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acm95a_notes - ACM 95a/100a: Complex Numbers Eric Tai June...

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Unformatted text preview: ACM 95a/100a: Complex Numbers Eric Tai June 20, 2007 Contents 1 Complex Variables 4 1.1 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Complex Functions 5 2.1 Complex Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Complex Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Complex Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Integer Powers of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Integer Roots of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.7 Complex Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.8 Branches of z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Periodicity of Complex Functions 7 3.1 Complex Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Complex Functions as Mappings 9 4.1 The Riemann Surface of z 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Stereographic Projection Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Branch Cuts 9 5.1 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Stereographically Projected Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Limits of Complex Functions 11 6.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2 The Complex Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2.1 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2.2 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2.3 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1 CONTENTS CONTENTS 7 Complex Integration 14 8 Contour Integration 15 8.1 Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 8.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Bound Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.4 The ML Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 9 Integration with Antiderivatives 18 10 Cauchy-Goursat Theorem 21 10.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10....
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This note was uploaded on 02/22/2010 for the course ACM 95A taught by Professor Nilesa.pierce during the Fall '06 term at Caltech.

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acm95a_notes - ACM 95a/100a: Complex Numbers Eric Tai June...

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