This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted. This course covers the algebra of complex numbers, the geometry of the complex plane, the spherical representation, the Analytic functions and the rational functions.

**Assessment**

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

**Certification**

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Course Credit: MIT

### Course Curriculum

Module 01 | |||

lecture1 The algebra of complex numbers | 00:05:00 | ||

lecture2 Exponential function and logarithm for a complex argument | 00:10:00 | ||

lecture3 Analytic functions; rational functions | 00:05:00 | ||

lecture4 Power series | 00:05:00 | ||

lecture5 Exponentials and trigonometric functions | 00:05:00 | ||

lecture6 Conformal maps; linear transformations | 00:05:00 | ||

lecture7 Linear transformations (cont.) | 00:10:00 | ||

lecture8 Line integrals | 00:05:00 | ||

lecture9 Cauchy-Goursat theorem | 00:05:00 | ||

lecture10 The special cauchy formula and applications | 00:05:00 | ||

lecture11 Isolated singularities | 00:05:00 | ||

lecture12 The local mapping; Schwarz’s lemma and non-Euclidean interpretation | 00:10:00 | ||

Module 02 | |||

lecture13 The general Cauchy theorem | 00:10:00 | ||

lecture14 The residue theorem and applications | 00:10:00 | ||

lecture15 Contour integration and applications | 00:10:00 | ||

lecture16 Harmonic functions | 00:10:00 | ||

lecture17 Mittag-Leffer’s theorem | 00:10:00 | ||

lecture18 Infinite products | 00:10:00 | ||

lecture19 Normal families | 00:05:00 | ||

lecture20 The Riemann mapping theorem | 00:05:00 | ||

lecture21_22 The prime number theorem | 00:20:00 | ||

lecture23 The extension of the zeta function to C, the functional equation | 00:05:00 | ||

Assessment | |||

Submit Your Assignment | 00:00:00 | ||

Certification | 00:00:00 |

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