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SE CSE SEM 3 – ENGINEERING MATHEMATICS III

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Module 1 – Laplace Transform 14 Topics

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1.1.a Definition of Laplace transform

1.1.b Condition of Existence of Laplace transform

1.2 Laplace Transform (L) of Standard Functions like ? ?? , ???(??), ???(??), ???ℎ(??), ???ℎ(??) and ? ? , ?ℎ??? ? ≥ 0

1.3.a Properties of Laplace Transform – Linearity

1.3.b Properties of Laplace Transform – First Shifting Theorem

1.3.c Properties of Laplace Transform – Second Shifting Theorem

1.3.d Properties of Laplace Transform – Change of scale Property

1.3.e Properties of Laplace Transform – Multiplication by t

1.3.f Properties of Laplace Transform – Division by t

1.3.g Properties of Laplace Transform – Laplace Transform of derivatives and integrals (Properties without proof)

1.4 Evaluation of real improper integrals by using Laplace Transformation

1.5.a Self-learning topics – Heaviside’s Unit Step function

1.5.b Self-learning topics – Laplace Transform. of Periodic functions

1.5.c Self-learning topics – Dirac Delta Function

Module 2 – Inverse Laplace Transform 7 Topics

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2.1.a Definition of Inverse Laplace Transform

2.1.b Linearity property

2.1.c Inverse Laplace Transform of standard functions

2.1.d Inverse Laplace transform using derivatives

2.2 Partial fractions method to find Inverse Laplace transform

2.3 Inverse Laplace transform using Convolution theorem (without proof)

2.4 Self-learning Topics: Applications to solve initial and boundary value problems involving ordinary differential equations

Module 3 – Fourier Series 8 Topics

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3.1.a Dirichlet’s conditions

3.1.b Definition of Fourier series and Parseval’s Identity (without proof).

3.2 Fourier series of periodic function with period 2π and 2l

3.3 Fourier series of even and odd functions

3.4 Half range Sine and Cosine Series.

3.5.a Self-learning Topics – Orthogonal and orthonormal set of functions

3.5.b Self-learning Topics – Complex form of Fourier Series

3.5.c Self-learning Topics – Fourier Transforms

Module 4 – Complex Variables 9 Topics

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4.1.a Function f(z)of complex variable

4.1.b Limit, Continuity and Differentiability off(z)

4.1.c Analytic function: Necessary and sufficient conditions for f(z) to be analytic (without proof).

4.2 Cauchy-Riemann equations in Cartesian coordinates (without proof).

4.3 Milne-Thomson method: Determine analytic function f(z)when real part (u), imaginary part (v) or its combination (u+v / u-v) is given

4.4 Harmonic function, Harmonic conjugate and Orthogonal trajectories.

4.5.a Self-learning Topics – Conformal mapping

4.5.b Self-learning Topics – Linear and Bilinear mappings

4.5.c Self-learning Topics: – cross ratio, fixed points and standard transformations

Module 5 – Statistical Techniques 5 Topics

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5.1 Karl Pearson’s coefficient of correlation (r)

5.2 Spearman’s Rank correlation coefficient (R) (with repeated and nonrepeated ranks)

5.3 Lines of regression

5.4 Fitting of first- and second-degree curves.

5.5 Self-learning Topics – Covariance, fitting of exponential curve

Module 6 – Probability 7 Topics

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6.1 Definition and basics of probability

6.2 conditional probability

6.3.a Total Probability theorem

6.3.b Baye’s Theorem

6.4 Discrete and continuous random variable with probability distribution and probability density function.

6.5 Expectation, Variance, Moment generating function, Raw and central moments up to 4th order.

6.6 Self-learning Topics: Skewness and Kurtosis of distribution (data).

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