tailieunhanh - Giáo trình toán kỹ thuật 9

Cung cấp cho người học cơ sở toán học chuyên sâu, là những công cụ hữu hiệu để học tập nghiên cứu chuyên ngành điện tử viễn thông. Gồm các nội dung sau: Hàm biến số phức, hàm giải tích, phép biến hình bảo giác. Tích phân phức. Khai triển hàm giải tích thành chuỗi Taylor, chuỗi Laurent. Thặng dư của hàm biến phức tại điểm bất thường cô lập. Phép biến đổi Z. | Table contd. The Laplace Transforms of Some Commonly Encountered Functions. Complementary error function erfc z 1 erf r 72 where -u O 0. THE HEAVISIDE STEP AND DIRAC DELTA FUNCTIONS Change can occur abruptly. We throw a switch and electricity sud denly nows. In this section we introduce two functions the Heaviside step and Dirac delta that will give US the ability to construct compli cated discontinuous functions to express these changes. Heaviside step function We define the Heaviside step function as 73 Note that this transform is identical to that for i 1 if a 0. This should not surprise US. As pointed out earlier the function f t is zero for all t 0 by definition. Thus when dealing with Laplace transforms i 1 and H t are identical. Generally we will take 1 rather than as the inverse of 1 jf. The Heaviside step function is essentially a bookkeeping device that gives us the ability to switch on and switch off a given function. For example if we want a function i to become nonzero at time t a we represent this process by the product a . On the other hand if we only want the function to be turned on when a t b the desired expression IS then fit 7 z a - ỈỈ t it . Fori a both step functions in the brackets have the value of zero. For a t b the first step function has the value of unity and the second step function has the value of zero so that we have i . For t b both step functions equal unity so that their difference is zero. Wl 1 y-------------------K 1 2 3 4 t Figure Graphical representation of . Consider Figure . We would like to express this graph in terms of Heaviside step functions. We begin by introducing step functions at each point where there is a kink discontinuity in the first derivative or jump in the graph in the present case at t 0 t 1 f 2 and . 3. Thus fit a0 t J7G ơiơ ơ-l đ2 W -2 -l-a3ự tfƠ-3 where the coefficients Uo i ai i i . . . are yet to be determined. Proceeding from left to right in Figure the .