# FOURIER INTEGRALS  1   sin v  dv jI3 j @ j f x 0 j   v L 1 in Visual Studio .NET Drawing QR Code in Visual Studio .NET FOURIER INTEGRALS  1   sin v  dv jI3 j @ j f x 0 j   v L 1

FOURIER INTEGRALS  1   sin v  dv jI3 j @ j f x 0 j   v L 1
Reading QR Code In VS .NET
Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications.
Denso QR Bar Code Generation In .NET
Using Barcode maker for VS .NET Control to generate, create Denso QR Bar Code image in VS .NET applications.
Also
QR Recognizer In .NET
Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications.
Making Barcode In Visual Studio .NET
Using Barcode creator for VS .NET Control to generate, create bar code image in VS .NET applications.
sin v dv both converge, we can choose L so large that jI2 j @ =3, jI3 j @ =3. v Also, we can choose so large that jI1 j @ =3. Then from (4) we have jIj <  for and L su ciently large, so that the required result follows. This result follows by reasoning exactly analogous to that in part (a). Since j f x j dx and
Bar Code Reader In VS .NET
Using Barcode decoder for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications.
QR Code Creation In C#.NET
Using Barcode maker for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET applications.
14.12. Prove Fourier s integral formula where f x satis es the conditions stated on Page 364.
QR-Code Creator In Visual Studio .NET
Using Barcode drawer for ASP.NET Control to generate, create QR image in ASP.NET applications.
QR Code ISO/IEC18004 Drawer In VB.NET
Using Barcode drawer for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications.
We must prove that lim 1 L!1  L 1
Drawing Linear In Visual Studio .NET
Using Barcode generator for Visual Studio .NET Control to generate, create 1D Barcode image in .NET applications.
Bar Code Drawer In VS .NET
Using Barcode creator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications.
0 u 1
GS1 128 Encoder In Visual Studio .NET
Using Barcode generator for .NET Control to generate, create EAN128 image in .NET framework applications.
Identcode Printer In VS .NET
Using Barcode creation for .NET framework Control to generate, create Identcode image in Visual Studio .NET applications.
f u cos x u du d
Bar Code Decoder In .NET Framework
Using Barcode Control SDK for ASP.NET Control to generate, create, read, scan barcode image in ASP.NET applications.
Bar Code Maker In None
Using Barcode maker for Software Control to generate, create bar code image in Software applications.
 1  1   f u cos x u du @ j f u j du, which converges, it follows by the Weierstrass test Since    1 1 1 f u cos x u du converges absolutely and uniformly for all . Thus, we can reverse the that
Code 3 Of 9 Generation In VS .NET
Using Barcode drawer for Reporting Service Control to generate, create Code 3/9 image in Reporting Service applications.
Print Code 3 Of 9 In Objective-C
Using Barcode drawer for iPhone Control to generate, create Code 39 Full ASCII image in iPhone applications.
f x 0 f x 0 2
GS1-128 Drawer In Objective-C
Using Barcode generation for iPhone Control to generate, create GTIN - 128 image in iPhone applications.
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications.
order of integration to obtain 1  L d
Paint UCC-128 In Java
Using Barcode creator for Java Control to generate, create USS-128 image in Java applications.
Draw EAN13 In Java
Using Barcode drawer for Java Control to generate, create EAN 13 image in Java applications.
0 u 1
f u cos x u du
L 1 1 f u du cos x u d  u 1 0 1 1 sin L u x du f u  u 1 u x 1 1 sin Lv dv f x v  u 1 v 0 1 sin Lv 1 1 sin Lv dv dv f x v f x v  1 v  0 v
where we have let u x v. f x 0 f x 0 Letting L ! 1, we see by Problem 14.11 that the given integral converges to as 2 required.
MISCELLANEOUS PROBLEMS 14.13. Solve @U @2 U 2 subject to the conditions U 0; t 0; U x; 0 @t @x bounded where x > 0; t > 0.
0<x<1 , U x; t is xA1
We proceed as in Problem 13.24, 2 13. A solution satisfying the partial di erential equation and the rst boundary condition is given by Be  t sin x. Unlike Problem 13.24, 13, the boundary conditions do not prescribe the speci c values for , so we must assume that all values of  are possible. By analogy with that problem we sum over all possible values of , which corresponds to an integration in this case, and are led to the possible solution 1 2 B  e  t sin x d 1 U x; t
where B  is undetermined.
By the second condition, we have & 1 1 0<x<1 B  sin x d f x 0 xA1 0
from which we have by Fourier s integral formula 2 1 2 1 2 1 cos  f x sin x dx sin x dx B   0  0 
FOURIER INTEGRALS
[CHAP. 14
so that, at least formally, the solution is given by   2 1 1 cos  2 t U x; t e sin x dx  0  See Problem 14.26.
14.14. Show that e x
Since e
p 1 x2 =2 is even, its Fourier transform is given by 2= e cos x dx. 0 p Letting x 2 u and using Problem 12.32, 12, the integral becomes p p 2  2 =2 2 1 u2 2 p e cos 2 u du p e =2 e  0  2
x =2
is its own Fourier transform.
which proves the required result.
14.15. Solve the integral equation y x g x
1 y u r x u du
where g x and r x are given.
Suppose that the Fourier transforms of y x ; g x ; and r x exist, and denote them by Y ; G ; and R , respectively. Then taking the Fourier transform of both sides of the given integral equation, we have by the convolution theorem Y G & p 2 Y R or Y 1 G p 2 R
Then
y x f 1
' G 1 1 G p p p e i x d 2 1 1 2 R 1 2 R
assuming this integral exists.
Supplementary Problems
THE FOURIER INTEGRAL AND FOURIER TRANSFORMS & 1=2 jxj @  14.16. (a) Find the Fourier transform of f x 0 jxj >  (b) Determine the limit of this transform as  ! 0 and discuss the result. Ans: 1 sin  ; a p 2  1 b p 2 jxj < 1 jxj > 1
& 1 x2 14.17. (a) Find the Fourier transform of f x 0  1 x cos x sin x x (b) Evaluate cos dx. 2 x3 0 r   2 cos sin 3 Ans: a 2 ; b  16 3
CHAP. 14] &