## Complex Analysis for Applications

What's new:

• Some basic info about the final.  (Mar 15)
• Regarding the question of notes etc. for the midterms: no notes, books or calculators are allowed for the mid-terms.  A 5x7 index card will be allowed for the final.  (Feb 4)

Course Information:

 Official stuff Stuff specific to 132/1 Winter 2000

Lecture notes:

• Week 1: Complex arithmetic, complex sets, limits, differentiation, Cauchy-Riemann equations. [pdf]
• Week 2: Complex analytic functions, harmonic functions, Möbius transforms.   [pdf]
• Week 3: Möbius transforms, complex exponential, trig, hyperbolic, and log functions.   [pdf]
• [Errata: in the last two displays on page 22, e^y and e^{iy} should be switched. Thanks to Andrew Solomon for this correction.]
• Week 4: Complex powers, inverse trig functions, review for first midterm.   [pdf]
• Week 5: Contour integration, Fundamental theorem of calculus, Cauchy theorems and applications.   [pdf]
• [Errata: On the third line of page 43, a "dt" is missing in the equation before "and we have". In the final display of page 50, e^z should be sin(z).]
• Week 6: Power series, Taylor series, Laurent series.   [pdf]
• Week 7: No lecture notes this week.
• Week 8: Zeroes, singularities, the point at infinity.   [pdf]
• Week 9: The residue theorem; trig integrals, rational integrals; trig-rational integrals.  [pdf]
• Week 10: Principal value integrals; integrals with branch cuts; argument principle; Rouche's theorem [pdf]

Errata in Week 6 (thanks to Nick Chan):

·        p.6, 2nd line, the 2nd term in the right hand side, w_n is placed wrongly

·        p.7, 3rd line, z_1 is missing in the series expression

·        p.10, in the comparison test, z_n is missing in the second summation

·        p.26, -3rd line, a_n * ( z - z_0 )^n is missing in the summation

Erratum in Week 10 (thanks to Chan-Ho Suh): z^5+z+1 should be z^5-z-1 throughout.

Sample Exams:

 Sample quiz:  DVI format PostScript format Solutions Sample first midterm:  DVI format PostScript format Solutions Sample second midterm:  DVI format PostScript format Solutions Sample final:  DVI format PostScript format Solutions

Java applications

 Applet 1: The complex plane Applet 2: Elementary complex maps Applet 3: Möbius transforms Applet 4: Multi-valued functions Applet 5: The complex derivative Applet 6: The complex integral Applet 7: Laurent series

These applets are not officially part of the course, but are meant to illustrate some geometric aspects of complex arithmetic and analysis.  Of course, your browser has to be Java-enabled to view these applets properly (for Netscape, this means checking the Enable Java box in "Edit/Preferences/Advanced"; for IE, check the Java JIT compiler enabled box in "View/Internet Options/Advanced").