Fourier Analysis:Among the choices, find the fourier integral representaion of the given function: Given: f(t)a.) f(t) = [[b.) f(t)S[[c.) f(t) = [[d.) f(t)Sol==t, 01-[sin w +-[sin w +-[sin w +-[sin w +1WIT1WIT1WIT1WIT1[ -cos w+ sh w]]|sin we]owWIT[-cos w+ sh w][sin we]owwt dw(cos(cos w - 1)-1) cos wt +1(cos w - 1) cos wt +-ππω-(cos w +1) cos wt +-(cos w --1)]cos wt ]dwWIT{[cos w + — sin w]]sh wit]dw

Given the function is ft=t0<t<10t<0, t>1

t, 01 1 a.) f(t) [m (sh w + (cos w - 1)] cos wt + [[ -[sin sin OLL - cos w + sh w]]sin wet dw - 1 shw]]sin weldw 1 1 b.) f(t) - -[sin w + (cos w - 1)]cos wt + [1/[(- -COS W+ wt ᏩᎫᎢ . ππω OLL 1 c.) f(t) = -[sin (shw + (cos w +1)]cos wt + [[cos w + — sin w]]sin wt]ow d.) f(t) - (sh w+ (cos w - 1)]cos wt Jow = -[sin WIT Given: f(t)

t, 01 1 a.) f(t) [m (sh w + (cos w - 1)] cos wt + [[ -[sin sin OLL - cos w + sh w]]sin wet dw - 1 shw]]sin weldw 1 1 b.) f(t) - -[sin w + (cos w - 1)]cos wt + [1/[(- -COS W+ wt ᏩᎫᎢ . ππω OLL 1 c.) f(t) = -[sin (shw + (cos w +1)]cos wt + [[cos w + — sin w]]sin wt]ow d.) f(t) - (sh w+ (cos w - 1)]cos wt Jow = -[sin WIT Given: f(t)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 55E

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Question

Fourier Analysis:
Among the choices, find the fourier integral representaion of the given function:

Given: f(t)
a.) f(t) = [[
b.) f(t)
S[[
c.) f(t) = [[
d.) f(t)
Sol
=
=
t, 0<t<1
0, t<0, t>1
-[sin w +
-[sin w +
-[sin w +
-[sin w +
1
WIT
1
WIT
1
WIT
1
WIT
1
[ -cos w+ sh w]]|sin we]ow
WIT
[-cos w+ sh w][sin we]ow
wt dw
(cos
(cos w - 1)
-1) cos wt +
1
(cos w - 1) cos wt +
-
ππω
-(cos w +1) cos wt +
-
(cos w -
-1)]cos wt ]dw
WIT
{[cos w + — sin w]]sh wit]dw

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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