# HW5 - 0MPLEX VARIABLE For 2 = x jy ﬁnd the image region...

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Unformatted text preview: 0MPLEX VARIABLE For 2 = x + jy ﬁnd the image region in the w plane corresponding to the semi-inﬁnite 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: IZ-2+Jl=|Z-j+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—l-jl=lz-3+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 half-plane 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 half-plane x > 0 in the zplane maps onto the half-plane 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 az-l—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)(1-j)—(Z+1)(-j- 01' wz—Jz—l as before. 1.0 11" ' "12: Show that if z = x + jy, the image of the half-plane 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, 1-J

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