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Course Code    : PMAT 41343

Title                   : Complex Analysis

Pre-requisites   : PMAT 22293

Learning Outcomes:

At the end of this course, the student should be able to

• identify regions in the complex plane
• explain the concepts of analytic functions and harmonic functions and then the important of Cauchy-Riemann equations
• explain the convergence power series and develop analytical capabilities in Taylor or Laurent series in a given domain
• describe the basic properties of singularities and zeros of analytic functions and calculate residues and use these to evaluate real integrals
• describe conformal mappings between various plane regions

 Course Contents:

Complex Numbers, Complex Valued Functions, Limits, Continuity, Differentiability, Analytic functions, Cauchy Riemann Equations, Elementary Functions, Line Integrals, Cauchy-Goursat Theorem, Morera’s Theorem, Cauchy’s Integral Formula, Cauchy’s Inequality, Lioville’s Theorem, Fundamental Theorem of Algebra, Maximum Modulus Principle, Minimum modulus Principle, Taylor and Laurent Series, Singularities, Argument Principle, Rouche's theorem, Residue Theorem, Maximum modulus theorem, Conformal mappings, Schwarz Christoffel Transformation.

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

 Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Brown, J.W. & Churchill, R.V. (9th Ed., 2014). Complex variables and applications, McGraw-Hill.
2. Spiegel, M., Lipschutz, S. & Schiller, J. (2nd Ed., 2009). Schaum’s Outline of Complex Variables, McGraw Hill.
3. Hann, L. & Epstein, B. (1st Ed., 1996). Classical Complex Analysis, Jones and Bartlett Publishers.
4. Ponnusamy, S. (2nd Ed., 2005). Foundation of Complex Analysis, Alpha Science.

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Course Code            : PMAT 32322

Title                          : Mathematical Methods

Pre-requisites          : PMAT 22293

 Learning Outcomes:

At the end of this course, the student should be able to

• classify partial differential equations using various techniques learned
• solve hyperbolic, parabolic and elliptic equations using fundamental principles
• apply range of techniques to find solutions of standard PDEs
• demonstrate accurate and efficient use of Fourier analysis techniques and their applications in the theory of PDE’s
• solve real world problems by identifying them appropriately from the perspective of partial derivative equations.

 Course Contents:

Partial Differential Equations (PDEs): Classification of PDE, First order partial differential equations: Lagrange’s method and Charpit’s method

Second order partial differential equations: Linear Partial Differential Equations with Constant Coefficients, Partial Differential Equations of Order two with Variables Coefficients, Classification of Partial Differential Equations Reduction to Canonical or Normal Form: Parabolic, elliptic and hyperbolic partial differential equations.

Fourier Series: Fourier Series, Even and Odd Functions, Half-Range Expansions, Fourier Integral, Fourier Cosine and Sine Transforms

Solutions of PDEs: Separations of variables, Fourier Transforms, Laplace Transforms, D’Alembert’s Solution of the Wave Equation

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.
2. 2. Zill, D.G. (7th Ed., 2020). Advanced Engineering Mathematics, Jones & Bartlett Learning.
3. 3. Raisinghania, M.D. (19th Ed., 2018). Advanced Differential Equations, S.Chands, India.

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Course code  :  PMAT 32332

Title               :  Geometry

Pre-requisites :  PMAT 22293

Learning outcomes:

Upon successful completion of the course students should be able to

• develop an intuitive understanding of the nature of general equation of second degree
• apply transformations and use symmetry to analyze mathematical situations.
• define conic sections and their properties
• explain and use various techniques for calculating tangents, normal, pair of tangents, pole and polar of conic sections
• compute lines, plane, cone, sphere, ellipsoid and hyperboloid in three dimensions
• analyze characteristics and properties of two dimensional and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
• demonstrate the knowledge of geometry and its applications in the real world.

 Course contents:

Analytical Geometry in Two Dimensions: Pairs of straight lines, General equation of second degree, Translations and rotations of axes, Invariants of transformations

Conic sections: Classifying general equation of second degree into Parabolas, Ellipses, Hyperbolas, eccentricity. Equation of tangent, Pairs of tangents and chords of contact, Harmonic conjugates, Pole and Polar, Parametric treatment, Degenerate conic, Properties of conic.

Analytical Geometry in Three Dimension: Coordinate system, Direction cosines and direction ratios of a line, Angle between two lines, Parallel lines, Perpendicular lines, Plane and Straight line, Shortest distance between two-non intersecting lines, Skewed lines, General Equation of the second degree, Sphere, Cone, Ellipsoid and hyperboloid, Tangent plane, Normal, Pole and Polar.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Chatterjee, D. (2008). Analytical Geometry, Alpha Science International.
2. Thomas, G.B. & Finney, R.L. (2008). Calculus and Analytic Geometry.
3. Maxwell, E.A. (1962). Elementary Coordinate Geometry, Oxford University press.
4. Jain, P.K. & Ahmad, K. (1994). Analytical Geometry of Two Dimensions, Wiley.
5. Kishan, H. (2006). Coordinate Geometry of Two Dimensions. Atlantic Publishers & Dist.
6. Jain, P.K. (2005). A Textbook of Analytical Geometry of Three Dimensions. New Age International.
7. McCrea, W.H. (2006). Analytical Geometry of Three Dimensions, Dover Publications, INC.

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Course Code            : PMAT 31212

Title                          : Mathematics for Computing III

Pre-requisites          : PMAT 12212

 Learning Outcomes:

At the end of the course the student should be able to

• demonstrate knowledge in basic mathematical concepts of calculus
• use basic mathematical concepts of calculus for further studies
• categorize ordinary differential equations
• solve linear ordinary differential equations using appropriate methods
• derive the ordinary differential equations for certain applications.

 Course Contents:

Limits and Derivatives: Limit of a Function, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity-Horizontal Asymptotes, Derivatives and Rates of Change, Derivative as a Function, Differential rules, Applications of Differentiation

Integrals: Definite Integral, Fundamental Theorem of Calculus, Substitution Rule

Applications of Integration: Areas between Curves, Volumes, Volumes by Cylindrical Shells, Arc Length, Area of a Surface of Revolution, Techniques of Integration

Ordinary Differential Equations (ODEs): Introduction to Differential Equations, ODEs, Order, Degree, classification of linear and non-linear ODEs, solution of a differential equation, Family of curves, first order ODEs: Separable ODEs, Exact ODEs, Integrating Factors, Linear ODEs, Higher Order Linear ODEs: Homogeneous Linear ODEs, Homogeneous Linear ODEs with Constant Coefficients, Differential Operators, Special types of ODEs: Bernoulli Equations, Euler–Cauchy Equations, Applications of ODEs.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.
2. Ross, K.S. (2nd Ed., 2015). Elementary Analysis: The Theory of Calculus, Springer.
3. Stewart, J. (8th Ed., 2015). Calculus Early Transcendentals, Thomson Learning, Inc.