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File: Fourier Transformation Pdf 179237 | Chapter 8pde
chapter 8 fourier transform we introduce the fourier transform a special linear integral transformation for dierential equations which are dened on unbounded domains the method is associated with the french ...

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              Chapter 8
              Fourier Transform
              We introduce the Fourier transform, a special linear integral transformation for
              differential equations which are defined on unbounded domains. The method is
              associated with the French physicist and mathematician Joseph Fourier whose
              work opened up a new way to solve linear PDEs. As we will see later, the Fourier
              transform reduces a partial differential equation in (t; x)-domain to an ordinary
              differential equation in (t;!)-domain.
              8.1 Definition
              The Fourier transform is a linear integral transformation of a function f(x) in
                                        ^
              x-domain to another function f(!) in !-domain. As we will see in sequel, ! can
                                                         ^
              be considered as the frequency variable and thus f(!) represents the frequency
              distribution embedded in function f(x).
                 1. (Admissible functions) We first introduce the admissible space of func-
                    tions.
                    Definition 8.1. A function f(x); ¡1 < x < 1 is called integrable if the
                    following relation holds Z
                                          1jf(x)jdx<1:                       (8.1)
                                         ¡1
                    Afunction f(x);¡1
                                              >p
                                         f(x)=< x 1    x>0;
                                              >
                                              > p
                                              :¡ ¡x x<0
                    is not admissible even it satisfies the following property
                                          lim Z Rf(x)dx=0;:
                                         R!1 ¡R
                                               1
               2                                                      Fourier Transform
                     Note that
                                  Z R                Z R 1           p
                               lim    jf(x)jdx=2 lim    p dx= lim 4 R=1:
                              R!1 ¡R             R!1 0    x     R!1
                     TheFouriertransformisusuallydefinedforadmissiblefunctions. However,
                     as we will see below, the definition can be extended for a wider class of
                     functions than the admissible ones.
                  2. (Definition of Fourier transform) Suppose f is a function (not neces-
                     sarily admissible) defined on R:(¡1;1). Its Fourier transform Fffg:=
                     ^
                     f(!) is defined by the following integral
                                            ^     Z 1     ¡i!x
                                   Fffg:=f(!)= ¡1f(x)e        dx;                 (8.2)
                     as long as the integral exists. The improper integral is understood in the
                     following sense
                                Z 1f(x)e¡i!xdx= lim Z Rf(x)e¡i!xdx:               (8.3)
                                 ¡1              R!1 ¡R
                     Example 8.1. Consider the following function
                                                81 00;
                                                                    ¡1 x<0
                           is not admissible, and the definition of the Fourier transform fails as
                                      ^      Z 0      ¡i!x      Z 1 ¡i!x             Z 1
                                     f(!)=        ¡e      dx+        e     dx=¡2i        sin(!x)dx:
                                               ¡1                 0                   0
                4                                                             Fourier Transform
                       Now, let us consider the following function
                                                 f (x)= e¡"x x>0 :
                                                  "          "x
                                                           ¡e    x<0
                       The function is admissible and its transform is
                             ^      Z 1        ¡i!x        Z 1 ¡"x                    !
                             f (!)=     f (x)e     dx=¡2i      e   sin(!x)dx=¡2i           :
                              "          "                                          "2+!2
                                     ¡1                      0
                                                              ^
                       Since f (x)!f(x) for "!0, we define f(!) through the following limit
                              "
                                                 ^         ^       ¡2i
                                                 f(!)=limf (!)=       :
                                                            "       !
                                                        "!0
                                                       ^
                    4. (Inverse transform) Suppose f(!) is a function defined on !2(¡1;1).
                                                          ^
                       The inverse Fourier transform of f is defined by the following integral
                                            ¡1 ^    1 Z 1 ^     i!x
                                          F (f)=2 ¡1f(!)e         d!;                      (8.4)
                       as long as the integral exists. The integral is understood in the following sense
                                     Z 1^      i!x          Z R ^     i!x
                                         f(!)e    d!= lim       f(!)e   d!:                 (8.5)
                                      ¡1               R!1 ¡R
                                                                                      ¡1
                       The following theorem establishes the relationship between F;F   . See the
                       appendix of this chapter for a proof.
                       Theorem 8.1. Assume that f is an admissible function, then
                                                           +       ¡
                                            ¡1 ^       f(x )+f(x )
                                          F (f)(x)=           2      ;                      (8.6)
                                 +          ¡
                       where f(x ) and f(x ) are the right and left limits of f at x respectively.
                                                                           ^   2(1¡cosw)
                       Example8.3. Letusverifythetheoremforfunctions f=           i!   . Wehave
                          ¡1 ^       1      Z R1¡cos! i!x         2     Z R1¡cos!
                        F (f)(x)=i lim            !     e   d!= lim          !     sin(!x)d!;
                                       R!1 ¡R                       R!1 0
                       where we used the symmetry property for the integration. The following
                       figures shows the value of the integral for R = 20; 100 respectively. It is
                       observed that for R!1 the integral approaches f(x).
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...Chapter fourier transform we introduce the a special linear integral transformation for dierential equations which are dened on unbounded domains method is associated with french physicist and mathematician joseph whose work opened up new way to solve pdes as will see later reduces partial equation in t x domain an ordinary denition of function f another sequel can be considered frequency variable thus represents distribution embedded admissible functions rst space func tions called integrable if following relation holds z jf jdx p...

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