ONLINE EXAMINATIONS [Mid 2 - M3] 1. the Machaurin's series for log (1+z)=

Transcription

1 ONLINE EXAMINATIONS [Mid 2 - M3] 1. the Machaurin's series for log (1+z)= Expand cosz into a Taylor's series about the point z= cosz= cosz= cosz= cosz= 4. Expand the function about z=π 5. Expand the function in Laurent's series, for

2 6. The Taylor's series for f(z0 at z=z o is given by f(z)=, where a n= 7. The Maclaurin's series for f(z) at z 0=0 is given by 8. The Taylor's series expansion of f(z0=e z about z=a is 9. The Maclaurin's series for e -z =

3 10. Expand f(z)=sinz in a Taylor's series about z=π/4 11. The Laurent's series of at z=0 is 12. Expand as Laurent's series about z=0 1+z+z 2 +z z n z+z 2 -z (-1) n z n The Laurent's series of f(z)= about z=1 is

4 14. The Laurent's series expansion of f(z)= about z=0 is 15. The Laurent's series expansiono of f(z0=e 1/z about z=0 is 16. the Laurent's series of f(z)= about z=0 is 17. Maclaurin's series for e z =

5 18. Maclaurin's series for sinz = 19. Standard maclaurin's series for cosz= 20. Standard Maclaurin's series for cosh z= 21. If then tqaylor's series expansion of log (1+z) about z=0 is

6 22. Taylor series expansion of in is 23. What kind of singularity have the function at z=2πi is a simple pole a double pole an isolated essential singularity non-isolated singularity 24. What kind of sigularity have the function at z= is a simple pole a double pole an isopated essential singularity non-isolated essential singularity 25. What kind of singularity have the function at z=a is a simple pole a double pole an isolated essential singularity non-isolated essential singularity 26. The zeros of are z=±i z=±1 z=0 z=±i, ±1

7 27. The singularity of are z=±i z=±1 z=0 z=±i, ±1 28. What kind of singularity have the function f(z)= at z= is a simple pole a double pole a non-isolated essential singularity an isolated essential singularity 29. What kind of singularity have the function f(z)= at z=1 is a simple pole a double pole a non-isolated essential singularity an isolated essential singularity 30. What kind of singularity have the function f(z)= at z=0 is a simple pole a double pole a non-isolated essential singularity an isolated essential singularity 31. Let f(z)= (classify the singularity) z=0 is a removable singularity z=0 is a pole z=0 is an essential singularity z=0 is a simple pole 32. Let f(z)= (classify the singularity) z=0 is a removable singularity z=0 is an essential singularity z=0 is a simple pole z=0 is a double pole 33. Let f(z)= (classify the singularity) z=0 is a removable singularity z=0 is an essential singularity z= 0 is a simple pole z=0 is not isolated singularity point 34. Find the zeros of f(z)=z 2 +1 f(z) has no zeros z=±1 z=±i z=1, i 35. Find the zeros of

8 z=0 36. Let a be an isolated singularity for f(z). then a is called a removable singularity if the principal part of f(z) at z=a has a finite number of terms no terms an infinite number of terms one term 37. Let a be an isolated singularity of f(z). The point a is called a pole if the principal part of f(z) at z=a has a finite number of terms no terms an infinite number of terms one term 38. Let a be an siolated singularity of f(z). The point a is called an essential singularity of f(z) at z=a if the principal part of f(z) at z=a has a finite bumber of terms no terms an infinite number of terms one term 39. A pole of order 1 is called a pole a simple pole a double pole a singularity 40. What kind of singularity have the function sec at z=0 is a simple pole a double pole an isolated essential singularity non-isolated essential singularity 41. What kind of singularity have the function at z= is a simple pole a double pole an isolated essential singularity non-isolated essential singularity 42. What type of singularity have the function is a simple pole a double pole an isolated essential singularity at z=2πi non isolated essential singularity at z=2πi 43. What kind of singularituy have the function cot z at z= is a simple pole a double pole an isolated essential singularity non-isolated essential singularity 44. The poles of are at z=

9 z=0, ±1 z=0, ±2 z=0,2 z=0,1 45. What kind of singularity have the function at z=0 is a simple pole a double pole an isolated essential singularity non-isolated essential singularity 46. What kind of singularity have the function at z=0 is a simple pole a double pole an isolated essential singularity non-isolated essential singularity 47. What type of singularity have the function is z=1 is a pole of 4 th order z=1 is a double pole z=1 is essential singularity z=1 is non-essential singularity 48. What type of singularity have the function is z=0 is a pole z=0 is a double pole z=0 is essential singularity z=0is non-essential singularity 49. Find the zeros of f(z)= z=±nπ, n=0, 1, 2, z=±2nπ, n=0, 1, 2, The residue of at z=-2 is 51. Residue at z= of Tanz is

10 The residue of at z=0 is 53. The residue of at z=2 is 54. The residue of at z=-1 is 55. If f(z) has a simple pole at z=a, then Res{f(z), a}=

11 56. If f(z) has a pole of order m at z=a, then Res{f(z), a}= 57. The value of is 0 4πi πi 2π 58. Residue of at z=i is i 59. Poles of are given by z=-1 z=i z=±1 z=±i 60. The residue of e z.z -5 at z=0 is If f(z)= then the residue of f(z) at z=0 is

12 62. the residue of z=iπ is 63. If f(z)= then the residue of f(z) at z=1 is e e 2 e-1 e If f(z)= then the residue of f(z) at z=0 is Residue at Z=1 of f(z)= is Residue at z=3 of is

13 67. If 'a' is a simple pole, then Ren is (z-a) Φ(a) 68. If f(z)=- then the residue of f(z) at z=1 is If f(z)= then the residue of f(z) at z=2 is The residue of at z=1 is 3e 2e 71. The residue of at z=1 is 72. The residue of at z=-2 is

14 73. The residue of z=-3 is The value of the integral, C is is πi 2πi The value of, C is is 0 2πi 8πi 2πie The value of where C is the circle x 2 +y 2 =4 is 0 πi -πi 2πi 77. The residue at z=0 of the function f(z)= is The residue of at z=-1 is

15 The value of where c is is 2πie 4 2πi The residue at z=0 of is The value of, where C is the circle, is 0 2πi -2πi πi 82. Residue of tanz at z= is Singular point of are z=1,2 z=π,2 z=0,1,2 z=-1, Res = If f(z) has a pole of order three at z=a, then Res[f(a)]=

16 86. Residue of at z=0 is The value of is 0-2πi 2πi πi 88. If C is 0-2πi 2πi πi 89. The value of, C being, is 0 2πie 3 2πie 90. The residue of at z=2 is 91. If f(z)= then residue of f(z) at z=0 is

17 1 92. The residue of at z=-1 is 93. The residue of at z=i is 94. the residue of at z=-i is 95. Residue of at z=i is

18 96. The residue of at z=-i is 97. The residue of at z=1 is 98. The residue at z=1 is e 2 e 4 -e 2 -e The residue of at z=2 is e 2 e 4 -e 2 -e Write the value of where c is the circle z = 1 0

19 2πi 101. The value of, where z = 1 is 0 2πi 102. If then write the value of 2π[4-15] 2π[4+ 15] 2π( 15) 103. If then find the value of 104. To evaluate the integrats of the type the contous used is any circle unit circle semi-circle rectangle 105. To integrate, we should use circular contour

20 indented semi-circular contour rectangular contour unit circle To integrate, we should use circular contour indented semi-circular contour rectangular contour unit circle 107. To integrate, we will use a contour circular contour z = R real axis and lower half of circle z =R real axis and upper half of circle z = R real axis and unit z = If then write the value of 109. The number of zeroes of f(z) = z 4-8z+10 which lie in the annules region 1< z < 3 is The number of zeros of f(z) = 2z 5 -z 3 +3z 2 -z+8 which lie inside c: z = 1 is The number of zeros of f(z) = 2z 4-2z 3 +2z 2-2z+9 which lie inside c: z = 1 is If f(z) = and C is z = 2, then = 0 2πi 6πi

21 4πi 113. If f(z) = and C is z = 10, then dz = 0-5πi -12πi Every polynomial of degree n has exactly (n-1) zeros (n+1) zeros n zeros 2 zeros 115. If f(z) and g(z) are analytic with in and an a closed curve c, and 1g(z) < f(z) on C, then f(z) and g(z) have same number of zeros inside C f(z) and f(z)+g(z) have same number of zeros inside C f(z) and f(z)-g(z) have same number of zeros insidec f(z) and f(z) g(z) have same number of zeros inside C 116. The number of zeros of f(z) = 2z 5 +8z-1, and c: z = 2 which lie in c is If f(z) = z 5-3iz 2 +2z-1+i, then evaluate = where C is encloses all zeros of C. 0 4πi 10πi 5πi 118. If f(z) = and C is z 2, then = -2πi -4πi -6πi -12πi 119. If (z) = z 7 +4z and C is z = 1 then the number of zeros which lie inside C is IF f(z) = and C is z = 4, then = 0-2πi -4πi -8πi 121. If f(z) = tanπz, and C is z = π, then = 0 2πi

22 4πi 6πi 122. If f(z) = sin πz, and C is z = π, then 0 7πi 14πi 10πi 123. If f(z) = cos πz, and C is z = π, then 0 6πi 12πi 14πi 124. If f(z) is analytic and bounded, then f(z) is zero a constant linear a polynomial of degree n 125. If z = a, is a pole of multiplicity p and z = b is a root of multiplicity n and f(z) 0 on c, then dz is equal to 0 2πip 2πi(n-p) 2πi n 126. If f(z) = z 7-5z and C is z = 2, then is equal to 2πi 7πi 4πi 14πi 127. If f(z) = z 7-5z and C is z = 1, then is equal to 0 πi 2πi 14πi 128. If f(z) = and c is z = 2, then is equal to 0 πi 2πi 14πi 129. The number of zeros of f(z)=2 7-4z 3 +z-1, which lie inside is 0 7 3

23 The number of zeros of f(z)=2 7 -z 3 +12, which lie in the annulur region is The number of zeros of f(z)=e z -4z n +1 which lie inside is 0 n-1 n n The number of zeros of f(z)=z 9-2z 6 +z 2-8z-2 which lie inside is The number of xeros of f(z)=az n -e z which lie inside C is 0 n-1 n+1 n 134. the number of zeros which lie inside of z 4 +3z 3 +6 is The number of zeros of z 4 -z 3 +9z 2 +2 which lie in the annulus is The number of zeros of z 4 -z 3 +9z 2 +2 in is The number of zeros of z 4 +3z 3 +6 which lie in the anmulus is The number of roofs of z 6-5z 4 +z 3-2z which lie in the annulus is

24 The image of the line y = under the transformation w = is u 2 + v 2 = 1 u 2 + (v+1) 2 u 2 + (v-1) 2 = 1 u 2 + v 2 = 140. The image of the line y = 2 under w = is v = 0 u = The image of the line x = 0 under the transformation w = is υ = 0 v = 0 u = 0 V = The image of the line y = x-1 under the transformation w = is u 2 + v 2 -u-v =0 u 2 + v 2 +u+v =0 v = 0 4v+1 = The image of the line x = 0 under the transformation w = is u = 0 υ = 0 u = 1 V = If z = r and w = R, then under the transformation w =, R is equal to r r 2 2r 145. If θ is argz and Φ is arg w, under the transformation w =, then Φ = θ -θ

25 2θ 146. If the circle passes through the origin, then the image of the circle under the transformation w = is a straight line a circle unit circle a rectangle 147. If the circle does not pass through the origin, then the image of the circle under the transformation w = is y = mx y = mx + c a rectangle a circle 148. The image of the point at infinity, under the transformation w = is ( 0,0) (0,1) (1,0) (i,i) 149. The image of line y=2x under the transformation W=z 2 is u=2v 4u=3V 4u=-3V 4u=V 150. The image of the line x=1 under the transformation W=Z 2 is V 2 =4u V 2 =u V 2 =-4u V 2 =4(u+1) 151. The image of the region under the transformation W=z 2 is 152. If and, then under the transformation w=z 2, R is equal to r r 2 2r 153. If θ is arg z and & is argw under a the transformation W=z 2, then Φ is equal to

26 θ 2θ 2θ The image of the straight line y=x under the transformation W=z 2 is u=0 u=v u=-v V= the image of the line y=-x under the transformation W=z 2 is u=0 u=v u=-v V= The critical point of W=z 2 is The image of the infinite strip between the lines x=1 and x=2 under the transformation W=e z is The region between the lines x=1 and x=2 The annular region between the circles and The real axis The upper half plane 158. the image of the region between the liones y=0 and y=π/2 under the transofrmation W=e z is The real axis The imaginary axis The first quadrant of W-plane The upper half plane 159. The image of the region o y under the transformation W=e z is The first quadrant of w-plane The region The region The line y=x 160. The image of the region between the lines y=0 and y=π under the transformation W=e z is lower half of the plane upper half of the plane the real axis the imaginary axis 161. The image of the strip -a<x<0, under the transformation W=e z is interior of the unit circle extension of the unit circle the real axis the imaginary axis 162. The image of the line x=0 under the transformation W=e z is

27 The unit circle The line W=1 The line V=1 The circle 163. The image of the real axis under the transformation W=e z is u=0 V=0 u=e 164. The image of the y-axis under the transformation W=e z is u=1 v=1 u 2 +v 2 =1 u 2 +v2= The critical points of W=e z are 1 and -1 1 and 0 2 and -2 N0 critical points 166. The image of the annular region between the circles and under the transformation W=logz is The region between the lines u=0 and u=logz The region between and The region between V=0 and V=2 The region u=1 and u= The image of argz=α under the transformation W=Z+ is u 2 +V 2 = The image of the circle under the transformation W=Z+3+2i is (u+3) 2 +(V+2) 2 =4 (u-3) 2 +(V-2),sup>2=4 u 2 +V 2 =4 x 2 +y the image of the line y=x under the transformation is

28 170. The image of a circle in Z-plane under the transformation W=Z+ is an ellipse a staight a circle a hyperbola 171. The image of, under the transformation W=logz is u=0 v=0 u= The image of the line y=0 under the transformation W=logz is u=0 V=0 u= The image of the line y=x, under the information W=logZ is V=0 V=Π 174. The image of, under the transformation W=logZ is u=0 u=log a V=0 V= loga 175. The critical points of W=Z+ are 1 and -1 1 and 0 2 and -2 0 and the image of the line o y=a, under the transformation W=sinz is an ellipse a hyperbole a circle a straight line 177. The image of the line x=a, under the transformation W=sinz is an ellipse a hyperbola a circle a straight line 178. The image of the line y=b, under the transformation W=cosz is an ellipse a hyperbola a circle

29 a straight line The critical points of W=cosZ are 0 nπ (2n+1)π 180. The image of the line x=a, under the transformation W=cosZ is an ellipse a hyperbola a circle a line 181. The critical point of the transformation W=sinz is 0 nπ (2n+1)π 182. If f(z) is an analytic function than the transformation W=f(z) is conformal if f(z)= The point at which is callled the fixed point the invariant point the critical point conformal point 184. The critical points of are a, -a -1, 1 0, If z=x+iy and W=, under the transformation W=e z, then R= e z e y x y 186. The bilinear transformation which maps the points z=1, -1, i to the points W=0,, i is

30 187. The bilinear transformation which maps the points &infini;, i, o into the points o, i, is W= 188. The invariant points of are 1, -1 i, -i 189. The bilinear transformation, which maps the points z=-1, 2, i-1 into the points w=,, i-1 is 190. The bilinear transformation, which maps the points i, 0, 1 into the points 1-i, 0 is W=

31 191. The invariant points of the bilinear transformation are (0, 0) -1, 1 1, 1 i, -i 192. The fixed points of W= are ±1 ±i 0-1, The fixed point of the transformation are z=i z=-i z=±i z=&plusdmn; The fixed points of the tranformation are z=3, 3 z=-3, 3 z=-3, -3 z=3i 195. A critical point of the transformation W=z 2 is z=i z=-i z=0 no critical points