240 CHAPTER 6 • COMPLEX INTEGRATION
7. Find fct(o) z -^3 sinh (z^2 ) dz.- Find fcz-^2 sinz dz along the following contours:
(a) The circle C{ ( ~).
(b) The circle Ct ( 7).- Find fct(O) z-" exp z dz, where n is a positive integer.
- Find fc z-^2 (z^2 - 16)-^1 expz dz along the following contours:
(a} The circle Ct (0).
(b} The circle Ct (4).11. Find fct(i+i) (z^4 + 4r
1
dz.- Find fc z -^1 (z -1)-^1 expz dz along the following contours:
(a} The circle ct (0).
(b) The circle Ci (0).
- Find fc (z^2 + lr^1 sinz dz along the following contours:
(a) The circle Ct (i).
(b} The circle ct ( -i).- Find fct<•l (z^2 + 1)-
2
dz. - Find fc (z^2 + 1)-
1
dz along the following contours:
(a.) The circle C{ (i).
(b} The circle Ct (-i).- Let P (z) = ao +a1z+<hz^2 +asz^3. Find fct(o) P (z)z-n d.z, where n is a positive
integer. - Let z1 and z2 be two complex numbers that lie interior to the simple closed contour
C with positive orientation. Evaluate fc (z -z 1 ) -^1 (z - z2)-^1 dz. - Let f be analytic in the simply connected domain D and let z 1 and Z2 be two
complex numbers that lie interior to the simple closed contour C having positive
orientation that lies in D. Show that
State what happens when z2 - z 1.- The Legendre polynomial Pn (z) is defined by
Pn (z} = 2 !n!:; [ (z^2 - l)"].