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## Applied Mathematics 3

Students studying Telecommunication Engineering will find this subject very useful. Hundreds of important topics on Applied Mathematics 3 are organized neatly into lessons below.

### Overview

#### Topics Covered

**1. Laplace Transform**

1.1 Laplace Transform (LT) of Standard Functions: Definition of Laplace transform, Condition of Existence of Laplace transform, Laplace transform of e Sin at at at , ( ),cos( ), sinh( ),cosh( ), at at t n Heaviside unit step function ...

Read more**1. Laplace Transform**

1.1 Laplace Transform (LT) of Standard Functions: Definition of Laplace transform, Condition of Existence of Laplace transform, Laplace transform of e Sin at at at , ( ),cos( ), sinh( ),cosh( ), at at t n Heaviside unit step function, Dirac-delta function, Laplace transform of Periodic function

1.2 Properties of Laplace Transform: Linearity, first shifting theorem, second shifting theorem, multiplication by t n , Division by t, Laplace Transform of derivatives and integrals, change of scale, convolution theorem, Evaluation of integrals using Laplace transform.

**2 Inverse Laplace Transform & its Applications**

2.1 Partial fraction method, Method of convolution, Laplace inverse by derivative

2.2 Applications of Laplace Transform: Solution of ordinary differential equations, Solving RLC circuit differential equation of first order and second order with boundary condition using Laplace transform (framing of differential equation is not included)

**3 Fourier Series**

3.1 Introduction: Orthogonal and orthonormal set of functions, Introduction of Dirchlet’s conditions, Euler’s formulae.

3.2 Fourier Series of Functions: Exponential, trigonometric functions of any period =2L, even and odd functions, half range sine and cosine series

3.3 Complex form of Fourier series, Fourier integral representation, Fourier Transform and Inverse Fourier transform of constant and exponential function.

**4 Vector Algebra & Vector Differentiation**

4.1 Review of Scalar and Vector Product: Scalar and vector product of three and four vectors, Vector differentiation, Gradient of scalar point function, Divergence and Curl of vector point function

4.2 Properties: Solenoidal and irrotational vector fields, conservative vector field

**5 Vector Integral**

5.1 Line integral

5.2 Green’s theorem in a plane, Gauss’ divergence theorem and Stokes’ theorem

**6 Complex Variable & Bessel Functions**

6.1 Analytic Function: Necessary and sufficient conditions (No Proof), Cauchy Reiman equation Cartesian form (No Proof) Cauchy Reiman Equation in polar form (with Proof), Milne Thomson Method and it application, Harmonic function, orthogonal trajectories

6.2 Mapping: Conformal mapping, Bilinear transformations, cross ratio, fixed points

6.3 Bessel Functions: Bessel’s differential equation, Properties of Bessel function of order +1/2 and -1/2, Generating function, expression of cos(xsinθ ), sin (x sinθ ) in term of Bessel functions

Read less### Syllabus

**1. Laplace Transform**

1.1 Laplace Transform (LT) of Standard Functions: Definition of Laplace transform, Condition of Existence of Laplace transform, Laplace transform of e Sin at at at , ( ),cos( ), sinh( ),cosh( ), at at t n Heaviside unit step function, Dirac-delta function, Laplace transform of Periodic function

1.2 Properties of Laplace Transform: Linearity, first shifting theorem, second shifting theorem, multiplication by t n , Division by t, Laplace Transform of derivatives and integrals, change of scale, convolution theorem, Evaluation of integrals using Laplace transform.

**2 Inverse Laplace Transform & its Applications**

2.1 Partial fraction method, Method of convolution, Laplace inverse by derivative

2.2 Applications of Laplace Transform: Solution of ordinary differential equations, Solving RLC circuit differential equation of first order and second order with boundary condition using Laplace transform (framing of differential equation is not included)

**3 Fourier Series**

3.1 Introduction: Orthogonal and orthonormal set of functions, Introduction of Dirchlet’s conditions, Euler’s formulae.

3.2 Fourier Series of Functions: Exponential, trigonometric functions of any period =2L, even and odd functions, half range sine and cosine series

3.3 Complex form of Fourier series, Fourier integral representation, Fourier Transform and Inverse Fourier transform of constant and exponential function.

**4 Vector Algebra & Vector Differentiation**

4.1 Review of Scalar and Vector Product: Scalar and vector product of three and four vectors, Vector differentiation, Gradient of scalar point function, Divergence and Curl of vector point function

4.2 Properties: Solenoidal and irrotational vector fields, conservative vector field

**5 Vector Integral**

5.1 Line integral

5.2 Green’s theorem in a plane, Gauss’ divergence theorem and Stokes’ theorem

**6 Complex Variable & Bessel Functions**

6.1 Analytic Function: Necessary and sufficient conditions (No Proof), Cauchy Reiman equation Cartesian form (No Proof) Cauchy Reiman Equation in polar form (with Proof), Milne Thomson Method and it application, Harmonic function, orthogonal trajectories

6.2 Mapping: Conformal mapping, Bilinear transformations, cross ratio, fixed points

6.3 Bessel Functions: Bessel’s differential equation, Properties of Bessel function of order +1/2 and -1/2, Generating function, expression of cos(xsinθ ), sin (x sinθ ) in term of Bessel functions