Table of Contents
Study Materials
Readings
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Saff, Edward B., and Arthur David Snider. Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002. ISBN: 0139078746. The problems bold-faced below seemed especially well suited for the students' practice and for discussion at the recitations. | lec # | TOPICS | readings | reading assignments |
|---|---|---|---|
| I. Complex Algebra and Functions | |||
| 1 | Algebra of Complex Numbers Complex Plane Polar Form | Read 1.1 - 1.3 | p5: 8, 9,24 p12: 6, 7fq,16 p22: 6, 7eq, 8, 11, 17 |
| 2 | cis(y) = exp(iy) Powers Geometric Series | Read 1.4, 1.5 | p31: 1, 3, 4, 11, 12a,14 p37: 4, 5d, 9, 10, 11, 21b |
| 3 | Functions of Complex Variable Analyticity | Read 2.1 - 2.3 | p56: 4, 5 p63: 6, 7, 11cd p70: 4, 7c, 13, 15 |
| 4 | Cauchy-Riemann Conditions Harmonic Functions | Read 2.4, 2.5 | p77: 1, 3, 6, 8, 11 p84: 2, 3bc, 6, 9, 12, 18 |
| 5 | Simple Mappings: az+b, z2, √z Idea of Conformality | Skim pp. 377-79, 383-87 | p57: 7, 8, 9, 11,13 p71: 8 p392: 1, 3ae |
| 6 | Complex Exponential | Reread 1.4 Begin 3.2 | p32: 20a, 23a p115: 3, 5ab, 9a, 14a, 17ab, 20 |
| 7 | Complex Trigonometric and Hyperbolic Functions | Finish 3.2 | p115: 5cdef, 11, 14bcd, 17c, 18b, 23 |
| 8 | Complex Logarithm | Read 3.3 | p123: 1, 4, 5, 6, 8, 11, 12, 19 |
| 9 | Complex Powers Inverse Trig. Functions | Read 3.5 | p136: 1, 4, 5, 7, 9, 11, 19 |
| 10 | Broad Review ... Probably focusing on sin-1z | ||
| II. Complex Integration | |||
| 11 | Contour Integrals | Read 4.1, 4.2 | p160: 2, 4, 5, 13 p171: 3b, 6c, 9, 10, 11, 14 |
| 12 | Path Independence | Read 4.3 Skim 4.4a | p178: 1aeh, 2, 5, 6, 7, 11 |
| Exam 1 | |||
| 13 | Cauchy's Integral Theorem | Read 4.4b | p200: 6, 9, 10, 13, 15, 17, 18, 20 |
| 14 | Cauchy's Integral Formula Higher Derivatives | Read 4.5 | p212: 1, 3abd, 5, 6, 8, 13 |
| 15 | Bounds Liouville's Theorem Maximum Modulus Principle | Read 4.6 | p219: 1, 7, 10, 14, 15, 16, 18 |
| 16 | Mean-value Theorems Fundamental Theorem of Algebra | Skim 4.7 | p225: 4, 6, 7, 11 + 14?! |
| 17 | Radius of Convergence of Taylor Series | Read 5.2, 5.3 | p249: 1ad, 4, 5be, 11ab, 18a p259: 7, 13a |
| III. Residue Calculus | |||
| 18 | Laurent Series | Read 5.5 | p276: 3, 5, 6, 7a, 10 |
| 19 | Poles Essential Singularities Point at Infinity | Read 5.6, 5.7 | p285: 1, 2, 3, 5, 8, 12 p117: 25 p290: 1, 4, 6 |
| 20 | Residue Theorem Integrals around Unit Circle | Read 6.1, 6.2 | p313: 1adq, 3abce, 5, 7 p317: 2, 6, 9 |
| 21 | Real Integrals From -∞ to +∞ Conversion to cx Contours | Read 6.3 | p325: 1, 4, 6, 10, 11, 13 |
| 22 | Ditto ... Including Trig. Functions Jordan's Lemma | Read 6.4 | p336: 1, 3, 7, 11, 12 |
| Exam 2 | |||
| 23 | Singularity on Path of Integration Principal Values | Read 6.5 | p344: 1a, 3, 5, 10, 12 |
| 24 | Integrals involving Multivalued Functions | Read 6.6 Skim 6.7 | p354: 1, 2, 4, 8, 10, 13 p364: 7, 8, 9 (all 3 w/o Rouche!) |
| IV. Conformal Mapping | |||
| 25 | Invariance of Laplace's Equation | Read 7.1 | p374: 1, 3, 4 |
| 26 | Conformality again Inversion Mappings | Read 7.2+3 to p389 | p382: 1b, 3, 7*, 11, 13 (* see also p57: 6 + p85: 11) |
| 27 | Bilinear/Mobius Transformations | Finish 7.3 Skim 7.4 | p392: 3cd, 6, 7b, 8, 11 p405: 18 |
| 28 | Applications I | Read 7.6 | 430: 1, 2, 3, 6, 10 p416: 4(!) |
| 29 | Applications II | Read 7.7 | p440: 3, 5, 6 |
| V. Fourier Series and Transforms | |||
| 30 | Complex Fourier Series | Read 8.1 to p453 | p459: 1a, 2ac, 5, 7ac |
| 31 | Oscillating Systems Periodic Functions | Skim 3.6 | p143: 1b, 3, 5 p393: 13(!) p461: 9, 10 |
| 32 | Questions of Convergence Scanning Function Gibbs Phenomenon | Finish 8.1 | p460: 3, 6, 8, 11 |
| 33 | Toward Fourier Transforms | Read 8.2 | p473: 1abd, 2, 3abc |
| 34 | Applications of FTs | Read 8.2 again | p473: 6abc, 7, 8 |
| Exam 3 | |||
| 35 | Special Topic: The Magic of FFTs I | Reread pp. 457-59 | p462: 12(!) |
| 36 | Special Topic: The Magic of FFTs II | ||
| Final Exam | |||