B.Sc Mathematics Honours 8th Semester
Courses in this semester
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8th Semester syllabus (NEP)
Unit-wise syllabus from the Odisha State Higher Education Council model curriculum. Tap a paper to open it.
Core XX · 4 creditsFunctional Analysis
Course objective
The objective of this course is to introduce students to Lp spaces, Banach spaces, Hilbert spaces, etc. Students will also be exposed to bounded linear operators on Hilbert spaces, which is required to study quantum mechanics, scattering theory and spectral theory, etc. Knowledge of real analysis, measure theory and linear algebra is a prerequisite for this course.
Unit I
Review of metric spaces (not a part of examination), Lp and lp spaces, inequalities (Hölder, Minkowski, Jensen), completeness of Lp, denseness and separability, normed linear spaces, properties of normed linear spaces, continuity of linear maps.
Unit II
Inner product spaces, examples, Hilbert spaces, examples, closed subspaces, existence of a unique element of smallest norm, orthogonal complements and properties, projection theorem, Riesz representation theorem, orthonormal sets, Gram-Schmidt orthonormalization.
Unit III
Orthonormal basis, Fourier expansion, Bessel's inequality, Riesz-Fischer theorem, Parseval's formula, Banach spaces, examples, Hahn-Banach theorem, Baire's category theorem.
Unit IV
Open mapping theorem, closed graph theorem, uniform boundedness principle, duals of Lp[a, b], bounded linear operators on Banach spaces, spectrum of a bounded operator and properties, resolvent set and examples.
Learning outcomes
- Handle inequalities in Lp spaces and normed linear spaces.
- Learn all basic results on Hilbert spaces for further application.
- Know about Fourier series with respect to an orthonormal basis and related results, and basic results of Banach spaces.
- Know more results on Banach spaces and bounded linear operators with spectrum on Banach spaces.
Books recommended
- B. V. Limaye, Functional Analysis, 3rd Ed., 2014.
- E. Kreyszig, Functional Analysis, Wiley-India, 2007.
Books for reference
- Goffman and Pedrick, A First Course in Functional Analysis, AMS, 2017.
- J. B. Conway, A Course in Functional Analysis, 2nd Ed., Springer, 2006.
- P. K. Jain and O. P. Ahuja, Functional Analysis, 2nd Ed., New Age International Publication, New Delhi, 2004.
- Markus Haase, Functional Analysis: An Elementary Introduction, American Mathematical Society, 2014.
Core XXI · 4 creditsAnalytic Number Theory
Course objective
The aim of this course is to study number theory by using analytic tools such as inequalities, limits, calculus, etc.
Unit I
The arithmetical functions and their relations, Möbius function, Euler totient function, Mangoldt function, Liouville's function, the divisor function, the Dirichlet product of arithmetical functions, Dirichlet inverses and Möbius inversion formula, multiplicative functions, the Bell series of an arithmetical function and Dirichlet multiplications, derivatives of arithmetical functions, the Selberg identity.
Unit II
The big oh notation, Euler's summation formula, some elementary asymptotic formulas, averages of arithmetical functions, the average order of divisor functions, the average order of Euler totient function, the average order of Möbius and Mangoldt functions, the partial sums of a Dirichlet product, applications to the Möbius and Mangoldt functions, some elementary theorems on distribution of prime numbers, Chebyshev's functions and their relations with π(x).
Unit III
Some equivalent forms of the prime number theorem, Shapiro Tauberian theorem, the partial sums of the Möbius function, brief sketch of an elementary proof of the prime number theorem, Ramanujan's sum and generalizations, quadratic residues, Legendre's symbol and its properties, Gauss's lemma, the quadratic reciprocity law, the Jacobi symbol.
Unit IV
The half-plane of absolute convergence of a Dirichlet series, Euler products, analytic properties of Dirichlet series, mean value formulas for Dirichlet series, an integral formula for the coefficients and the partial sums of a Dirichlet series, the Riemann zeta function and the L-function, properties of the gamma function, integral representation for the Hurwitz zeta function, analytic continuation of the Hurwitz zeta function, analytic continuation of the Riemann zeta function and the L-function.
Learning outcomes
- Understand the arithmetical functions and their relations.
- Find the average order of multiplicative functions and know the distribution of prime numbers.
- Know the prime number theorem and Ramanujan's sum.
- Know the basic theory of Riemann zeta function and related L-function.
Books recommended
- T. M. Apostol, Introduction to Analytic Number Theory, Springer International Edition, 2010.
- Jean-Marie De Koninck, Florian Luca, Analytic Number Theory: Exploring the Anatomy of Integers, American Mathematical Society, 2012.
Books for reference
- Jeffrey Stopple, A Primer of Analytic Number Theory: From Pythagoras to Riemann, Cambridge University Press, 2003.
Core XXII · 4 creditsComplex Analysis-II
Course objective
This course introduces the basic concepts of conformal mappings, entire functions, Weierstrass infinite products, Hadamard's factorization theorem, Gamma function, Zeta function and normal family.
Unit I
Mappings of elementary functions and cross ratio, bilinear transformations and its properties, mapping of some elementary functions, mappings of z², eᶻ, sin z, log z, z + 1/z, etc., conformal mappings.
Unit II
Maximum modulus theorems, Schwarz lemma, argument principle, Rouché's theorem, applications to fundamental theorem of calculus, uniqueness and identity theorems, Hurwitz's theorem, harmonic functions, mean value theorem, Poisson integral formula, Harnack's inequality and theorem, Hadamard three circle theorem.
Unit III
Weierstrass' factorization theorem, Gamma function and its properties, Riemann zeta function, Riemann's functional equation, Mittag-Leffler's expansion theorem and its applications, analytic continuation, uniqueness of direct analytic continuation, uniqueness of analytic continuation.
Unit IV
Canonical products, Jensen's formula, Poisson-Jensen formula, Hadamard's three circles theorem, order of an entire function, exponent of convergence, Borel's theorem, Hadamard's factorization theorem, equicontinuity, normal family, families of analytic functions.
Learning outcomes
- Solve problems involving conformal mappings.
- Understand the applications of Cauchy integrals and properties of harmonic functions.
- Handle Gamma function, Riemann zeta function and become familiar with analytic continuations.
- Solve problems involving infinite products, equicontinuity and normal family.
Books recommended
- L. V. Ahlfors: Complex Analysis, McGraw Hill, 3rd Edition, 2017.
- S. Ponnusamy and Herb Silverman: Complex Variables with Applications, Birkhauser, 2006 (Indian Edition 2012).
Books for reference
- J. Bak and D. J. Newman: Complex Analysis (3rd Edition), Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997.
- H. A. Priestley: Introduction to Complex Analysis, Oxford University Press, 2008.
- D. Sarason: Complex Function Theory, AMS, Second Edition, 2007.
- E. M. Stein and R. Shakarchi: Complex Analysis, Princeton University Press, Princeton, New Jersey, 2003.
- John B. Conway: Functions of One Complex Variable, Springer International Student Edition, Narosa Publishing House, Second Edition, 2002.
- R. V. Churchill, J. W. Brown and R. F. Verhey: Complex Variables and Applications, McGraw Hill, 9th Edition, 2013.
Core XXIII · 4 creditsDifferential Equations-III
Course objective
The objective of this course is to: (a) understand the basic methods for qualitative behavior of solutions of ordinary differential equations and boundary value problems; (b) understand the basic methods to solve system of differential equations; (c) expose students to some real-life problems using the system of differential equations; (d) expose the students to canonical forms of hyperbolic, elliptic and parabolic PDEs.
Unit I
Oscillation of Second Order Linear Differential Equations: fundamental results, Sturm's Comparison Theorem and Hille-Wintner type oscillation. Second Order Boundary Value Problem: Sturm-Liouville differential equation, eigenvalue problems, Green's function and Picard's Theorem.
Unit II
System of first order equations, existence and uniqueness theorems, fundamental matrix, homogeneous and non-homogeneous linear systems with constant coefficient, mathematical formulation of predator-prey model, epidemic model of influenza, battle model and their solutions.
Unit III
One dimensional heat equation and its origin, heat conduction problem for an infinite rod and finite rod, existence and uniqueness of solution, two dimensional heat equation and Laplace equation, boundary value problems, maximum and minimum principles, uniqueness and continuity theorems, Dirichlet problem for a circle, Dirichlet problem for annulus, Neumann problem for a circle.
Unit IV
Solution of heat equation, wave equation, Laplace equation and Helmholtz equation by Green's function method and examples.
Learning outcomes
- Handle oscillation properties of ordinary differential equations and Sturm-Liouville differential equations.
- Know existence and uniqueness theorems and application to mathematical modelling.
- Learn solutions of heat equation and various boundary value problems for Laplace equation.
- Solve various PDEs using Green's function.
Books recommended
- Deo and Raghavendra, Text Book of Ordinary Differential Equations, Tata McGraw-Hill Pub. Company Ltd., New Delhi, 2017.
- Belinda Barnes and Glenn R. Fulford, Mathematical Modeling with Case Studies, A Differential Equation Approach using Maple and Matlab, 2nd Ed., Taylor and Francis Group, London and New York, 2009.
- Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, 4th edition, Birkhauser, Indian reprint, 2014.
Books for reference
- K. Shankar Rao, Introduction to Partial Differential Equations, PHI Learning Private Ltd., 2011.
- J. N. Sharma, K. Singh, Partial Differential Equations for Engineers and Scientists, Narosa, 2nd Edition, 2009.
- Robert C. McOwen, Partial Differential Equations, Pearson Education Inc., 2002.
- Martin Braun, Differential Equations and their Applications, Springer International Student Ed., 1978.
- S. L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, India, 2014.
Previous year questions (PYQ)
PYQs for this semester will be uploaded soon. Ask on WhatsApp for PYQ solutions in the AIFM app.
Demo classes
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What you get in the full course
Frequently asked questions
What is the syllabus of B.Sc Mathematics Honours 8th Semester in Odisha (NEP)?
It has 4 papers: Functional Analysis, Analytic Number Theory, Complex Analysis-II, Differential Equations-III. The unit-wise syllabus is on this page, and the syllabus PDF can be opened or downloaded.
What is the fee for the B.Sc Mathematics Honours 8th Semester course at AIFM EDUCATION?
The 8th Semester course is coming soon. Message AIFM EDUCATION on WhatsApp to be notified.
Where can I find B.Sc Mathematics Honours 8th Semester previous year question papers (PYQ)?
PYQs for this semester will be added to this page soon. PYQ solutions are available in the AIFM EDUCATION app.
Is this course for all universities in Odisha?
Yes. It follows the NEP 2020 model curriculum of the Odisha State Higher Education Council, used by universities across Odisha. Classes are explained in Odia, with notes in English.