Solutions to homework # 2 I.15. The area of a triangle is equal to half the area of the parallelogram dened by two sides of the triangle and the angle between them. The latter area is the absolute value of the cross product of the two vectors correspondin

If you want to shorten the project by two weeks, which tasks should be crashed (and
by
how
much)
while
minimizing
direct
costs?
A.
Task
A
by
two
weeks.
B.
Task
B
by
two
weeks.
C.
Task
B
by
one
week
and
Task
E
by
one
week.
D.
Task
A
by
one
week
and
Task
F

If you want to shorten the project by two weeks, which tasks should be crashed (and
by
how
much)
while
minimizing
direct
costs?
A.
Task
A
by
two
weeks.
B.
Task
B
by
two
weeks.
C.
Task
B
by
one
week
and
Task
E
by
one
week.
D.
Task
A
by
one
week
and
Task
F

If you want to shorten the project by two weeks, which tasks should be crashed (and
by
how
much)
while
minimizing
direct
costs?
A.
Task
A
by
two
weeks.
B.
Task
B
by
two
weeks.
C.
Task
B
by
one
week
and
Task
E
by
one
week.
D.
Task
A
by
one
week
and
Task
F

Answers and comments on homework # 7 IV.2. (a) f (x) = 11 2 sin(2nx) . n n=1
(b)
2
n=1
(1)n
sin(nx) . n
(c) The series for [0, 1] converges faster to the function f on [0, 1] than the series for [1, 1]. On the other hand, the second series approximates

Solutions to homework # 5 II.31. (a) Consider the rectangular contour CR consisting of the intervals IR :=[R, R], + SR :=[R, R + 2 i/b], JR :=[R + 2 i/b, R + 2 i/b], SR :=[R + 2 i/b, R]. First note that
JR
eaz dz = 1 + ebz
IR
ea(x+2i/b) dx = e2ia/b 1 + eb

Solutions to homework # 4 II.27. (a) Res f (i) = lim(z i)f (z ) = lim(z i)
z i z i
z2 i2 z2 = lim = . (z i)(z + i) z i z + i 2i
(b) Since e1/z 1 = e1 e1/z , we get e1/z 1 = e1 (1 + 1 1 + + ). 1!z 2!z 2
The coecient c1 of 1/z in this Laurent series, i.e.,

Solutions to homework # 3 II.16. (a) f (z ) = z 2 sin z . Since sin z = ei(x+iy) ei(x+iy) ey (cos x + i sin x) ey (cos x i sin x) eiz eiz = = 2i 2i 2i cos x(ey ey ) (ey + ey ) sin x +i = 2 2
and z 2 = (x2 y 2 ) + 2ixy , we get f (x, y ) = u(x, y ) + iv (x

Solutions to homework # 1 35. First note u1 + u2 = u3 , i.e., the vectors u1 , u2 , u3 are linearly dependent, i.e., they lie in the same plane. The vectors u1 and u2 , on the other hand, are not not proportional, hence linearly independent. So, it is eno

If you want to shorten the project by two weeks, which tasks should be crashed (and
by
how
much)
while
minimizing
direct
costs?
A.
Task
A
by
two
weeks.
B.
Task
B
by
two
weeks.
C.
Task
B
by
one
week
and
Task
E
by
one
week.
D.
Task
A
by
one
week
and
Task
F