This document discusses Fourier integrals, which extend the method of Fourier series to non-periodic functions defined on the whole real number line. It contains sections on the introduction, formulas, applications, Fourier cosine integrals, and Fourier sine integrals. The introduction explains that Fourier integrals represent functions using integrals rather than sums, allowing functions that are non-periodic or defined on an infinite interval to be analyzed. The formulas section gives the general representations for Fourier integrals and their connection to the function being represented. Applications include solving ordinary and partial differential equations. Fourier cosine and sine integrals simplify the general Fourier integral representation for even and odd functions, respectively.

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Gujarat Technological University
SUBMITTED TO:-
PROF. BIJAL KHUMAN

2. 130110107054-SHUKLA DEVANSHI
130110107055-SONAGARA CHIRAG
130110107056-SUTARIYA ASHISH
130110107057-SUTARIYA YASH
130110107058-SUTHAR HEMANT
130110107059-SHRINAND THAKKAR
130110107060-VIRALI THAKKAR
PRESENTED BY:-
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3. Contain:-
 1. introduction
 2. formulas
 3. application
 4. fourier cosine integral
 5. fourier sine integral
 6. reference
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4. 1:-INTRODUCTION
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5. Fourier series are powerful tools for problems involving
functions that are periodic or are of interest on a finite
interval only.
 many problems involve functions that are nonperiodic and
are of interest on the whole x-axis, we ask what can be done
to extend the method of Fourier series to such functions. This
idea will lead to “Fourier integrals.”
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6. 2:-FORMULAS
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7.  











wvdvvfwBand
wvdvvfwA
where
dwwxwBwxwAxf
sin)(
1
)(
cos)(
1
)(
,
sin)(cos)()(
0


This is called a representation of
f(x) by a Fourier integral.
integral from 0 to which represents f(x), namely,
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8. 3:-APPLICATIONS
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9. Applications of Fourier Integrals:-
 The main application of Fourier integrals is in solving ODEs and
PDEs.
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12. 4:-FOURIER
COSINE
INTEGRAL
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13. Fourier Cosine Integral
Just as Fourier series simplify if a function is even or
odd, so do Fourier integrals, and you can save
work. Indeed, if f has a Fourier integral
representation and is even. This holds because
the integrand of is odd. Then reduces to a Fourier
cosine integral
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14. For even function f(x): B(w)=0,




0
wvdvcos)v(f
2
)w(A



0
dwwxcos)w(A)x(f
Therefor,
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15. 5:-FOURIER
SINE
INTEGRAL
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16. Fourier Sine Integral
Note the change in : for even f the integrand
is even, hence the integral from to equals
twice the integral from 0 to infinite. Similarly, if
f has a Fourier integral representation and is
odd. This is true because the integrand of is
odd. Then a Fourier sine integral
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17. For odd function f(x): A(w)=0




0
wvdvsin)v(f
2
)w(B



0
dwwxsin)w(B)x(f
Therefor,
17

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22. 6:-REFERENCE
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23. 1.Advanced Engineering Mathematics (10th Edition) by Erwin
Kreyszig
2.http://googleimages.co.in
3.http://wikiepedia.com
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