C:\Documents and Settings\00002

Chapter 4 Integrals Exercise 1( Integration of analytic function)

1. Evaluate the integral

a.



c

sin²z dz, C from −i toialong|z|  in the right half plane.

b.



c

ze^z2dz, C from 1 along the axis to i.

c.



c

coszdz, Cis the semicircle|z|  , x ≥ 0,from−itoi..

d.



i

i 2

e^zdz.

e.



0

2i

cos^z₂dz.

Exercise 2 ( Line Integral)

1. Find a parametric representation z _ z_t_for the following:

a. The straight line segment from 0 to 4-7i b. |z_3−i_{| } 5, counterclockwise.

c. 4x² ₋9y² _ 36,counterclockwise.

d. y _ x³, from (-2,-8) to(3,27).

2. Use parametric representation for C to evaluate



c

f_z_dz.

a. f_z_{ } ^z_2

z ^{and C is}

i. the semicircle z _ 2e^i (0≤  ≤ 

ii. the circlez _ 2e^i_0≤  ≤ 2.

b. f_z_{ } z₋1 and C is the arc fromz _ 0 toz _ 2 consisting of i. the semicircle z _ 1e^i_{ ≤  ≤ 2.}

ii. the segment 0 ≤ x ≤ 2 of the real axis.

c. f_z_{ } ⁴y when y _ 0

1 when y  0 and C is the arc from z-1-i to z1i along the curvey  x³

d. fz  Rez²,and C is the unit circle ,counterclockwise.

3. Let C be the arc of the circle|z|  2from z2 to z2i that lies in the first quadrant. Without evaluating the integral, show that



c

dz

z² ₋1 ≤  3

Exercise 3 ( C.I.Theorem- C.I.Formula- Derivative of analytic function) 1. Let C denote the positively oriented boundry of the square whose sides lie along the linesx  2 and y  2.Evaluate each of these integrals

a.



c

e^z

z−^i₂ dz, b.



c

cosz

zz²8dz; c.



c

z 2z1 dz,

d.



c

tan ₂^z

z−x0² dz _{−2 } x0  2, e.



c

coshz z⁴ dz.

2. Find the value of the integral of g(z) around the circle |z−i|  2in the positive sense when

a. gz  1

z² _4, b. gz  1

z²_4^{2 .}

3. Let C denote the boundary of the domain between the circle |z_{| } 4 and the square whose sides lie along the linesx  1,y  1assuming that C is oriented so that the points of the domain lie to the left of C ,point out why



c

f_z_dz _ 0when

a. f_z_{ } ¹

3z²1, b. f_z_{ } z_2

sin^z₂, c.f_z_{ } z 1−e^{z .}

4. Show that



c

fzdz  0where C is the circle|z|  1,and when

a. f_z_{  z−3}^z2 ; b.f_z_{ } ze^−z; c. fz  1

z² 2z2. 5. Integrate z² _1

z² −1 on the unit circle with center at:

a. z  1; b.z  ¹₂; c.z  i. 6. Evaluate the following integrals:

a.



c

2e^z 2 cosz−4

z dz; c : |z|  1.

b.



c

z³_2z² _z_2

z−i dz; c : |z|  2.

c.



c

1

zz−2dz; c : |z|  3.

d.



c

e^2z

z1⁴ dz; c : |z|  3.