HW8 - W 1.6 CONTOUR INTEGRATION 71 where y is a circle...

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Unformatted text preview: W 1.6 CONTOUR INTEGRATION 71 where y is a circle centred at z = 1. Writing f,(z) = Z4, then _ 11(2) lewd: _ it (ze 1de . and, sincef1(z) is analytic within and on the circle 9!, it follows from (1.48) that jg f(z)dz = 2mg isfmz) = flj(1222)z=l 2=l so that Z4 3 dz = 121:} C (2 e 1) Evaluate fc(zz + 3x) dz along the following contours I‘ Using the Cauchy integral theorem, evaluate the C in the complex 2 plane: contour integral (a) the straight line joining 2 +j0 t0 0 +j2; 22 dz (b) the straight lines from 2 +j0 to 2 + j2 and then m to 0 + j2; C (c) the circle [2] = 2 from 2 +j0 to 0 +j2 in an Where C is anticlockwise direction. I (a) the elrcle |z| = 1 Evaluate Msz“ — 23 + 2) dz around the following (13) the circle Izl = 3 closed contours C in the 2 plane: Using the Cauchy integral theorem, evaluate the (a) the CirCle lZI = 1; contour integral (b) the square with vertices at 0 + jO, 1 + jO, l+jland0+jl; 5zdz (c) the curve consisting of the parabolas :x2 from a 0+;0tol+JIandyzzxfromI+Jt00+Jll L where C is Generalize the result of Example 1.30, and show that _ (a) the Circle |z| = 3 dz jg“ (n =1) (b) the circle |z| = 5 C(z—zu)" 0 (incl) h Using the Cauchy integral theorem, evaluate the where C is a simple closed contour surrounding following contour integrals: the point z = 20. 23 +2 . (a) ——. dz Evaluate the contour integral C (22 + 1)’ dz where C is the unit circle Jz| = 1; Cz—4 4 (b) fil Z 2 dz where Cis any simple closed curve and z : 4 is C (2 - 1N2 + 2) (a) outside C (b) inside C where C is the circle lzl = 3. 73 FUNCTIONS OF A COMPLEX VARIABLE so, by the residue theorem, . 1 _21t [—27C][J_—\§]#\3 Thus Using the residue theorem, evaluate the following contour integrals: Evaluate the integral % 2 dz 2 '3 H +1 (20% (3z-+2)dz where c is c (2—1)<zl+4)' ' ' = l ' v = (a) the Cerle |z[ 2 (b) the Circle H 2 i (i) the circle IE 72‘ = 2 where C15 ” . Evaluate the integral (11) the Guide lzl = 4 z2 + 3jz — 2 3+9 dz (b) (22-22)dz Z Z 5 ( C(z+l)2(22+4) (i) the circle lzl = 3 where C is (a) the circle 12] 2 1 (b) the circle lzl = 4 where (:15 (ii) the circle Iz+jl = 2 Calculate the residues at all the poles of the function 2 2 + 2 + 4 ~ flz) = W ( ) 1 dA 1 (z +1)(z +6) (_(z+l)'(z—l}(z-2) Hence calculate the integral (i) the circle Izl :lE jgflzmz whereas (ii) the circle 12+ 1 | :1 C (iii) the rectangle with vertices where Cis at ij, 3 ij (a) the circle |z| = 2 (b) the circle |z —jl = 1 (c) the c1rcie|z|= 4 (d) § (z_1)dz 2 4 ' Evaluate the integral 5 (-7 r 4“: + 1} § dz (1) the circle lzl : g 2 2 2 CZU”) whereas (ii) mecircte|z+§ :2 where C is (iii) the triangle with vertices at —%+i, -%—j, 3 +10 (a) the circle 121 = (b) the circle |z| = 2 —————_ 1.7 ENGINEERING APPLICATION: ANALYSING AC CIRCUITS 79 Using a suitable contour integral, evaluate the °° (f) 2 xdx ____ “(x2+1)2(x2+2x+2) following real integrals: co °° 23: m dx dx d5 (L[ b (a) I_mx2+x+l ()I_M(x2+l)2 (g) In 3~2c056+sin6 (h) I0x4+1 m fix . w dx c ———__ () I1, (xl+1)(xz+4)2 (I) I_m()cz+4x+5)2 Zn 211 2 c0539 4d9 . c056 0”] 5—4c056d9 mi 5+4sins (“In 3+2cost9d9 1.7 Engineering application: analysing AC circuits 1 ] 1 Z—R+JQJC, Y—Z Writing 1_1+ijR Z‘ R we clearly have R Z_ (1.50) Equation (1.50) can be interpreted as a bilinear mapping with Z and C as the two vari- ables. We examine what happens to the real axis in the C plane (C varies from 0 to 00 and, of course, is real) under the inverse of the In apping given by (1.50). Rearranging (1.50), we have R e Z C h (1.51) i. Taking Z = x + j y c: R—x-iy _ x+iy—R _(x+iyeR)(y+ix) . . — . — 1.52 with +Jy) (MO—ix) wR(x2+y2) ( ) Equating imaginary parts, and remembering that C is real, gives 0=x2+y2—Rx (1.53) which represents a circle, with centre at GR, 0) and of radius éR. Thus the real axis in the. C plane is mapped onto the circle given by (1.53) in the Z plane. Of course, C is positive. If C = 0, (1.53) indicates that Z: R. The circuit of Figure 1.34 confirms that the'transformation w = 1/2, w = u +jv, transfonns the circle 3:2 +y2 = 2m: in the . state the straight line it = l/2a in the w plane. tang conducting wires of radius a are placed and parallel to each other, so that their eetion appears as in Figure 1.41. The are-separated at O by an insulating gap of ble dimensions, and carry potentials i V0 1"inherited. Find an expression for the potential "general point (x, y) in the plane of the cross- and sketch the equipotentiais. , the points A(—1, 0), 13(0, 1), C( 3—; g and — 33(3, 0) in the 2 plane, _ straight liney = 0, the circlex2+y2 =1. te your answer with a diagram showing the . w planes and shade on the w plane the region 'poridington-i~y2 <1. “semicircular disc of unit radius, [(x, y): ' s I, y 2-: 0], has its straight boundary at rature 0 “C and its curved boundary at l00 °C. a that the temperature at the point (x, y) is _ Zfltanef 2y ] 1_x2 _y2 ifihow that the function GCIJ) = 2-750 — y) . satisfies the Laplace equation and construct its-harmonic conjugate H(x, y) that satisfies '0, 0) =0. Hence obtain, in terms of z, where fit-=3: + jy, the function F such that W: F(z) here W: G + jH. how that under the mapping w = 1112, the ‘ harmonic function 60:, y) defined in (a) is mapped into the function Gm, o) = 2e"cos v — e2" sin 21) Verify that 004, L2) is harmonic. (c) Generalize the result (b) to prove that under the mapping w =f(z), Wheref’(z) exists, a harmonic function of (x, y) is transfonned into a harmonic function of (u, U). Show that ifw : (z + 3)/(z — 3), w = u +jv, z at +9, the circle a2 + v2 = k2 in the w plane is the image of the circle 1 + k2 1 7 k2 x2+y2+6 x+9=0 (kzil) in the 2 plane. Two long cylindrical wires, each of radius 4 mm, are placed parallel to each other with their axes 10 mm apart, so that their cross—section appears as in Figure I .42. The wires carry potentials 1V0 as shown. Show that the potential m, y) at the point (x, y) is given by =£iln [(x+3)2+y2]-h1[(x—3)2+fl} Figure 1.42 Cylindrical wires of Exercise 69. Find the image under the mapping w =j(1ez) l+z z:x+jy,w=u+jv,of (a) the points AU, 0), 8(0, 1), C(0, —l) in the 2 plane, (b) the straight line y = 0, (c) the circle 3:2 +y2 : I. A circular plate of unit radius, [(x, y): x2 + y2 S 1], has one half (withy > 0) of its rim, x2 +y2 = l, at temperature 0 °C and the other half (with y < 0) at temperature 100 °C. Using the above mapping, prove that the steady-state temperature at the point (x, y) is ...
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HW8 - W 1.6 CONTOUR INTEGRATION 71 where y is a circle...

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