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Contents

**Integral transforms with applications 7B**

**Learning Outcomes:**

Students after successful completion of the course will be able to

1. Evaluate Laplace transforms of certain functions, find Laplace transforms of derivatives and of integrals.

2. Determine properties of Laplace transform which may be solved by application of special functions namely Dirac delta function, error function, Bessel function and periodic function.

3. Understand properties of inverse Laplace transforms, find inverse Laplace transforms of derivatives and of integrals.

4. Solve ordinary differential equations with constant/ variable coefficients by using Laplace transform method.

5. Comprehend the properties of Fourier transforms and solve problems related to finite Fourier transforms.

#### Unit – 1: Laplace transforms- I

1. Definition of Laplace transform, linearity property-piecewise continuous function.

2. Existence of Laplace transform, functions of exponential order and of class A.

3. First shifting theorem, second shifting theorem and change of scale property.

**Unit – 2: Laplace transforms- II**

1. Laplace Transform of the derivatives, initial value theorem and final value theorem. Laplace transforms of integrals.

2. Laplace transform of tn. f (t), division by t, evolution of integrals by Laplace transforms.

3. Laplace transform of some special functions-namely Dirac delta function, error function,Bessel function and Laplace transform of periodic function.

**Unit – 3: Inverse Laplace transforms**

1. Definition of Inverse Laplace transform, linear property, first shifting theorem, second shifting theorem, change of scale property, use of partial fractions.

2. Inverse Laplace transforms of derivatives, inverse, Laplace transforms of integrals, multiplication by powers of ‘p’, division by ‘p’.

3. Convolution, convolution theorem proof and applications.

**Unit – 4: Applications of Laplace transforms**

1. Solutions of differential equations with constants coefficients, solutions of differential equations with variable coefficients.

2. Applications of Laplace transforms to integral equations- Abel’s integral equation.

3. Converting the differential equations into integral equations, converting the integral equations into differential equations.

**Unit – 5: Fourier transforms**

1. Integral transforms, Fourier integral theorem (without proof), Fourier sine and cosine integrals.

2. Properties of Fourier transforms, change of scale property, shifting property, modulation theorem. Convolution.

3. Convolution theorem for Fourier transform, Parseval’s Identify, finite Fourier transforms.

**5th semester Degree mathematics Integral transforms with applications 7B Download Here Materials**

Name of the Unit | Download material |

Unit – 1: Laplace transforms- I | Click Here |

Unit – 2: Laplace transforms- II | Click Here |

Unit – 3: Inverse Laplace transforms | Click Here |

Unit – 4: Applications of Laplace transforms | Click Here |

Unit – 5: Fourier transforms | Click Here |