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Unformatted text preview: 0MPLEX VARIABLE For 2 = x + jy ﬁnd the image region in the w plane
corresponding to the semiinﬁnite strip 3: > 0, 0 < y < 2 in the 2 plane under the mapping w = jz + 1. Illustrate the regions in both planes. ‘Zthe cartesian form y = mx + c (m and c real
'ts) the equations of the following straight in thezpiane,z=x+jy: IZ2+Jl=Zj+3l ' __ a: _
(b) 2+ 2* + 41(2 Z )_ 6 Find the images of the following curves under
where * denotes the complex conjugate. the mapping I Find the point of intersection and the angle of w : (J + "mg +553 _ 1 intersection of the straight lines (a) y = 0 (b) x = 0 lz—ljl=lz3+j (c)x2+y2=l (d)x2+y2+2y:l fz_l+ji=iz_3_j wherez=x+jy. . The ﬁmction w = jz + 4 — 3j is a combination of
translation and rotation. Show this diagrammatically,
following the procedure used in Example 1.2. Find
the image of the line 6x +y = 22 (z = x +jy) in the
w plane under this mapping. The mapping w = 062 + 13 (a, ,3 both constant
complex numbers) maps the point z = l + j to
the point w = j and the point z = —l to the point
w : 1 + j. (a) Determine a and ,8. (b) Find the region in the w plane
corresponding to the upper halfplane
Irn(z) > 0 and illustrate diagrammatically. (c) Find the region in the w plane corresponding to
the disc z < 2 and illustrate diagrammatically. (d) Find the ﬁxed point(s) of the mapping. Show that the mapping w = (l — j)z, where w = u +
jv and z = x +jy, maps the region y > 1 in the 2 plane onto the region at + v > 2 in the w plane.
Illustrate the regions in a diagram. Under the mapping w =jz +j, where w = u +_jv
andz = x + jy, show that the halfplane x > 0
in the zplane maps onto the halfplane v > 1 in the In (b)—(d) use the values of 0: and B determined w plane. in (a). 1.2.3 Inversion The inversion mapping is of the form
1 w = _ (1.7)
Z and in this subsection we shall consider the image of circles and straight lines in the
2 plane under such a mapping. Clearly, under this mapping the image in the w plane of the general circle '2 — Zoi 7“ V
in the 2 plane, with centre at 20 and radius r, is given by
1 ———z
0
w = r (1.8) but it is not immediately obvious what shaped curve this represents in the w plane. To
investigate, we take w = u + jv and 20 = x0 + jyo in (1.8), giving and from (1 .l7b) c =ja. Thus azl—a 1
w:.—.=T
Jaz—Ja J z+l_
zelﬁ ‘J 1.2 COMPLEX FUNCTIONS AND MAPPINGS 23 .z+l 2—1 Alternatively, using (1.16) we can obtain (W J')(1* 0) _ (Z  0)(j + 1) (Wr0)(1j)—(Z+1)(j 01' wz—Jz—l as before. 1.0 11" ' "12: Show that if z = x + jy, the image of the halfplane
y > c (c constant) under the mapping w = 1/2 is the
interior of a circle, provided that e > 0. What is
the image when G = 0 and when 0 < 0? Illustrate
with sketches in the w plane. :14 Determine the image in the w plane ofthe circle
3 ' 7'
Z + 3 +_] ‘: 3 under the inversion mapping w = 1/2.
Show that the mapping w = 1/2 maps the circle z u a [ : a, with a being a positive real constant,
onto a straight line in the w plane. Sketch the corresponding curves in the z and w planes, indicating the region onto which the interior
of the circle in the 2 plane is mapped. Find a bilinear mapping that maps 2 2 0 to w = j,
z=—j to w: l and z : —l to w = 0. Hence sketch
the mapping by ﬁnding the images in the w plane
ofthe lines Re (2) = constant and Im(z) : constant in
the 2 plane. Verify that z = %(j — ]){—l : V93) are
ﬁxed points of the mapping. The two complex variables w and z are related
through the inverse mapping W Z (21) Find the images ofthe points z = l, l —j and
0 in the w plane. (b) Find the region of the w plane corresponding
to the interior of the unit circle z < l in the
2 plane. = l + j 0) (c) Find the curves in the w plane corresponding
to the straight lines x = y and x +y = l in the
2 plane. (d) Find the ﬁxed points of the mapping. Given the complex mapping _z+l
_z—l W where w = u +jv and z = x + jy, determine the
image curve in the w plane corresponding to the
semicircular arc x2 +}12 = l (x 53 0] described from
the point (0, —l) to the point (0, l). (a) Map the region in the 2 plane (2 = x + jy) that
lies between the lines at = y and y = 0, with x < 0,
onto the w plane under the bilinear mapping z+j
2—3 (Hz'nr: Consider the point w = § to help identify
corresponding regions.) (b) Show that, under the same mapping as in (a),
the straight line 3): + y : 4 in the 2 plane
corresponds to the unit circle w = I in the
w plane and that the point w = 1 does not
correspond to a ﬁnite value of z. w: If w : (z —j)/(z +j), ﬁnd and sketch the image in
the w plane corresponding to the circle z = 2 in the
2 plane. Show that the bilinear mapping jﬂv z — zo w=e
z—z’g ...
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 Fall '10
 ChinC.Lee
 1 w, ZOI, 1J

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