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# Fourier Transforms

Consider a function that is periodic in with period . In other words,

 (698)

for all . Recall, from Section 6.4, that we can represent such a function as a Fourier series: that is,

 (699)

where

 (700)

[We have neglected the term in Equation (699), for the sake of convenience.] Equation (699) automatically satisfies the periodicity constraint (698), because and for all and (with the proviso that is an integer). The so-called Fourier coefficients, and , appearing in Equation (699), can be determined from the function by means of the following readily demonstrated (see Exercise 1) results:

 (701) (702) (703)

where , are positive integers. Here, is a Kronecker delta function. In fact,

 (704) (705)

(See Exercise 1.) Incidentally, any periodic function of can be represented as a Fourier series.

Suppose, however, that we are dealing with a function that is not periodic in . We can think of such a function as one that is periodic in with a period that tends to infinity. Does this mean that we can still represent as a Fourier series? Consider what happens to the series (699) in the limit , or, equivalently, . The series is basically a weighted sum of sinusoidal functions whose wavenumbers take the quantized values . Moreover, as , these values become more and more closely spaced. In fact, we can write

 (706)

In the continuum limit, , the summations in the previous expression become integrals, and we obtain

 (707)

where , , and . Thus, for the case of an aperiodic function, the Fourier series (699) morphs into the so-called Fourier transform (707). This transform can be inverted using the continuum limits (i.e., the limit ) of Equations (704) and (705), which are readily shown to be

 (708) (709)

respectively. (See Exercise 5.) The previous equations confirm that and . The Fourier-space (i.e., -space) functions and are known as the cosine Fourier transform and the sine Fourier transform of the real-space (i.e., -space) function , respectively. Furthermore, because we already know that any periodic function can be represented as a Fourier series, it seems plausible that any aperiodic function can be represented as a Fourier transform. This is indeed the case. Equations (707)-(709) effectively enable us to represent a general function as a linear superposition of sine and cosine functions. Let us consider some examples.

Consider the top-hat'' function,

 (710)

(See Figure 50.) Given that and , it follows from Equations (708) and (709) that if is even in , so that , then , and if is odd in , so that , then . Hence, because the top-hat function (710) is even in , its sine Fourier transform is automatically zero. On the other hand, its cosine Fourier transform takes the form

 (711)

Figure 50 shows the function , together with its associated cosine transform, .

As a second example, consider the so-called Gaussian function,

 (712)

As illustrated in Figure 51, this is a smoothly varying even function of that attains its peak value at , and becomes completely negligible when . Thus, is a measure of the width'' of the function in real space. By symmetry, the sine Fourier transform of the preceding function is zero. On the other hand, the cosine Fourier transform is readily shown to be

 (713)

where

 (714)

(See Exercise 2.) This function is a Gaussian in Fourier space of characteristic width . The original function can be reconstructed from its Fourier transform using

 (715)

This reconstruction is simply a linear superposition of cosine waves of differing wavenumbers. Moreover, can be interpreted as the amplitude of waves of wavenumber within this superposition. The fact that is a Gaussian of characteristic width [which means that is negligible for ] implies that in order to reconstruct a real space function whose width in real space is approximately it is necessary to combine sinusoidal functions with a range of different wavenumbers that is approximately in extent. To be slightly more exact, the real-space Gaussian function falls to half of its peak value when . Hence, the full width at half maximum of the function is . Likewise, the full width at half maximum of the Fourier-space Gaussian function is . Thus,

 (716)

since . We conclude that a function that is highly localized in real space has a transform that is highly delocalized in Fourier space, and vice versa. Finally,

 (717)

(See Exercise 3.) In other words, a Gaussian function in real space, of unit height and characteristic width , has a cosine Fourier transform that is a Gaussian in Fourier space, of characteristic width , and whose integral over all -space is unity.

Consider what happens to the previously mentioned real-space Gaussian, and its Fourier transform, in the limit , or, equivalently, . There is no difficulty in seeing, from Equation (712), that

 (718)

In other words, the real space Gaussian morphs into a function that takes the constant value unity everywhere. The Fourier transform is more problematic. In the limit , Equation (713) yields a -space function that is zero everywhere apart from (because the function is negligible for ), where it is infinite [because the function takes the value at ]. Moreover, according to Equation (717), the integral of the function over all remains unity. Thus, the Fourier transform of the uniform function is a sort of integrable spike'' located at . This unusual function is known as the Dirac delta function, and is denoted . Thus, one definition of a delta function is

 (719)

As has already been mentioned, for , and . Moreover,

 (720)

Consider the integral

 (721)

where is an arbitrary function. Because of the peculiar properties of the delta function, the only contribution to the previous integral comes from the region in -space in the immediate vicinity of . Furthermore, provided is well-behaved in this region, we can write

 (722)

where use has been made of Equation (720).

A change of variables allows us to define , which is a spike'' function centered on . The previous result can be generalized to give

 (723)

for all . Indeed, this expression can be thought of as an alternative definition of a delta function.

We have seen that the delta function is the cosine Fourier transform of the uniform function . It, thus, follows from Equation (708) that

 (724)

This result represents yet another definition of the delta function. By symmetry, we also have

 (725)

It follows that

 (726)

which yields

 (727)

where use has been made of Equation (724), and a standard trigonometric identity. (See Appendix B.) Likewise,

 (728) (729)

(See Exercise 4.) Incidentally, Equations (727)-(729) can be used to derive Equations (708) and (709) directly from Equation (707). (See Exercise 5.)

Next: General Solution of 1D Up: Wave Pulses Previous: Wave Pulses
Richard Fitzpatrick 2013-04-08