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Unformatted text preview: Complex Variable Final Exam June 27, 2006 (21 %) True or false (If it is false, explain brieﬂy why it isn’t true) 1. A. If w =f{z) is analytic within a domain D of z-plane and maps D to the domain 0’ of w—plane, then P(u, v)=P(w) is a harmonic function in D” will result in p(x, y)=p(z) to be harmonic function in D. B: ﬂz)=P(z)/Q(z), P(x) is a complex polynomial of order n, and Q(x) is a complex polynomial of power m, ifm > n, a > 0, and the contour CR is {212 = Remﬂ S 8 S 271:}, then { f(z)el”‘dz approach 0 as R approaches inﬁnity. f(2) = 1 possess a pole of order 2 atz = 0. 22 e: —1 For f(z) : i/(z k 3), the Laurent series valid for ‘2‘ > 3 is z"l + 32'2 + 92'"3 + Since there are an inﬁnite number of negative powers of z = z - O, z = 0 is an essential singularity. l .- . . . . f z 2 has an inﬁnlte number of Singular pornts and all of the smgular pornts are ( ) sinlr/Zl isolated singularities. z — 1 z _ 2 the linear fractional transformation has the so-called circle preserving property. The image of the circle ‘2 — 1| =1 under the complex mapping T(z) 2 is a circle since The complex function f (z) = z + % is analytic except at z = 0, therefore the mapping w :ﬂz) from w—plane to Z—plane is conformal except at z I 0. i (8 0/o) Find the Laurent series of f(2) = m , centered at z = 0, in the following domains: A. lzl<1 B. 1<|zl<3 . C. yzl>3 (24 %) Evaluate the following: A. B. C. D. x-dx ”lx2+llx3+2x+2l Ismzng'd‘ganzl,2,3,...;hint: (x+y)(::x"y"“k[ n ] k=0 H—k cscz - dz , where C is the clockwise contour deﬁned by 9):2 + 25y2 = 900 coth (as/2) Solve the inverse Laplace transform of F (s) = 2 1 ' S + .Hint: f(t)=%P.V. i'”F(s)ds 7:: -Im (15 %) In Fig. 1, ﬁnd an appropriate conformal mapping and solve the Dirichlet problem. E Hint: ﬁrst consider mapping Fig. l to the upper . 2 DE 1 V V = O=R = lle < AWL?) < 71/4} complex plane; then try to solve the new BC 2 ”(11): 1: 11(13): 0 problem using a linear combination of F, = {zllzl <1,Arg(z)= 0,Arg(z) = 3/4} %2' Arg(w + a). 11 : {z|1z| > 1, Arg(z) = 0,Arg(2) = 3/4} 5. (l6 %) (a) Find the linear fractional transformation that map the points F0, x:l, x=oo in the Z—plane to uz-l , u=0, and u=1 in the W-plane. This is in fact, the normalized Smith chart in transmission theory and microwave engineering. (b) Please ﬁnd the mapping lines of x=0, x=l, yzl, and y=-l, in the right-half of the Z—plane to W-plane; mark each line accordingly. 6. (16 %) SchwarZ—Christoffel Transformation ~ Find the mapping function from the upper-half complex plane x-y (Fig. 2) to u-v (Fig. 3) plane. A. First, find f ‘ (z) . Map the x-y plane using the Schwarz—Christoffel transformation to Fig. 4, ﬁrst, then extent an to inﬁnity. Require that f (—1) = It1 , f (0) = 721' / 2 , and f (1) = ul + m' . B. From A, ifwe require that Im(f(t)) = O for t < —] , Im(f(t))= It, 1‘ >1, and Im(f(0 )= :ri/2,f1ndf(z). J. y ’ Vﬂ’wﬁﬁg’ﬁgx‘ff/ . " ' .‘ .r A? a: “ y‘Wﬂi t . . ' 2* ~ 2‘5 Fig. 3 Fig. 4 ...
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