Course Syllabus
This syllabus outlines the modules of Engineering Mathematics – II.
- Sequences: convergence, divergence, and boundedness; Cauchy sequences
- Infinite series of positive terms; comparison and limit comparison tests
- Ratio test (D'Alembert) and root test (Cauchy); alternating series; Leibniz test
- Absolute and conditional convergence; power series; radius of convergence
- Uniform convergence of sequences and series of functions; Weierstrass M-test
- Functions of a complex variable; limits, continuity, and differentiability
- Analytic functions; Cauchy–Riemann equations in Cartesian and polar forms
- Harmonic functions; conjugate functions and conformal mappings
- Complex integration; Cauchy's integral theorem; Cauchy's integral formula
- Taylor series and Laurent series; classification of singularities
- Residues; Cauchy's residue theorem; evaluation of real definite integrals
- First-order ODEs: separable, exact, homogeneous, and linear equations
- Integrating factor; Bernoulli's equation and Clairaut's equation
- Second-order linear ODEs with constant coefficients; auxiliary equation
- Method of undetermined coefficients; variation of parameters
- Euler–Cauchy equation; simultaneous linear ODEs; series solution
- Definition and existence conditions; Laplace transforms of standard functions
- Properties: linearity, first and second shifting theorems, scaling
- Laplace transform of derivatives and integrals; multiplication by $t^n$
- Inverse Laplace transform; partial fractions and convolution theorem
- Solution of initial-value problems using Laplace transforms
- Periodic functions; Dirichlet conditions; Euler's formulae
- Fourier series of functions defined on $[-\pi,\pi]$ and $[0,L]$
- Even and odd functions; half-range cosine and sine series
- Parseval's identity; convergence at points of discontinuity
- Harmonic analysis from tabulated data (practical Fourier analysis)