Problem Set - 11
MATHEMATICS-I(MA10001) Autumn 2018
1. (a) Evaluate Z2+i
1โi
(2x+iy + 1)dz, along the paths
(i) x=t+ 1, y = 2t2โ1
(ii) the straight line joining 1 โiand 2 + i
(b) Evaluate Z1+i
0
(xโy+ix2)dz along
(i) the straight line from z= 0 and z= 1 + i
(ii) real axis from z= 0 and z= 1 and then a line parallel to
imaginary axis from z= 1 to z= 1 + i
(c) Find the value of the integral ZC
(z+ 1)2dz where C is the bound-
ary of the rectangle with vertices at the points 1+ i,โ1+ i,โ1โi
and 1 โi
(d) Compute Zฮ
|z|dz where ฮ is the left half of the unit circle |z|= 1
from z=โito z=i
(e) Find the value of ZC
(z2โiz)dz along the curve C:y=x3โ3x2+
4xโ1 joining points (1,1) and (2,3)
2. (a) Verify that the value of the integral ZC
z2dz is same in all case:
(i) Cis the straight line joining the point A(0,0) and B(1,2)
(ii) Cis the straight line path from A(0,0) to P(1,0) followed by
the straight line path from P(1,0) to B(1,2)
(iii) Cbe the parabolic path y= 2x2joining the point A(0,0) and
B(1,2)
(b) Integrate xz along the straight line from A(1,1) to B(2,4)in the
complex plane.Is the value same if the path of integration from A
to B is along the curve x=t,y=t2?
(c) Evaluate the function f defined by the integral f(z) = H|w|=1
ew2โ1
wโzdw
