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Unformatted text preview: lass Eaten %:%jx% June 9, 2009 10:20~12: 10 A. Deﬁnitions: write down the deﬁnitions or the theorems and answer the questions. (10%) 1. State Laurent’s theorem. 2. Deﬁnition of singularity and isolated singularity. B. True or false (If it is false, explain brieﬂy why it isn’t true). (15%) l -3 possess a pole of order 4 at 2:0. 22(el‘ )— 1) 1. Function f(z) = 2. If w = f (z) is analytic within a domain D of z-plane and maps D to the domain D” of w—plane, then that P(u,v)= P(w) is a harmonic ﬁmction in D“ will result in p(x,y)=p(z) =P(}‘(z)) to be a harmonic function in D. 3. For f(z)- — 3 the Laurent series valid for [2‘ > 3 is z‘l + 3Z_2 + 92‘3 +....... Since there are 2— inﬁnite number of negative power of z = z — 0 , so 2 z 0 is an essential singularity. 20 C. Find the Laurent series of f (z) = centered at 2:2, in the following domains: (10%) (22 -—4.z+8)(z2 —4z-12) 1. jz-2}<2 2. 2<jz—2|<4 3. |z—2l>4 D. Evaluate the following: (25%) 1. ——dz, C is the clockwise contour deﬁned by lz| = (2 +623 2. éCﬂ—_dZC C: [2+3]:Z ()z+2 (z— 1)2 2 3Jsin2" 6d6, n=1, 2, 3.. .(Hint: (x+y)” =Zx" H 338]) COtl'l(7£'S/2) 4. Evaluate the inverse Laplace transform of F(s)= 1 52 + 5. Evaluate P. V j— dx “l +1) E. (a) Find the linear transformation that map the points F0, x=1, x=oo in the z-plane to u=— 1, u=0, and url in the w-plane. This is indeed the normalized Smith chart in Electromagnetic theory. (2%) (b)Please find the mapping lines of x=0, x=1, y=1 , and y=-1, in the right —half of the z-plane to w-Plane; sketch each line accordingly. (8%) ' (10g x22) 1 dx (Hint: Use the contour in Figure 1) (15%) 0 "l- x2 F. Evaluate P.V.OD G. Find the mapping function from the upper-half plane x—y (Fig. 2) to u-v (Fig. 3) plane by following the steps below. 1. Find f ‘(2) which map the x-y plane using the Schwarz-Christoffel transformation to Fig. 4. f 711' should satisfy f(—1)= u], f(0) = 3, f(1) = u1 + 721'. Then extent ul to inﬁnity. (5%) 2. Let Im(f(t))=0 forf<-1, Im(f(t))=7r fort>1,and Im(f(0))=—7;—i,ﬁndﬂz).(10%) Fig. 1 Fig. 3 ...
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