SOLUTIONS TO HW7
Stein and Shakarchi, Chapter 6
1
Exercise 1. Let = limN N n log(N ) denote the Euler-Mascheroni constant. Recall
n=1
that by Theorem 1.7 and the Hadamard Factorization Theorem, we have
s s/n
1
1+
= ses
e
(s)
n
n=1
N
= ses lim
N
1+
n=1
s
COMPLEX ANALYSIS. HOMEWORK 6
Problem 1. Complex Variables Introduction and Applications Second Edition MARK J. ABLOWITZ,
ATHANASSIOS S. FOKAS
Problems for Section 4.6: 5
Problem 2. Complex Variables Introduction and Applications Second Edition MARK J. ABL
SOLUTIONS TO HW6
Stein and Shakarchi, Chapter 5
Exercise 1. Let f (z) be a holomorphic function on the open unit disc D, and let cfw_z1 , . . . , zN
be the zeros of f (z) inside D (counted with multiplicity). For D, we let (z) denote the
function
z
(z)
SOLUTIONS TO HW5
Stein and Shakarchi, Chapter 3
z
Problem 1. (a) Consider the following sequence of functions: fn (z) = n . We then have
1
fn (0) = 0 for all n. Now let r > 0 be a positive real number, and choose n such that n < r,
so that Dr (0) D1/n (0)
Math 215A HW5
Solutions
November 16, 2011
1
Problem 6.1
We have 1/(s) = es s
(1+s/n)es/n for all s. Thus, (s) = es 1/s
n=1
n/(n+s)es/n
n=1
whenever 1/(s) = 0 s = 0, 1, 2, . Now, note that = limN (
log(N ), so es = limN (N s es
s)es/n ) = limN
2
N sN !
s(
SOLUTIONS TO HW4
Stein and Shakarchi, Chapter 3
iz
Exercise 3. Let f (z) = z2e+a2 , with a > 0, and let R = 1 2 , where 1 denotes the contour
from R to R along the real axis, and 2 is the semicircular contour Reit for t [0, ]. Take
R > a. On 2 , we have (
COMPLEX ANALYSIS. HOMEWORK5
Problem 1. Evaluate
P.V.
0
3
xi 4
dx,
x
where R, x is real, and > 0 is also real.
Problem 2. Complex Variables Introduction and Applications Second Edition MARK J. ABLOWITZ,
ATHANASSIOS S. FOKAS
Problems for Section 4.3: P13
Pr
SOLUTIONS TO HW3
Stein and Shakarchi, Chapter 2
Problem 1. (a) We let
n
z2 .
f (z) =
n=0
Since all of the coecients are either 0 or 1, it is clear that the radius of convergence is 1.
Now let = 2p , with p and k positive integers. We then have, for r < 1,
SOLUTIONS TO HW2
Stein and Shakarchi, Chapter 1
Exercise 13. Let f be a holomorphic function on an open set , and write f (z) = u(x, y) +
iv(x, y) as usual, with u and v dierentiable.
(a) Assume (f ) = u(x, y) is constant, so that
u
= 0 and
x
u
= 0.
y
Sin
SOLUTIONS TO HW1
Stein and Shakarchi, Chapter 1
Exercise 1.(a) We could write this out in real and imaginary parts, but its easier just to
think about what this means: it is the set of points z in C such that z is equidistant from
z1 and z2 . If the point
Math 70300
Homework 7
Due: within 72 hours
1. Let u be harmonic in a region G and suppose that the closed disc D(a, R) is contained
in G. Show that
1
u(a) =
u(x, y) dxdy.
R2 D(a,R)
Hint: Use polar coordinates.
For every r between 0 and R we have by the me
MATH 207, HOMEWORK 4
Remark 1. The answers to this HW are all from Rubens HW4, there is a very short addition
in problem 15,b.
Problem 2. Use Rouchs theorem to establish the fundamental theorem of algebra.
e
Answer. Consider degree n polynomial p(z) = n a
MATH 207, HOMEWORK 5
The solutions are from Reubens homework set
Problem 1 (Chapter 5, Exercise 6). Prove Walliss product formula
22 44
2m 2m
=
.
2
13 35
(2m 1) (2m + 1)
Answer. First note that by continuity of ln (as a real function) (1 + an ) converges
Math 70300
Homework 5
Due: November 14
1. Calculate the integrals using contour integration. Complete explanations are required.
(i)
0
(iv)
0
dx
3+1
x
(ii)
0
/2
cos(ax)
dx (a, b > 0),
(x2 + b2 )2
cos x
dx,
x2 + 1
(v)
0
(iii)
0
d
a + sin2
(log x)2
dx
1 +
Math 70300
Homework 6
Due: December 5
1. Let f (z) be a holomorphic function in the disc |z| < R1 and set
M (r) = sup |f (z)|,
A(r) = sup (f (z),
|z|=r
0 r < R1 .
|z|=r
(a) Show that M (r) is monotonic and, in fact, strictly increasing, unless f is a
cons
Math 70300
Homework 7
Due: within 96 hours
1. Let u be harmonic in a region G and suppose that the closed disc D(a, R) is contained
in G. Show that
1
u(a) =
u(x, y) dxdy.
R2 D(a,R)
Hint: Use polar coordinates.
2. Prove Hadamards three circles theorem: Let
Math 70300
Homework 6
Due: December 5
1. Let f (z) be a holomorphic function in the disc |z| < R1 and set
M (r) = sup |f (z)|,
A(r) = sup (f (z),
|z|=r
0 r < R1 .
|z|=r
(a) Show that M (r) is monotonic and, in fact, strictly increasing, unless f is a
cons
Math 70300
Homework 3
Due: October 19, 2006
1. Let f (z ) be an analytic function with nonzero derivative. Let f (z ) = u(x, y )+ iv (x, y )
and consider the level curves of u and v , i.e., the sets
cfw_z = x + iy C : u(x, y ) = u0 ,
cfw_z = x + iy C : v
Math 70300
Homework 4
Due: within 72 hours
1. (a) Let z1 and z2 be two points on a circle C. Let z3 and z4 be symmetric with respect
to the circle. Show that the cross ratio (z1 , z2 , z3 , z4 ) has absolute value 1.
Use a linear fractional transformation
Math 70300
Homework 5
Due: November 14
1. Calculate the integrals using contour integration. Complete explanations are required.
(i)
0
(iv)
0
dx
3+1
x
(ii)
0
cos x
dx,
x2 + 1
/2
cos(ax)
dx (a, b > 0),
(x2 + b2 )2
(v)
0
(iii)
d
a + sin2
0
(log x)2
dx
1 +
Math 70300
Homework 2
Due: September 28, 2006
1. Suppose that f is holomorphic in a region , i.e. an open connected set. Prove that
in any of the following cases
(a)
(f ) is constant; (b)
(f ) is constant; (c) |f | is constant; (d) arg(f ) is constant;
we
Math 70300
Homework 3
Due: October 19, 2006
1. Let f (z) be an analytic function with nonzero derivative. Let f (z) = u(x, y)+iv(x, y)
and consider the level curves of u and v, i.e., the sets
cfw_z = x + iy C : u(x, y) = u0 ,
cfw_z = x + iy C : v(x, y) =
Math 70300
Homework 4
Due: within 72 hours
1. (a) Let z1 and z2 be two points on a circle C . Let z3 and z4 be symmetric with respect
to the circle. Show that the cross ratio (z1 , z2 , z3 , z4 ) has absolute value 1.
az + b
(b) Let ad bc = 1, c = 0 and c
Math 70300
Homework 1
September 12, 2006
The homework consists mostly of a selection of problems from the suggested books.
1. (a) Find the value of (1 + i)n + (1 i)n for every n N.
We will use the polar form of 1 + i = 2(cos(/4) + i sin(/4) and De Moivres
Math 70300
Homework 2
Due: September 28, 2006
1. Suppose that f is holomorphic in a region , i.e. an open connected set. Prove that
in any of the following cases
(a)
(f ) is constant; (b)
(f ) is constant; (c) |f | is constant; (d) arg(f ) is constant;
we
Math 185 - Spring 2015 - Homework 9 - Solution sketches
Problem 1. Show that
1
xs dx =
0
1
s+1
for s C such that Re s > 1.
Fix > 0. We have xs = es log x for x [, 1]. Using Homework 3, Problem 6, we see that
1
xs dx
F (s) =
denes a holomorphic function fo
Math 185 - Spring 2015 - Homework 8 - Solution sketches
Problem 1. Let f : H C be holomorphic and satisfy |f (z)| 1 for z H and f (i) = 0. Show
that
zi
for z H.
|f (z)|
z+i
Recall g(z) = i z+i is a biholmorphism from D to H and g 1 (z) = i(zi) . Then F =
Math 185 - Spring 2015 - Homework 1
Hard copy due:
Tuesday, February 3
at 11am.
Notation. D = cfw_z C : |z| < 1, D = cfw_z C : |z| = 1.
Problem 1. For all z C\cfw_0 there exists a unique w C\cfw_0 such that zw = 1, which we denote
1
by z or z 1 . Given z
Math 185 - Spring 2015 - Homework 3
Hard copy due: Tuesday, February 24
at 11am.
Problem 1. Show that the relation is homotopic to is an equivalence relation. That is,
(i) any curve is homotopic to itself,
(ii) if 0 is homotopic to 1 , then 1 is homotopic