Calculus, Laplace Transform and Numerical Techniques

Course objectives
1. Develop the ability to solve differential equations using series methods and special functions applicable to engineering problems. 2. Introduce Laplace and Fourier transform techniques for analyzing differential equations and signal-processing applications. 3. Build a strong foundation in complex variable theory and its applications in engineering and applied sciences. 4. Understand the concepts of sequences and series, convergence tests, and their role in mathematical modeling. 5. Enable students to apply advanced mathematical techniques to solve real-world problems in computer science, electronics, communication, and engineering.

Course content
Series Solution & Special Functions: Ordinary and singular point, Series solution about an ordinary point, Solution of equation by Frobenius method to solve second order linear differential equations, formation of Indicial equation, 1. Determination of indicial roots, cases of indicial roots: Distinct roots, equal roots, roots differ by an integer and not differ by an integer. Applications: Fluid mechanics, quantum mechanical wave equation, Heat transfer, Solving electromagnetic field equation. Laplace Transform: Definition of Laplace Transform, Existence Conditions, Linearity Property, Laplace Transform of some standard functions, I and II Shifting theorems (without proof), Change of Scale Property, Laplace transform of derivatives and integrals, Periodic Functions, Unit Step Function, Inverse Laplace Transform, Partial Fraction Method, Convolution theorem (without proof), Application of Laplace transform to solve ordinary differential equation. Applications : Transfer Function Determination, Circuit Equations, Step Response. Fourier Series& Fourier Transform: Periodic function, even and odd function, Dirichlet’s Condition for the Existence of a Fourier Series, Fourier series expansion of a function in the interval c to c+2l , Change of Interval, Half Range Series: Half range sine and cosine series in the interval -l to l, Parseval’s Identity, Fourier Integral theorem, Fourier transform, Inverse Fourier Transform, Finite Fourier Sine and Cosine transform. Applications : Computer Graphics, Cyber Security, Digital Image Processing, Machine Learning Feature Extraction, Image compression. Complex Analysis: Functions of Complex Variables, Limit Continuity and Differentiability, Analytic Function, Cauchy-Riemann Equations (Cartesian and Polar Form), Harmonic Function, Milne- Thomson Method, Complex integrals, Cauchy- Integral theorem (without proof), Cauchy integral formula (without proof), Taylor’s series and Laurent’s series, Singularities, Residue of a function, Cauchy Residue Theorem, Evaluating the Real definite integrals. Applications : Network Theory, Electromagnetic Waves, Transmission line. Unit V -Sequence and Series: Definition of a Sequence, Monotonic Sequence, Convergence and Divergent Sequence, Cauchy Sequence, Necessary 5. Condition for Convergence, Tests of Convergence: Comparison Test, Ratio Test, Raabe’s Test, Logarithmic Test Applications: Discrete Time Signal Representation. Course Outcomes CO Course Outcome Statement KL CO1 Apply series solution and Frobenius methods to solve second- K3 order linear differential equations arising in engineering applications. CO2 Use Laplace transform techniques to solve ordinary K3 differential equations and analyze engineering systems such as electrical circuits and control systems. CO3 Apply Fourier series and Fourier transform methods to K3 analyze periodic functions and solve problems in signal processing, image processing, and machine learning. CO4 Solve complex-valued functions using complex analysis K3 techniques and evaluate complex integrals and residues for engineering applications. CO5 Solve the convergence of sequences and infinite series using K3 appropriate convergence tests and apply these concepts to mathematical modeling and discrete-time systems.