General: Course info General Syllabus Daily Syllabus Homework: HW 1 due Fri 1/29 HW 2 due Fri 2/5 HW 3 due Mon 2/15 HW 4 due Fri 2/19 HW 5 due Mon 2/29 HW 6 due Fri 3/11 HW 7 due Fri 3/25 HW 8 due Mon 4/4 HW 9 due Fri 4/8 HW 10 due Fri 4/15 HW 11 due Mon 4/25 HW 12 due Wed 5/4 Exam Reviews: Review Exam 1 Review Exam 2 Review Exam 3 Final Review Return to: 313 Home Department Home UNM MATH 313: Lecture Summaries and Homework Lecture 1: Complex numbers.   Wed, Jan 20, 2016.                     Definitions, algebra in cartesian form, geometric interpretation                     in the complex plane, some properties. Triangle inequality.                     Lecture Summary,   HW 1: 1-4   Due: Fri 1/29 Lecture 2: Euler's formula.   Fri, Jan 22, 2016.                     Polar representation, Euler's formula, algebra in exponential form.                     Lecture Summary,  HW 1: 5-6  Due: Fri 1/29 Lecture 3: Powers and roots. Fundamental Theorem of Algebra.   Mon, Jan 25, 2016.                     Algebra in exponential form: multiplication, division, powers, roots. Fundamental Theorem.                     Lecture Summary,  HW 1: 7-10   Due: Fri 1/29 Lecture 4: Sets of numbers in the complex plane.   Wed, Jan 27, 2016.                     Several definitions and examples.                     Lecture Summary,  HW 2: 1-7   Due: Fri 2/5 Lectures 5-6: Functions of a complex variable.   Fri-Mon, Jan 29-Feb 1, 2016.                     Mappings/transformations from x-y to u-v plane. Examples. e^z. Multivalued log(z).                     Lecture Summary,  HW 2: 8-14   Due: Fri 2/5 Lectures 7-8: Limits of functions of a complex variable.   Wed-Fri, Feb 3-5, 2016.                    Mappings/transformations from x-y to u-v plane. Examples. e^z. Multivalued log(z).                     Lecture Summary,  HW 3.   Due: Mon 2/15 Lecture 9: Continuity and Differentiability.   Mon Feb 8, 2016.                    Definitions. Examples. Rules. Lecture Summary Lecture 10-11: Cauchy-Riemann Equations. Analytic functions   Wed-Fri Feb 10-12, 2016.                    Differentiability and C-R: theorem and proof. Examples. C-R in polar coordinates.                   Analytic and entire functions. Lecture Summary,  HW 4.   Due: Fri 2/19 Lecture 12: Exponentials and Logs. Harmonic functions.   Mon Feb 15, 2016.                    Derivatives of exponentials and logs. Harmonic functions, harmonic conjugates.                   Lecture Summary Lectures 13-14: Review and Exam 1. Lecture 15: Integrals int_a^b f(t) dt, where f=x(t)+iy(t).   Mon Feb 22, 2016.                    Integrating f(t). Integrating sin(mx)cos(nx). Inequality for integrals.                   Lecture Summary,   HW 5.   Due: Mon 2/29 Lecture 16: Line Integrals of real valued functions.   Wed Feb 24, 2016.                    Line integrals type I and type II, review. Lecture Summary,   Lecture 17: Line Integrals of functions of a complex variable.   Fri Feb 26, 2016.                    Computing Line integrals int_C f(z) dz. Examples. Lecture Summary   Lecture 18: Cauchy-Goursat.   Mon Feb 29, 2016.                    Path-independence of line integrals of analytic functions Lecture Summary   Lecture 19: More on Cauchy-Goursat. Antiderivatives and path-independence   Wed Mar 2, 2016.                    Path-independence of line integrals of analytic functions Lecture Summary   Lecture 20: Summary so far. Cauchy integral theorem.   Fri Mar 4, 2016.                    Summary and examples. Cauchy integral formula and applications.                   Lecture Summary   Lecture 21: Using Cauchy integral.   Mon Mar 7, 2016.                    Evaluating integrals around curves containing multiple singular points.                   Lecture Summary   Lecture 22: Derivatives of Analytic functions.   Wed Mar 9, 2016.                    Derivatives of all orders, using Cauchy integrals. Morera's Theorem. Applications.                   Lecture Summary   Lecture 23: Computing integrals. Maximum principle.   Fri Mar 11, 2016.                    Integrals with higher order singularities. Maximum principle for analytic functions.                   Lecture Summary   Lectures 24-25: Sequences, Series, Taylor Series.   Wed,Fri Mar 21,23, 2016.                    Convergence of sequence. Convergence of series. Derivation of Taylor series,                   including convergence. Examples. Resulting properties of analytic functions.                   Lecture Summary   Lectures 26-27: Review and Exam 2 Lectures 28-29: Laurent Series.   Fri April 1st, 2016.                    Derivation of Laurent Series. Examples. Lecture Summary   Lectures 30: Examples.   Mon April 4th, 2016.                    Finding Taylor and Laurent Series. Lecture Summary   Lectures 31: Dividing series. Zeros. Singular points.   Wed April 6th, 2016.                    Lecture Summary Lectures 32: The Residue Theorem.   Fri April 8th, 2016.                    Lecture Summary Lectures 35-36: Evaluating real integrals. Lecture Summary Lectures 37-38: Mappings. Lecture Summary Lectures 39-40: Review and Exam 3. Lectures 41: Conformal maps. Harmonic functions. Lecture Summary