| I. Complex Algebra and Functions |
| 1 |
Algebra of Complex Numbers
Complex Plane
Polar Form |
Read 1.1 - 1.3
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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
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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
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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 |
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| 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 |
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Exam 1 |
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| 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 |
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Exam 2 |
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| 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 |
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Exam 3 |
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| 35 |
Special Topic: The Magic of FFTs I |
Reread pp. 457-59 |
p462: 12(!) |
| 36 |
Special Topic: The Magic of FFTs II |
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Final Exam |
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