points/sets in the complex plane; generalization of the triangle inequality to sums of more than two complex numbers [Sec. 5; skip Example 3]
  Homework: Sec. 3 / 1(a,c) (page 7); Sec. 5 / 1, 3, 5, 8 (pages 13, 14).
  The complete Homework 1 (problems given on January 19 and 21) is due on January 25 (Tuesday).
• Lecture 3 (Tue, Jan 26): Complex conjugates: definition and geometric meaning of a complex conjugate; complex conjugates of sums, products, differences, and quotients of two complex numbers; expressing Re(z) and Im(z) in terms of z and its conjugate; examples [Sec. 6]
  Exponential form: exponential form of a complex number; argument arg(z) and principal value Arg(z) or the argument of a complex number z; Euler's formula; examples [Sec. 7]
  Products and powers in exponential form: derivation of a formula for multiplication, e^iθ₁e^iθ₂=e^{i(θ₁+θ₂)}, division, e^iθ₁/e^iθ₂=e^{i(θ₁−θ₂)}, and integer powers, (e^iθ)ⁿ=e^inθ of complex numbers in exponential form; examples de Moivre's formula; deriving trigonometric identities by using de Moivre's formula; examples [Sec. 8]
  Arguments of products and quotients: a recap of all properties derived earlier; the main point is that arg(z) is a whole set, and arg(z₁z₂)=arg(z₁)+arg(z₂), but Arg(z₁z₂) may differ from Arg(z₁)+Arg(z₂) by 2πn for n∈Z [Sec. 9]
  Homework: Sec. 6 / 1(a,b), 2, 9, 10(a), 13, 14 (pages 16, 17); Sec. 9 / 1, 2, 4, 6, 9, 10 (pages 23-25).
• Lecture 4 (Thu, Jan 28): Roots of complex numbers: derivation of a formula for the nth root of a complex number, principal root [Sec. 10]
  Examples: examples of computing roots [Sec. 11]
  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 interior of a set; the closure of a set; connected sets; domain; region; bounded and unbounded sets; accumulation point of a set. [Sec. 12]
  Homework: Sec. 11 / 1(b), 3, 6, 7 (pages 30, 31); Sec. 12 / 1, 4(d), 5 (pages 34, 35).
  The complete Homework 2 (problems given on January 26 and 28) is due on February 2 (Tuesday).
• Lecture 5 (Tue, Feb 2): Functions and mappings: domain and range of a function ƒ of a complex variable; real-valued functions; real and imaginary parts of a function of a complex variable; examples; polynomials and rational functions of a complex variable; multiple-valued functions; inverse image of a point or a set under a mapping; elementary mappings: translation, rotation (about the origin), reflection (with respect to the x-axis) [Sec. 13]
  The mapping w=z²: geometric methods of visualising functions of complex variables; example of w=z²: constant-u and constant-v lines; mapping the region between the positive x-axis, the positive y-axis, and the parabola xy=1 to the infinite strip 0≤v≤2; mapping a sector of a circle (centered at the origin) by the mapping w=z² [Sec. 14]
  Limits: definition of a limit; uniqueness of the limit; generalization to the case of boundary points; an example: the limit of the function ƒ(z)=5iz+15 as z→1+2i equals 3+5i [Sec. 15]
  Reading assignment (mandatory): (1) read the proof of the Theorem of uniqueness of limits on page 45; (2) read Example 2 on pages 46-47.
  Homework: Sec. 14 / 1(c,d), 3, 4, 6, 8 (pages 43, 44).
• Lecture 6 (Thu, Feb 4): Limits (cont.): examples of computing limits or demonstrating that a limit does not exist [Example 2 from Sec. 15]
  Theorems on limits: relating the existence of a limit of a complex-valued function ƒ(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 (where the denominator does not vanish) [Sec. 16]
  Limits involving points at infinity: Riemann sphere, stereographic projection; 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 ∞; Theorem on equivalent conditions for limits involivng ∞ with the proof of part (2) done in class; examples of using this theorem [Sec. 17]
  Continuity: a definition of continuity in terms of limits; an equivalent "ε-δ" definition of continuity (definition (4)) [page 52 of Sec. 18]
  Reading assignment (mandatory): prove parts (1) and (3) of the Theorem from Sec. 17 (before reading them from the book, try to do prove them by yourselves).
  Homework: Sec. 18 / 1(a,b), 2(a), 3(a,b), 5, 7, 10(a,b) (pages 54, 55).
  Food for Thought*: Sec. 18 / 8, 13 (pages 54, 55).
  *Food for Thought problems are not to be turned in with the regular homework, but solving these problems will be very instructive for you, so please spend some time thinking about them.
  The complete Homework 3 (problems given on February 2 and 4) is due on February 9 (Tuesday).
• Lecture 7 (Tue, Feb 9): Limits (cont.): a detailed solution of Exercise 1(c) from Sec. 18 [page 54]
  Limits involving points at infinity (cont.): a detailed proof that the limit of ƒ(z)=5/z as z→0 is equal to ∞ (directly from the definition).
  Continuity (cont.): continuity of a composition of two continuous functions (Theorem 1, with a proof); if a continuous function is nonzero at a point, then it is nonzero in a neighborhood of this point (Theorem 2, with a proof); the continuity of ƒ(z) at z₀=(x₀,y₀) is equivalent to the continuity of its real u(x,y) and imaginary v(x,y) parts at (x₀,y₀) (Theorem 3, no proof); boundedness of |ƒ(z)| on a closed bounded region S (Theorem 4, no proof), think of examples when some condition in Theorem 4 is violated and the conclusion of the theorem does not hold [Sec. 18]
  Derivatives: definition of derivative; an example of a function without derivative: ƒ(z) equal to the complex conjugate of z (Example 2) [pages 55-57 of Sec. 19]
  Reading assignment (mandatory): read Example 2 [page 56 of Sec. 19]
  Homework: Sec. 20 / 1, 8(b) (page 61); additional problem*.
  * Additional problems are regular problems that should be turned in with the regular homework.
• Lecture 8 (Thu, Feb 11): Derivatives (cont.): examples of finding derivatives directly from the definition of derivative (derivatives of ƒ(z)=z, ƒ(z)=z³), examples of functions that have no derivatives (ƒ(z)=the complex conjugate of z, ƒ(z)=Re(z)), an example of a function that has derivative at one point only (ƒ(z)=|z|²); remarks about derivatives of complex-valued functions of a complex variable [pages 56-59 of Sec. 19]
  Differentiation formulas: derivative of a constant; derivative of ƒ(z)=z²; linearity of derivatives; products and quotients of two functions (with derivation of the formula for derivative of a quotient); chain rule for the derivative of composition of functions; example - deriving the formula for derivative of z⁻² in two ways - by using the quotent formula, and by using the chain rule [Sec. 20]
  Cauchy-Riemann equations: Cauchy-Riemann equations for the real and imaginary parts of a differentiable function [pages 62, 63 of Sec. 21]
  Homework: Sec. 20 / 2, 3, 6(b), 7 (page 61).
  Food for Thought: Sec. 20 / 9 (page 62).
  The complete Homework 4 (problems given on February 9 and 11) is due on February 16 (Tuesday).
• Lecture 9 (Tue, Feb 16): Cauchy-Riemann equations (cont.): Cauchy-Riemann equations for the real and imaginary parts of a differentiable function (finishing the proof) [Sec. 21]
  Suffiient conditions for differentiability: the Cauchy-Riemann equations at (x₀,y₀) and the continuity of the partial derivatives of u and v in an open neighborhood of (x₀,y₀) are sufficient for a function ƒ=u+iv to be differentiable at z₀=(x₀,y₀) [Sec. 23]
  Examples: Example 1: ƒ(z)=x³−3xy²+i(3x²y−y³): u and v satisfy the Cauchy-Riemann equations and their parital derivatives are continuous, so that ƒ is differentiable everywhere, which is not surprising because ƒ(z)=z³, whose derivative is ƒ'(z)=3z²;
  Example 2: ƒ(z)=|z|²=x²+y²: the Cauchy-Riemann equations are satisfied only at (0,0), the partial derivatives of u and v exist and are continuous in a neighborhood of (0,0), so that ƒ is differentiable at (0,0);
  Example 3: ƒ(z)=(complex conjugate of z)²/z for z≠0 and ƒ(z)=0 for z=0: the Cauchy-Riemann equations are satisfied at (0,0), but ƒ'(0) does not exist [Sec. 22]
  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 ƒ(z)=u(x,y)+iv(x,y) using Cartesian coordinates z=x+iy; expressing the real and imaginary parts of a function ƒ(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)) [pages 68, 69 of Sec. 24]
  Reading assignment (mandatory): Examples 1, 2, 3 on pages 67, 68 of Sec. 23.
  Homework: Sec. 24 / 1(a,c), 2(a,b), 3(a,b) (pages 70, 71).
• Lecture 10 (Thu, Feb 18): Polar coordinates (cont.): differentiating the defining relations U(r,θ)=u(X(r,θ),Y(r,θ)), V(r,θ)=v(X(r,θ),Y(r,θ)) and using the chain rule and the defining relations x=X(r,θ)=rcosθ, y=Y(r,θ)=rsinθ to find the partial derivatives of U(r,θ) and V(r,θ): 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. 24]
  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; the sum, the product, the ratio, and the composition of two analytic functions is an analytic function (with sketches of proofs); if ƒ'(z)=0 everywhere in a domain, then ƒ(z) is constant throughout the domain (a discussion of the importance of the condition of connectedness for the conclusion of the theorem) [Sec. 25]
  Reading assignment (optional): the proof of the theorem that if ƒ'(z)=0 everywhere in a domain, then ƒ(z) is constant throughout the domain [pages 73-74 of Sec. 25]
  Homework: Sec. 24 / 4(b), 5, 6, 7 (page 71); Sec. 26 / 1(a,c), 2, 4(a,b) (page 76).
  The complete Homework 5 (problems given on February 16 and 18) is due on February 23 (Tuesday).
• Lecture 11 (Tue, Feb 23) Harmonic functions: Laplace operator (Laplacian) Δ on functions u:Rⁿ→R defined by Δu(x₁,...,x_n):=∂²u/∂x₁²+...+∂²u/∂x_n²; examples of application in physics: heat equation in R³: u_t(t,x,y,z)=α²Δu(t,x,y,z), where u(t,x,y,z) is the temperature at the point (x,y,z) at time t, Δ=∂_xx+∂_yy+∂_zz is the Laplacian, and α is a positive constant (depending on the heat conductivity of the material; wave equation in R³: u_tt(t,x,y,z)=c²Δu(t,x,y,z), where u(t,x,y,z) is a function describing some wave process (i.e., the pressure change in the air due to acoustic waves), and c is the speed of the wave; stationary (steady-state) heat equation in two spatial dimensions: u_xx+u_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; examples [Sec. 27]
  Homework: Sec. 27 / 1, 2 (page 79); additional problem.
• Lecture 12 (Thu, Feb 25) The exponential function: a reminder from real-variable calculus - e is defined as the limit (1+1/n)ⁿ as n→∞, or as e=exp(1), where exp(x) is defined throught its Fourier series for any x∈R; basic property of e^x for x∈R: e^x₁e^x₂=e^x₁+x₂; definition of e^z for z∈C: if z=x+iy, then e^z:=e^xe^iy, where e^iy:=cos(y)+isin(y); proof ot the property e^z₁e^z₂=e^z₁+z₂ for z∈C; derivative of e^z; unexpected properties of e^z: e^z can take any nonzero value in C, e^z is periodic with complex period 2πi; examples [Sec. 30]
  The logarithmic function: definition of log(z); log(z) as a multivalued function; principal value Log(z) of log(z) [Sec. 31]
  Examples: examples illustrating the functions log(z) and Log(z); warning: log(z₁z₂)=log(z₁)+log(z₂), but in general Log(z₁z₂)≠Log(z₁)+Log(z₂) [Sec. 32]
  Homework: Sec. 30 / 3, 5, 7, 12 (pages 89, 90); Sec. 33 / 3 (page 95).
  Food for Thought: Sec. 30 / 2, 10, 11 (pages 89, 90); Sec. 33 / 2 (page 95); show by observing the Cauchy-Riemann equations (and/or considering a simple particular example) that "harmonic conjugate" is not a symmetric relation between two functions, i.e., that if v(x,y) is a harmonic conjugate of u(x,y), then u(x,y) is generally not a harmonic conjugate of v(x,y) (recall that v(x,y) is said to be a harmonic conjugate of u(x,y) if f(z)=u(x,y)+iv(x,y) is analytic function, i.e., if u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations).
  Fun reading: How the General Assembly of the State of Inidiana tried to establish mathematical truth by legislative fiat by redefining the value of π!
  Fun reading: Slide rule -- based on the formula ln(xy)=ln x ln y.
  The complete Homework 6 (problems given on February 23 and 25) is due on March 1 (Tuesday).
• Lecture 13 (Tue, Mar 1) Branches and derivatives of logarithms: multiple-valued versus single-valued functions; branch points and branch cuts; principal branch of log; idea of Riemann surfaces; derivative of log (alghough log is a multiple-valued function, its derivative is a single-valued function - why?) [Sec. 33]
  Some identities involving logarithms: arg vs. Arg, log vs. Log; identities for log of a product and a ratio of complex numbers; proving that log(zⁿ)=nlog(z) by induction and using the basic identity log(z₁z₂)=log(z₁)+log(z₂); defining the nth root of a complex number through log and comparison with the "old" definition [Sec. 34]
  Homework: Sec. 33 / 4, 6, 10, 11 (pages 95-97).
  Food for Thought: Sec. 33 / 5 (page 96).
• Lecture 14 (Thu, Mar 3): Exam 1 [on Sections 1-25, 27, 30-32 covered in Lectures 1-12, Quizzes 1-4, and Homework assignments 1-6]
  Links to quizzes: Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5.
• Lecture 15 (Tue, Mar 8): The power function: evolution of the concept of exponent for real numbers: (1) definition of xⁿ for x∈R and n∈N; (2) establishing the identities xⁿx^m=x^n+m and (xⁿ)^m=x^nm for x∈R, n∈N, and m∈N; (3) defining x⁻ⁿ:=xⁿ for x∈R and n∈N; (4) defining x^1/n by (x^1/n)ⁿ=x for x≥0, n∈N; (5) defining x^m/n:=(x^m)^1/n for x>0, m∈Z, and n∈N; (5) defining x^α:=e^α ln(x); definition of z^c for complex z and c; the principal value of z^c; derivative of z^c [Sec. 35]
  Examples: more examples of complex powers of complex numbers; cautionary examples of violation of some identities like z^cw^c=(zw)^c when principal values of the powers are taken [Sec. 36]
  The trigonometric functions sin z and cos z: definitions and basic properties of the trigonometric and the hyperbolic functions [Sec. 37]
  Homework: Sec. 34 / 3 (page 99); Sec. 36 / 1, 2(b), 3, 7, 9 (page 103); Sec. 38 / 3, 7 (pages 107-108).
  Food for Thought: Sec. 36 / 8 (page 103); Sec. 38 / 1 (page 107).
• Lecture 16 (Thu, Mar 10): The power function (cont.): exponential function with base c∈C, c≠0; derivatives of z^c and c^z; derivative of z^z=e^z log z [Sec. 35]
  The trigonometric functions sin z and cos z (cont.): periodicity of trigonometic functions; expressions for sin(x+iy), cos(x+iy), |sin(x+iy)|, |cos(x+iy)| (pages 104-105 of Sec. 37]
  Zeros and singularities of trigonometric functions: derivation of the locations of the zeros and singularities of sin, cos, tan, sec [Sec. 38]
  Hyperbolic functions: definitions of hyperbolic functions; derivatives of sinh and cosh; relations between hyperbolic and trigonometric functions; basic identity for hyperbolic functions: cosh²z−sinh²z=1; zeros and singularities of hyperbolic functions [Sec. 39]
  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 arcsin(z) in terms of log(z) [Sec. 40]
  Homework: Sec. 38 / 9(b) (page 108); Sec. 39 / 6 (page 111); Sec. 40 / 2, 5, 6 (page 114).
  Food for Thought: Sec. 38 / 5, 14, 16 (pages 108-109); Sec. 39 / 1, 7, 10, 11 (page 111); Sec. 40 / 1(a,d) (page 114).
  The complete Homework 7 (problems given on March 1, 8, 10) is due on March 22 (Tuesday).
• Lecture 17 (Tue, Mar 22): 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); examples; basic rules for differentiation: [z₀w(t)]'=z₀w'(t), (e^z₀t)'=z₀e^z₀t [Sec. 41]
  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. 42]
  Contours: definitions of an arc in C, a simple arc (Jordan arc), a simple closed curve, a positively (negatively) oriented simple closed curve; examples; linear reparameterization taking τ∈[α,β] into t∈[a,b]; a differentiable arc; length of a differentiable arch; unit tangent vector to a curve; a smooth arc; a contour (a piecewise smooth arc); a simple closed contour; Jordan curve theorem [Sec. 43]
  Reading/thinking assignment: 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; the length of a differentiable arch does not depend on the parameterization (think how you can prove this!) [pages 121-122 of Sec. 43]
  Homework: Sec. 42 / 2, 3, 4 (page 119); Sec. 43 / 3 (page 124).
  Food for Thought: Sec. 43 / 1, 4, 5 (pages 123-124).
• Lecture 18 (Thu, Mar 24): Contour integrals: definition of a contour integral; invariance of the value of a contour integral with respect to change of parameterization of the contour (i.e., independence of the value of the contour integral of the way the contour is parameterized); 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. 44]
  Some examples: Example 1 - computing a contour integral by direct parameterization; Example 2 - computing a contour integral by using the Fundamental Theorem of Calculus for contour integrals [pages 127-129 of Sec. 45]
  Reading assignment (mandatory): Example 3 on pages 129-130 of Sec. 45.
  Homework: Sec. 46 / 4, 5, 10, 11 (pages 133-134). Hint for problem 11: in part (b) you may use without proving the following indefinite integrals: integral of (4−y²)^1/2 equals (1/2)y(4−y²)^1/2+2 arcsin(y/2), integral of y²(4−y²)^−1/2 equals −(1/2)y(4−y²)^1/2+2 arcsin(y/2).
  Food for Thought: Sec. 46 / 12 (page 134).
  The complete Homework 8 (problems given on March 22, 24) is due on March 29 (Tuesday).
• Lecture 19 (Tue, Mar 29): Examples with branch cuts: examples of contour integrals in the presence of a branch cut; Problem 47/9(b) [Sec. 46]
  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. 47]
  Homework: Sec. 46 / 6, 9(a), 13 (pages 133-135); Sec. 47 / 1, 2, 4 (pages 138-139).
  Food for Thought: Sec. 47 / 3 (page 139).
• Lecture 20 (Thu, Mar 31): 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) [Sec. 48, Sec. 49]
  Reading assignment (mandatory): Examples 2-4 of Sec. 48.
  Reading assignment (optional): read the ε-δ argument in the computation of the derivative of F(z) (the end of Sec. 49 on page 147).
  Homework: Sec. 49 / 2, 4, 5 (page 147).
  The complete Homework 9 (problems given on March 29, 31) is due on April 5 (Tuesday).
• Lecture 21 (Tue, Apr 5): Cauchy-Goursat theorem: statement and proof of the theorem (assuming that the derivative of ƒ is continuous); an example of application [Sec. 50]
  Simply connected domains: definition of a simply connected domain; Theorem: 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: a function analytic in a simply connected domain D has an antiderivative everywhere in D; Corollary: entire functions have antiderivatives everywhere in C [Sec. 52]
  Homework: Sec. 53 / 1(a,c,d,f) (be specific!), 4, 7 (pages 159-161).
• Lecture 22 (Thu, Apr 7): Multiply connected domains: Theorem claiming the vanishing of the sum of the integrals of a function ƒ on a big contour (in positive direction) and small contours (in negative direction) inside the big one if the function ƒ is analytic on all contours and analytic in the space between the big and the small contours (with proof); Corollary about equality of integrals over identically oriented contours between which the integrand is analytic (Principle of deformation of paths); examples [Sec. 53]
  Cauchy integral formula: statement and proof of Cauchy integral formula; an example of application [Sec. 54]
  Homework: Sec. 53 / 2 (be specific!), 3, 6 (pages 159-161); Sec. 57 / 1(a,b,c), 2(a) (page 170).
  The complete Homework 10 (problems given on April 5, 7) is due on April 12 (Tuesday).
• Lecture 23 (Tue, Apr 12): An extension of the Cauchy integral formula: an extension of the Cauchy integral formula for the ƒ⁽ⁿ⁾(z); a formal "proof" (math majors may read the complete proof in Sec. 56) [Sec. 55, skip Example 3]
  Some consequences of the extension: Theorem 1 (if ƒ(z) is analytic at z₀, then all derivatives of ƒ(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. 57]
  Liouville's theorem and the fundamental theorem of algebra: statement and proof of Liouville's theorem [page 172 of Sec. 58]
  Homework: Sec. 57 / 1(d,e), 2(b), 4, 5, 7 (page 170, 171).
  Food for Thought: Sec. 57 / 3, 10 (page 170, 172).
• Lecture 24 (Thu, Apr 14): Liouville's theorem and the fundamental theorem of algebra (cont.): the fundamental theorem of algebra for the existence of a root of a polynomial equation of arbitrary degree n≥1, with a proof (skipping only the proof of the bound (6) from Sec. 5, pages 12-13); proof that each polynomial of degree n≥1 has exactly n complex roots (counted with their multiplicities) [Sec. 58]
  Convergence of sequences: sequences as functions N→C; definition and elementary properties of convergent sequences (z_n)_n∈N [Sec. 60]
  Convergence of series: definition of a series; partial sums S_N of a series; definition of a convergent series; sum of a series; if a series converges, its nth term converges to zero as n→∞; absolute convergence of a series; absolute convergence implies convergence; remainder ρ_N of a series; a series converges if and only if the sequence of its remainders ρ_N tends to 0 as N→∞; power series [Sec. 61]
  Homework: Sec. 59 / 1 (page 177); 61 / 1, 3, 4 (in Problem 61/3, use directly the definition of convergence from Sec. 60) (page 185)
  Food for Thought: Sec. 59 / 8 (page 178).
  The complete Homework 11 (problems given on April 12, 14) is due on April 19 (Tuesday).
• Lecture 25 (Tue, Apr 19): Taylor series: statement of Taylor's theorem; proof of Taylor's theorem for z₀=0 [Sec. 62, 63]
  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 1/(1+z²) about z₀=0 for |−z²|<1, (i.e., for |z|<1); using the Taylor expansion of 1/(1+z²) to find the Taylor expansion of arctan(z) by indefinite integration and determining the arbitrary additive constant C from arctan(0)=0; discussion of the strange fact that in Calculus the function arctan(x) is well-defined and smooth on R, but its Taylor expansion about x₀=0 only converges on (−1,1) - the reason for this paradox is that the singularities of arctan'(z)=1/(1+z²) at z=±i (not on the real axis!) prevent the disk of convergence of the Taylor series to have radius greater than the distance from 0 (the point about which we expand) and the nearest singularity [Sec. 64]
  Negative powers of (z−z₀): Examples of using Taylor's theorem to obtain expansions containing negative powers of (z−z₀) [Sec. 65]
  Reading assignment: read the proof of Taylor's theorem for z₀≠0 [page 189 of Sec. 63]
  Reading assignment: Example 1 from Sec. 64, illustrating using the geometric series formula to obtain the Taylor series of 1/(1−z) about z₀=i, determining the radius of convergence of the Taylor series by calculation and by the geometry of the problem (from the locations of the nearest point where the function ƒ is non-analytic and the point z₀ around which ƒ is expanded) [pages 190-192 of Sec. 64]
  Homework: Sec. 65 / 2, 4, 6, 9, 11 (pages 196-197).
  Food for Thought: Sec. 65 / 3, 5, 8, 10 (pages 196-197).
• Lecture 26 (Thu, Apr 21): Examples: expanding the function 1/(z+2) about z₀=0 in a Taylor series in the disk of radius 2 centerd at 0 by using the geometric series formula, and using this expansion to find the 527th derivative of 1/(z+2) at 0; expanding the function 1/(2+z) about z₀=3i in a Taylor series in the disk of radius 13^1/2 centerd at 3i by using the geometric series formula, and using this expansion to find the 527th derivative of 1/(2+z) at 3i; using the extension of the Cauch integral formula from Sec. 55 to compute the integrals of the function e^7z/[(z−1)(z−5)²] over a small contour encircling 1 (but not 5), over a small contour encircling 5 (but not 1), over a big contour encircling both 1 and 5, and over a big contour encircling 1 and 5 multiple times.
  Laurent series: statement of the Laurent's theorem; obtaining the Taylor's theorem as a particular case of Laurent's theorem [Sec. 66]
  Examples: expanding the function 1/(z+2) about z₀=0 in a Laurent series in the annulus 2<|z|(<∞) by rewriting the function as 1/(z+2)= z⁻¹/[1−(−2/z)] and using the geometric series formula to expand 1/[1−(−2/z)] in powers of (−2/z) (valid for |−2/z|<1, i.e., for 2<|z|); a sketch of expanding the function 1/[(z−3)(z²+1)] about z₀=1 in the following three domains: (1) in a Taylor series in the disk |z−1|<2^1/2, (2) in a Laurent series in the annulus 2^1/2<|z−1|<2, (3) in a Laurent series in the annulus 2<|z−1|.
  Reading assignment (mandatory): Examples 1-3 from Sec. 68.
  Homework: Sec. 68 / 1, 2, 3, 4 (pages 205-206).
  The complete Homework 12 (problems given on April 19, 21) is due on April 26 (Tuesday).
• Lecture 27 (Tue, Apr 26): Integration and differentiation of power series: statemens of Theorems 1 and 2 - a power series can be differentiated and integrated term by term; Example 2 [pages 213, 215, 216 of Sec. 71]
  On the side: Using complex analysis in applied problems in planar domains:
  • computing the steady temperature [Sec. 118-121];
  • finding the electrostatic potential [Sec. 122-123];
  • determining the two-dimensional fluid flow of an inviscid fluid [Sec. 124-126];
  • using Schwarz-Christoffel transformation to map a polygonal domains into a simpler one and solving the problem in the simpler domain (examples: fluid flow in a chanel through a slit and with an offset, electrostatic potential about an edge of a conducting plate) [Chapter 11].
  "Homework" (not to be turned in): Sec. 68 / 7 (page 206); Sec. 72 / 1, 2, 3, 6 (pages 218, 219); Sec. 73 / 5 (page 225).
• Lecture 28 (Thu, Apr 28): Exam 2 [on Sections 33-48, 50, 52-55, 57, 59-62, 64-66, 68 covered in Lectures 13, 15-26, Quizzes 6-10, and Homework assignments 7-12]
  Links to quizzes: Quiz 6, Quiz 7, Quiz 8, Quiz 9, Quiz 10.
• Lecture 29 (Tue, May 3): Isolated singular points: singular points; isolated singular points; examples [Sec. 74]
  Residues: a definition of a residue of a function ƒ at a point z₀ as the value of the coefficient c₋₁ in its Laurent expansion about z₀; expression for the residue of a function as a contour integral; examples of applications of residues to computations of contour integrals [Examples 1-3] [Sec. 75]
  Cauchy Residue Theorem: statement of Cauchy Residue Theorem; an example of application (please read the example from the book) [Sec. 77]
  "Homework" (not to be turned in): Sec. 77 / 2 (page 237).
• Lecture 30 (Thu, May 5): Applications of residues to compute definite integrals: examples of applying residues to evaluate the values of definite integrals [Sec. 85-92]
• Final exam: 8-10 a.m. on FRIDAY, THE 13TH OF MAY ☺ in 117 PHSC.

Good to know:

• The greek_alphabet.
• Some useful notations.