## Applied Mathematics 3

### Question Papers

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### Syllabus

**1. Laplace Transform**

**1.1 Laplace Transform of Standard Functions:**Introduction, Definition of Laplace transform, Laplace transform of at 1, e$^{at}$sin(at), cos(at), sinh(at), cosh(at),t$^n$ erf (t) , Heavi-side unit step, dirac-delta function, LT of periodic function.**Properties of Laplace Transform:**Linearity, first shifting property, second shifting property, multiplication by t$^n$ , division by t , Laplace Transform of derivatives and integrals, change of scale property. (without proof)

**2. Inverse Laplace Transform**

**2.1 Inverse Laplace Transform**by Partial fraction method, Convolution theorem**2.2**Application to solve initial and boundary value problem involving ordinary differential equations with one dependent variable and constant coefficients.

**3. Fourier Series**

Dirichlet‟s conditions, Fourier series of periodic functions with period 2$\pi$ and 2L , Fourier series for even and odd functions.

Half range sine and cosine Fourier series, Parsevel‟s identities (without proof)

Complex form of Fourier series, Orthogonal and Orthonormal set of functions.

**4. Complex Variable & mapping**

- Functions of a complex variable, Analytic functions, Cauchy- Riemann equations in Cartesian co-ordinates & Polar co-ordinates.
- Harmonic functions, Analytic method and Milne Thomson methods to find f(z), Orthogonal trajectories.
**Mapping:**Conformal mapping, bilinear transformations, cross ratio, fixed points, bilinear transformation of straight lines and circles.

**5. Z-transform**

- Z-transform of standard functions such as Z(a$^n$), Z(n$^p$).
- Properties of Z-transform :Linearity, Change of scale, Shifting property, Multiplication of K, Initial and final value, Convolution theorem ( without proof)
**Inverse Z transform:**Binomial Expansion and Method of Partial fraction

**6. Correlation & regression, Curve Fitting**

- Scattered diagrams, Karl Pearson‟s coefficient of correlation, covariance, Spearman's Rank correlation(non-repeated and repeated ranks)
- Regression coefficient & Lines of Regression.
- Fitting of curves: Least square method. Fitting of the straight line y = a + bx , parabolic curve, y = a + bx + cx$^2$ & exponential curve y=ab$^x$