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(b) jz ij < 1 c z 1 i
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16.59. Prove that the series
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zn is n n 1 n 1
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(a) absolutely convergent,
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(b) uniformly convergent for jzj @ 1.
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16.60. Prove that the series
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converges uniformly within any circle of radius R such that jz ij < R < 2.
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16.61. Locate in the nite z plane all the singularities, if any, of each function and name them: a z 2 ; 2z 1 4 b z ; z 1 z 2 2 c z2 z2 1 ; 2z 2 1 d cos ; z e sin z =3 ; 3z  f cos z : z2 4 2
Ans. (a) z 1, pole of order 4 2 (b) z 1, simple pole; z 2, double pole (c) simple poles z 1 i
(d) z 0, essential singularity (e) z =3, removable singularity ( f ) z 2i, double poles
16.62. Find Laurent series about the indicated singularity for each of the following functions, naming the singularity in each case. Indicate the region of convergence of each series. a cos z ;z  z  a b z2 e 1=z ; z 0 c z2 ;z 1 z 1 2 z 3
Ans:
1 z  z  3 z  5 ; simple pole, all z 6  4! 6! z  2! 1 1 1 1 ; essential singularity, all z 6 0 2! 3! z 4! z2 5! z3
b z2 z
CHAP. 16]
FUNCTIONS OF A COMPLEX VARIABLE
1 7 9 9 z 1 ; double pole, 0 < jz 1j < 4 256 4 z 1 2 16 z 1 64
RESIDUES AND THE RESIDUE THEOREM 16.63. Determine the residues of each function at its poles: a Ans. 2z 3 ; z2 4 b z3 z 3 ; 5z2 c ezt ; z 2 3 d z : z2 1 2
(a) z 2; 7=4; z 2; 1=4 (b) z 0; 8=25; z 5; 8=25
(c) z 2; 1 t2 e2t 2 (d) z i; 0; z i; 0
16.64. Find the residue of ezt tan z at the simple pole z 3=2. Ans: e3t=2 z2 dz , where C is a simple closed curve enclosing all the poles. C z 1 z 3 8i
16.65. Evaluate Ans:
16.66. If C is a simple closed curve enclosing z i, show that zezt dz 1 t sin t 2 2 1 2 C z 16.67. If f z P z =Q z , where P z Q z are polynomials such that the degree of P z is at least two less than and the degree of Q z , prove that
f z dz 0, where C encloses all the poles of f z .
EVALUATION OF DEFINITE INTEGRALS Use contour integration to verify each of the following 1 2 x dx  16.68. p 16.75. 4 2 2 0 x 1 1 16.69. dx 2 ; x6 a6 3a5 1 dx  x2 4 2 32 a>0 16.76.
2
p d 4 3 9 2 cos  2
sin2   d 5 4 cos  8 d 3 p 1 sin2  2 2 2 cos n d 1 2a cos  a2 n 0; 1; 2; 3; . . . ; 0<a<1 a > jbj
1 16.70.
2 16.77.
2 1 p x  16.71. dx 3 3 0 x 1 1 16.72.
2an ; 1 a2 a>0 2 16.79.
dx 3 p a 7 ; a4 2 8 2
d 2a2 b2  2 ; 3 a b cos  a b2 5=2 x sin 2x e 4 dx 2 4 4 x cos 2x e  dx 8 x4 4
1 16.73.
dx  2 2 2 9 1 x 1 x 4 d 2 p 2 cos  3
1 16.80.
2 16.74.
1 16.81.
1 16.82.
FUNCTIONS OF A COMPLEX VARIABLE x sin x 2 e  dx 2 1 2 4 x sin x  2e 3 dx 4e x x2 1 2 cos x  dx . cosh x 2 cosh =2 Hint: Consider ! Then let R ! 1: 1 16.84.
[CHAP. 16
sin2 x  dx 2 x2 sin3 x 3 dx 8 x3
1 16.83.
1 16.85.
1 16.86.
eiz dz, where C is a rectangle with vertices at R; 0 , C cosh z
R; 0 ; R;  r R;  .
MISCELLANEOUS PROBLEMS 16.87. If z ei and f z u ;  i v ;  , where  and  are polar coordinates, show that the Cauchy-Riemann equations are @u 1 @v ; @  @ @v 1 @u @  @
16.88. If w f z , where f z is analytic, de nes a transformation from the z plane to the w plane where z x iy and w u iv, prove that the Jacobian of the transformation is given by @ u; v j f 0 z j2 @ x; y 16.89. Let F x; y be transformed to G u; v by the transformation w f z . Show that if all points where f 0 z 6 0, @2 G @2 G 0. @u2 @v2 @2 F @2 F 0, then at @x2 @y2
az b , where ad bc 6 0, circles in the z plane are trans16.90. Show that by the bilinear transformation w cz d formed into circles of the w plane. 16.91. If f z is analytic inside and on the circle jz aj R, prove Cauchy s inequality, namely j f n a j @ where j f z j @ M on the circle. n!M Rn
[Hint: Use Cauchy s integral formulas.]
16.92. Let C1 and C2 be concentric circles having center a and radii r1 and r2 , respectively, where r1 < r2 . If a h is any point in the annular region bounded by C1 and C2 , and f z is analytic in this region, prove Laurent s theorem that f a h where an 1 2i
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