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1 Calculate the Laplace transform of cos t 2 Find the Laplace transform of cosh at where a is a constant 3 Using the Bromvich inversion integral, nd the inverse Laplace transform e k s of s 4 Using the Bromvich inversion integral, nd the inverse Laplace transform s of 2 s + 2 5 Using the Bromvich inversion integral and ! = e t t dt , nd the inverse 0 Laplace transform of F (s) = s
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In this chapter, we will introduce a few of the techniques that can be used to transform a region of the complex plane into another different region of the complex plane You may want to do this because it will be convenient for a given problem you re solving There are many types of transformations that can be applied in the limited space we have, we won t be able to cover but a small fraction of them Our purpose here is to introduce you to a few of the common transformations used and get you used to the concepts involved Let us de ne two complex planes The rst is the z plane de ned by the coordinates x and y We will now introduce a second plane, which we call the w plane, de ned by two coordinates that are denoted by u and v Mapping is a transformation between points in the z plane and points in the w plane This is illustrated in Fig 91
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z plane
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Mapping is a transformation of points in the z plane to points in a new w plane
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Linear Transformations
A linear transformation is one that relates w to z by a linear equation of the form w = z + (91)
where and are complex constants Consider for a moment the transformation w = z To see the effect of this transformation, we can write each factor in the polar representation Let
= aei
and as usual, denote z = rei Then w = z = (aei )(rei ) = arei ( + ) If a > 1, then the transformation expands the radius vector of z through the transformation r ar If a < 1, then the transformation contracts the radius vector of z as r ar = (1/ b) r , where b > 1 The transformation rotates the point z by an angle given by
= arg( )
about the origin The w plane is de ned by the coordinate w = u + iv EXAMPLE 91 Explain what the transformation w = iz does to the line y = x + 2 in the x-y plane
SOLUTION Note that
Mapping and Transformations
w = iz = i ( x + iy) = y + ix So we have the relations u = y v=x Hence y = x + 2 u = v + 2 That is, the line is transformed to v = u 2 This linear transformation maps one line into another one, as illustrated in Fig 92 EXAMPLE 92 Consider the transformation w = (1 + i ) z on the rectangular region shown in Fig 93 SOLUTION Notice that 1+ i 1+ i = 2 = 2 (cos /4 + i sin /4) = 2 ei / 4 2
y 5 4 3 2 1 3 2 1 1 z plane 1 2 3 x 3 2 1 1 1 2 3 4
v 1 2 3 x
5 w plane
Figure 92 The transformation w = iz described in Example 91
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Figure 93 A rectangular region to be transformed by w = (1 + i) z in Example 92
This tells us that the transformation will stretch lengths by 2 and rotate points in a counterclockwise direction about the origin by the angle /4 The transformed points are w = (1 + i ) z = (1 + i )( x + iy) = x y + i ( x + y) u= x y v=x+y So the points on the rectangle are transformed according to (1, 0) (1, i ) (1, i ) (0, 2i ) (0, i ) ( 1, i ) (0, 0) (0, 0) The transformed rectangle is illustrated in Fig 94
Figure 94 The rectangle in Fig 93 transformed by w = (1 + i) z It is rotated by /4 about the origin and lengths are increased by 2