Advanced Engineering Mathematics

Overview

This course is a basic course offered to UG/PG students of Engineering/Science background. It contains Analytic Functions, applications to the problems of potential flow, Harmonic functions, Harmonic conjugates, Milne’s method, Complex integration, sequences and series, uniform convergence, power series, Hadamard’s formula for the radius of convergence, Taylor and Laurent series, zeros and poles of a function, meromorphic function, the residue at a singularity, Residue theorem, the argument principle and Rouche’s theorem, contour integration and its applications to evaluation of a real integral, integration through a branch cut, conformal mapping, application to potential theory, review of unilateral and bilateral Z-transforms and their properties, application of calculus of residues for the inversion formula of Z- transforms and Laplace transforms, review of Fourier integrals and Fourier transforms, Finite Fourier transforms, discrete Fourier transforms and applications, basic concepts of probability, Bayes theorem, probability networks, discrete and continuous probability distribution, joint distribution, correlation coefficient, applications to problems of reliability, queueing theory, service time for a customer in a facility and life testing, testing of hypotheses. This course has tremendous applications in diverse fields of Engineering and Sciences such as Signal processing, Potential theory, Bending of beams etc.

INTENDED AUDIENCE : UG and PG students of technical institutions/ universities/colleges.

Syllabus 12

Week 1
- Analytic Function
- Cauchy-Riemann Equations
- Harmonic Functions, Harmonic Conjugates and Milne's Method
- Applications to the Problems of Potential Flow-I
- Applications to the Problems of Potential Flow-II
Week 2
- Complex Integration
- Cauchy's Theorem-I
- Cauchy's Theorem-II
- Cauchy's Integral Formula for the Derivatives of Analytic Function
- Morera's Theorem, Liouville's Theorem and Fundamental Theorem of Algebra
Week 3
- Winding Number and Maximum Modulus Principle
- Sequences and Series
- Uniform Convergence of Series
- Power Series
- Taylor Series
Week 4
- Laurent Series
- Zeros and Singularities of an Analytic Function
- Residue at a Singularity
- Residue Theorem
- Meromorphic Functions
Week 5
- Evaluation of real integrals using residues-I
- Evaluation of real integrals using residues-II
- Evaluation of real integrals using residues-III
- Evaluation of real integrals using residues-IV
- Evaluation of real integrals using residues-V
Week 6
- Bilinear Transformations
- Cross Ratio
- Conformal Mapping-I
- Conformal Mapping-II
- Conformal mapping from half plane to disk and half plane to half plane-I
Week 7
- Conformal mapping from disk to disk and angular region to disk
- Application of Conformal Mapping to Potential Theory
- Review of Z-transforms-I
- Review of Z-transforms-II
- Review of Z-transforms-III
Week 8
- Review of Bilateral Z-transforms
- Finite Fourier Transforms
- Fourier Integral and Fourier Transforms
- Fourier Series
- Discrete Fourier Transforms-I
Week 9
- Discrete Fourier Transforms-II
- Basic Concepts of Probability
- Conditional Probability
- Bayes Theorem and Probability Networks
- Discrete Probability Distribution
Week 10
- Binomial Distribution
- Negative Binomial Distribution and Poisson Distribution
- Continuous Probability Distribution
- Poisson Process
- Exponential Distribution
Week 11
- Normal Distribution
- Joint Probability Distribution-I
- Joint Probability Distribution-II
- Joint Probability Distribution-III
- Correlation and Regression-I
Week 12
- Correlation and Regression-II
- Testing of Hypotheses-I
- Testing of Hypotheses-II
- Testing of Hypotheses-III
- Application to Queuing Theory and Reliability Theory

Advantages and disadvantages

Advantages

Taught by IIT and IISc professors, and it follows the Indian university syllabus closely.
All videos and assignments are free on NPTEL and SWAYAM.
The certificate is recognised by many Indian universities for credit transfer and by GATE aspirants.
Great for GATE and semester exam preparation.
From Indian Institute of Technology Roorkee, a well-regarded name.
Completely free.
Self-paced: start any time.
A clear syllabus (12 parts) you can see before you start.

Disadvantages

The certificate needs a proctored exam at a centre, which has a fee.
Recorded classroom lectures: thorough, but slower than made-for-online courses.
New runs start on fixed dates (January and July).
Learning is free, but the certificate costs money.

Some points apply to every course of this kind; see how we rank.

Free to learn

Free: Every video and assignment is free on NPTEL and SWAYAM. Enrol when the next run opens.
Certificate: Optional. It needs a proctored exam at a centre, which has a fee.