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The focus of this course is to analyze the functions of a complex variable and the calculus of residues. Along with these it also covers subjects such as ordinary differential equations, partial differential equations, Bessel and Legendre functions, and the Sturm-Liouville theory. If you are keen to gather some knowledge about Calculus then this course is the perfect diploma course for you.

**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 Number Systems and Algebra of Complex Numbers | 00:10:00 | ||

lecture2 Elementary Complex Functions, Part 1 | 00:10:00 | ||

lecture3 Elementary Complex Functions, Part 2 | 00:10:00 | ||

lecture4 Branch Points and Branch Cuts | 00:10:00 | ||

lecture5 Analytic Functions | 00:15:00 | ||

lecture6 Complex Integrals | 00:10:00 | ||

lecture7 Cauchy’s Formula, Properties of Analytic Functions | 00:05:00 | ||

lecture8 Taylor Series, Laurent Series | 00:10:00 | ||

lecture9 Laurent Series (cont.) | 00:10:00 | ||

lecture10 Properties of Laurent Series, Singularities | 00:10:00 | ||

lecture11 Singularities (cont.) | 00:10:00 | ||

lecture12 Residue Theorem | 00:10:00 | ||

Module 02 | |||

lecture13 Evaluation of Real Definite Integrals, Case I | 00:20:00 | ||

lecture14 Evaluation of Real Definite Integrals, Case II | 00:10:00 | ||

lecture15 Evaluation of Real Definite Integrals, Case III | 00:10:00 | ||

lecture16 Evaluation of Real Definite Integrals, Case IV | 00:15:00 | ||

lecture17 Theorems for Contour Integration | 00:05:00 | ||

lecture18 Series and Convergence | 00:05:00 | ||

lecture19 Ordinary Differential Equations | 00:15:00 | ||

lecture20 Singular Points of Linear Second-order ODEs | 00:10:00 | ||

lecture21 Frobenius Method | 00:10:00 | ||

lecture22 Frobenius Method – Examples | 00:10:00 | ||

lecture23 Frobenius Method (cont.) and a particular type of ODE | 00:15:00 | ||

lecture24 Bessel Functions | 00:15:00 | ||

Module 03 | |||

lecture25 Properties of Bessel Functions | 00:05:00 | ||

lecture26 Modified Bessel Functions | 00:05:00 | ||

lecture27 Differential Equations Satisfied by Bessel Functions | 00:10:00 | ||

lecture28 Introduction to Boundary-Value Problems | 00:05:00 | ||

lecture29 Eigenvalues, Eigenfunctions, Orthogonality of Eigenfunctions | 00:10:00 | ||

lecture30 Boundary Value Problems for Nonhomogeneous PDEs | 00:15:00 | ||

lecture31 Sturm-Liouville Problem | 00:10:00 | ||

lecture32 Fourier Series | 00:15:00 | ||

lecture33 Fourier Sine and Cosine Series | 00:15:00 | ||

lecture34 Complete Fourier Series | 00:15:00 | ||

lecture35 Review of Boundary Value Problems for Nonhomogeneous PDEs | 00:15:00 | ||

Assessment | |||

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

Certification | 00:00:00 |

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