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Giáo trình toán kỹ thuật 7

Tác giả cho rằng nắm vững các kiến thức toán học trong cuốn sách này cũng sẽ rất hữu ích cho các sinh viên muốn học cao lên sau đại học. | An important function in transform methods is the Heaviside step function. ff i ỉ 3.2.16 In terms of the sign function it can be written H i 1 lsgn t . 3.2.17 Because the Fourier transforms of 1 and sgn t are 2tt u and 2 iw respectively we have that Z fl 0 5ri w d-_ 3.2.18 tư These transforms are used in engineering but the presence of the delta function requires extra care to ensure their proper use. Of the many functions that have a Fourier transform a particularly important one IS the Dirac delta function 4 For example in Section 3.6 we will use it to solve differential equations. We define it as rhe inverse of the Fourier transform F w 1. Therefore 1 r . ỏw e dư. 3.1.13 To give some insight into the nature of the delta function consider another band-limited transform F H Ỉ 3.1.14 where Í1 is real and positive. Then . Ề 1 - ĩ T- 131151 Z7T J_Q T 55 56 if a lQ b. This follows from the Law of the mean of integrals. Wc may also use several other functions with equal validity to represent the delta function. These include the limiting case of the following rectangular or triangular distributions 0 t linjH 3.1.22 -0 0 i or r 5 0 limp 0 Ị Ị e 3.1.23 H 0 1 1 and the Gaussian function limeiCp - e . 3.1.24 f e 0 y Ẽ Note that the delta function is an even function. FOURIER TRANSFORMS CONTAINING THE DELTA FUNCTION In the previous section we stressed the fact that such simple functions as cosine and sine are not absolutely integrable. Does this mean that these functions do not possess a Fourier transform In this section we shall show that certain functions can still have a Fourier transform even though we cannot compute them directly. The reason why we can find the Fourier transform of certain functions that are not absolutely integrable lies with the introduction of the delta function because ỉ ó w - w0 e tai dw 3.2.1 J oc for all t. Thus the inverse of the Fourier transform 0 w - Wo is the complex exponential or eÍWoí 2tt0 w-wo . 3.2.2 .