234 CHAPTER 6 • COMPLEX INTEGRATION
- f c cos z dz, where C is the line segment from - i to 1 + i.
- fc expz dz, where C is the line segment from 2 to i~.
- fc zexpz dz, where C is the line segment from - 1 -i~ to 2 + itr.
5. f c 1¥ dz, where C is the line segment from 1 to i.
- f c sin~ dz, where C is the line segment from 0 to tr - 2i.
- fc (z^2 + z-^2 ) dz, where C is the line segment from i to 1 + i.
- fc zexp (z^2 ) dz, where C is the line segment from l - 2i to 1+2i.
- f 0 zcosz dz, where C is the line segment from 0 to i.
- f c sin^2 z dz, where C is the line segment from 0 to i.
- fc Log z dz, where C is the line segment from 1 to 1 + i.
- f c ,f:_,, where C is the line segment from 2 to 2 + i.
- f c ~C! dz, where C is the line segment from 2 to 2 + i.
- f c ,'2-_^2 , dz, where C is the line segment from 2 to 2 + i.
- Show that fc 1 dz = z2 - z 1 , where C is the line segment from z 1 to Z2, by
parametrizing C. - Let z1 and z2 be points in the right half-plane and let C be t he line segment joining
them. Show t hat J. c~ a• = .!. z1 - .!. z:2. - Let z! be the principal branch of the square root function.
(a) Evaluate fc 4, where C is the line segment joining 9 to 3 + 4i.
2•}(b) Evaluate fc zt dz, where C is the right hall of the circle Ci (0) joining
- 2i to 2i.
- Using partial fraction decomposition, show that if z lies in the right half-plane and
C is the line segment joining 0 to z, then
[ ~ 2 ~ 1 = Arctan z = ~ Log(z+i)-~ Log(z-i) + ~-
- Let f^1 and g' be analytic for all z and let C be any contour joining the points z 1
and z2. Show that
[1 (z)g' (z)dz = J (z2)g(z2)- f (z1)g(z1)-[J' (z)g(z)dz.
- Compare the various methods for evaluating contour integrals. What are the
limitations of each method? - Explain how the fundamental theorem of calculus studied in complex analysis and
the fundamental theorem of calculus studied in calculus are different. How are
they similar? - Show that fc z'dz = (i - 1) I±r·, where C is the upper half of Ci (0).