Course info
General Syllabus
Daily Syllabus

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

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            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