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Engineering mathematics

Course Description: This course primarily aims to provide the master's student with the following: • In-depth and specialized knowledge and understanding of the fundamental concepts of analytic functions. • Fundamental knowledge and understanding related to the Cauchy integral formula, power series, Cauchy's theorems, Taylor, the maximum modulus principle, Laurent series, residues, Rouche's theorem, the argument principle, and conformal mapping. • Applications of harmonic functions, Laplace's equation in fluid mechanics, and applications of residue theory. • Classifications of partial differential equations and their analytic solutions. • Applications of the Fourier transform.
Credit hours: 3
Objectives of the course :

This course primarily aims to provide the master's student with the following:

• In-depth and specialized knowledge and understanding of the fundamental concepts of analytic functions.

• Basic knowledge and understanding of the integral formula, power series, Cauchy, Taylor, Extreme Value, Laurent, Residue, Routh, Intermediate Value, and Conformal Mapping theorems.

• Applications of harmonic functions, Laplace's equation in fluid mechanics, and residue theorem applications.

• Classifications of partial differential equations and their analytical solutions.

Fourier transform applications.

Course outputs :

Knowledge and understanding:

1.1 Identifying the mathematical concepts of complex analysis, integration, Maclaurin and Taylor series, power series, and conformal mappings in engineering fields in depth.

1.2 Explanation of Mathematical Concepts for Fourier Analysis and Partial Differential Equations, Complex Numbers and Functions, Laurent Series, and Residue Integration.

1.3 Description of technologies related to Fourier series, partial differential equations, and potential theory and their applications.

Skills:

Application of Fourier series, partial differential equations, complex analysis, integration, Maclaurin and Taylor and power series, and conformal mapping concepts in engineering fields.

2.2 Solving advanced problems in engineering fields, including Fourier series, partial differential equations, complex analysis, integration, Maclaurin and Taylor series and powers, and conformal transformations.

2.3 Process Engineering: Quantitative mathematical data analysis in complex and advanced contexts, suitable for the engineering field.

Values:

3.1 Demonstrate integrity and professionalism in all engineering mathematics activities.

Additional information:

Course Content:

1. Derivative; Analytic functions; Cauchy-Riemann equations; Laplace's equation; Exponential function 3.
2. Trigonometric and hyperbolic functions; the complex logarithm; complex integration: complex line integrals 3.
3. Cauchy's Integral Theorem and Formula; Derivatives of Analytic Functions.
Power Series and Taylor Series: Sequences, Series, Convergence Tests; Power Series; Functions Defined by Power Series.
5. Taylor and Maclaurin Series; Laurent Series 3.
6. Singular points and zeros; residue integration method; residue integration for real integrals 3.
7. Conformal Mapping: Geometry of Analytic Functions; Möbius Transformations 3.
8. Special Linear Fractional Transformations; Potential Theory: Electrostatic Fields; Use of Conformal Mapping; Heat Problems; Fluid Flow Poisson Integral Formula 3.
9. Fourier Analysis: Fourier Series, Forced Oscillations; Approximation by Trigonometric Polynomials 3.
10. Sturm-Liouville problems; orthogonal functions; generalized Fourier series 3.
Fourier Integrals, Fourier Transforms; Partial Differential Equations: Modeling 3.
12. Solution by Separation of Variables; D'Alembert's Solution to the Wave Equation 3.
13. Modeling; Heat Equation; Membranes 3.
14. Laplace in Polar, Cylindrical, and Spherical Coordinates; Complex Analysis: Introduction; 6.

References and Learning Resources:

• E. Kreyszig, Advanced Engineering Mathematics, Publisher: Wiley; 10th edition (16 August 2011), ASIN: 0470458364, ISBN-13: 978-0470458365

• Advanced Engineering Mathematics, 6th Edition, by Dennis G. Zill (Author), Publisher: Jones & Bartlett Learning; 6th edition (September 14, 2016), ISBN-10: 1284105903, ISBN-13: 978-1284105902
Differential Equations and Boundary Value Problems: Computing and Modeling, 5th Edition, by C. Edwards, David Penney, David Calvis, Publisher: Pearson; 5th edition (September 4, 2014), ISBN-10: 0321796985, ISBN-13: 978-0321796981

 

  • Last updated: 2026-05-05

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