Transcribed Text
The homework is from John B. Conway book Functions of One Complex Variable I Second edition.
P6768  #5, 713,  16, 19, 22
5. Let r(t) = exp << (1+i)t~¹)  for 0 < t 1 and >(0) = 0. Show that y is
a rectifiable path and find V(y). Give a rough sketch of the trace of y.
7. Show
that y: [0, 1] C, defined by >(t) = t + it sin 1 / 1 for t * 0 and
(0) = 0, is a path but is not rectifiable. Sketch this path.
8. Let 7 and a be the two polygons [1, i] and [1, 1 + i, i]. Express y and a as
paths and calculate f, f and 1. f where f(z) = 12/2.
9. Define y: [0, 2=] C by x(t) = exp (int) where n is some integer (positive,
negative, or zero). Show that Z 1 dz = 2win.
y
10. Define x(t) = e" for 0 t 2w and find Sr z" dz for every integer
n.
11. Let y be the closed polygon [1  i, 1+i,  1+i,  1  1  i]. Find
Z  1 dz.
y
12. Let dz where y: [0, w) C is defined by x(t) re". Show
=
that lim I(r) = 0.
00
13. Find fr z+ dz where: (a) y is the upper half of the unit circle from
+ 1 to  1: (b) y is the lower half of the unit circle from + 1 to  1.
16. Show that if y and a are equivalent rectifiable paths then V(y) = V(o).
19. Let v(t) = 1 + e" for 051521 and find 
22. Let Y be a closed rectifiable curve in an open set G and a E G. Show
that for n 2. i.(z  a) n dz = 0.
P68  #16, 20, 21 and P73  #3, 4, 5, 7,9
16. Show that if y and a are equivalent rectifiable paths then V(y) = V(o).
20. Let Y(1) = 2e" for  11 SIS IT and find fy (=2  1) 1 dz.
21. Show that if F1 and F2 are primitives for f: G
C
and G is open and
connected then there is a constant e such that F1 (z) = c + F2(z) for each Z
in G.
3. Suppose that y is a rectifiable curve in C and & is defined and continuous
on (y). Use Exercise 2 to show that
g(z) =
is analytic on C(y)  and
=
4. (a) Prove Abel's Theorem: Let a, (za)" have radius of convergence 1
and suppose that E an converges to A. Prove that
=
(Hint: Find a summation formula which is the analogue of integration by
parts.)
(b) Use Abel's Theorem to prove that log 2 = 1 I+++
5. Give the power series expansion of log z about z = i and find its radius of
convergence.
7. Use the results of this section to evaluate the following integrals:
iz
(a) 2 dz,
x11==", =
0<1<2m;
dz
(b)
,
= 05152m;
za
sin
(c) z³ Z dz,
x(t) = e",
051521
(d) flow log dz,
x(1) = 1+1e", and n 0.
9. Evaluate the following integrals:
ee
(a)
dz where n is a positive integer and x(t) = e", 0 t 2w;
2"
y
dz
(b)
where n is a positive integer and Y(t) = +e",Osis2
;
n

y
(c) / 2²+1 dz
where x(t) = 2e", 0 t 2w. (Hint: expand (2²+1)  1 by
means of partial fractions);
sin Z
(d)
dz where x(t) = e", OSIS21;
z
Z 1/m
(e)
dz where x(t) = 1 the",
(z1)" 
7
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