# DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE.

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DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE

FOURIER COSINE AND SINE TRANSFORMS [Chapter – 10.9] TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION] MATHEMATICS - II LECTURE :17

Where ‘c’ stands for cosine Equation (1) is called Fourier cosine transform of f(x) Equation (2) is called inverse Fourier cosine transform of Where ‘c’ stands for cosine Equation (1) is called Fourier cosine transform of f(x) Equation (2) is called inverse Fourier cosine transform of Fourier Cosine Transform Fourier Cosine IntegralFourier Cosine Transform Set where Then we have and (1) (2)

(1) (1) Where ‘s’ stands for sine Equation (1) is called Fourier sine transform of f(x) Equation (2) is called inverse Fourier sine transform of (1) (1) Where ‘s’ stands for sine Equation (1) is called Fourier sine transform of f (x) Equation (2) is called inverse Fourier sine transform of Fourier Sine Transform Fourier Sine Integral Fourier Sine Transform where Set Then we have and (1) (2)

Example 1: Find the cosine and sine transform of the function PROBLEMS ON FOURIER SINE & COSINE TRANSFORMS Solution:

PROBLEMS ON FOURIER SINE & COSINE TRANSFORMS

respectively. respectively. Properties of Fourier Cosine & Sine Transforms Linearity Property Notation: For any function f(x) Its Fourier cosine and Fourier sine transforms are denoted by and 1. 2.

Properties of Fourier Cosine & Sine Transforms

Proof:

Properties of Fourier Cosine & Sine Transforms

Proof: By Fourier cosine transform of derivative of a function, we have (a)

Properties of Fourier Cosine & Sine Transforms

(b) By Fourier sine transform of derivative of a function, we have

Some More Problems involving Fourier Cosine & Sine Integrals