Kusse B.R., Westwig E.A. Mathematical Physics: Applied Mathematics for Scientists and Engineers
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206
INTRODUCTION
TO
COMPLEX
VARIABLES
Identify the region of the z-plane where
5
can be placed
so
that
this
series
will
be
a valid representati6n
of
the function on the contour
C
and determine
the
cn.
(b)
Now consider the Laurent series expansion for
~(g)
about
z
=
z,,
Identify the region of the g-plane where
z,
can be placed
so
that
this
series
will be a valid representation of the function on the contour
C
and determine
the
gn.
17.
Consider the complex function
1
1
w(z)
=
-
+
-
2-3
2-4'
-_
(a)
Find the Taylor series expansion for
~(2)
about
2
=
1
and identify its region
(b)
Find the Laurent series expansion for
~(g)
about
z
=
0,
which converges in
(c)
Find the Laurent series expansion for
~(z)
about
z
=
1,
which converges in
(d)
Compare the
g-
coefficients
in
the series solutions to parts
(b)
and (c) above
(e)
Find the Laurent expansion
for
~(g)
about
z
=
1,
which converges in the
(f)
How
many different Taylor and Laurent series expansions for
~(z)
can be
of convergence.
the annulus
3
<
lgl
<
4.
the
annulus
2
<
12
-
11
<
3.
and comment.
region
3
<
I?
-
11.
made about the point
2
=
3.5?
18.
Consider the function
(a)
Find the function that generates the coefficients of the Taylor series expansion
for
this
function about the general point
z
=
5.
Indicate the region of
convergence for
this
series.
(b)
Find the function that generates the coefficients of the Laurent series expan-
sion for
this
function about the general point
g
=
5.
Indicate the region of
convergence for
this
series.
19.
Consider the complex function
EXERCISES
207
20.
21.
22.
23.
(a)
Find the Laurent series for
~(g)
expanded about the point
z
=
1
and identify
(b)
Find the Taylor series
for
~(g)
expanded about the point
z
=
1
and identify
(c)
Find the Laurent series for
I&)
expanded about the point
z
=
0
and identify
(d)
Why is there no part
(d)
to this problem?
the region of convergence.
the region
of
convergence.
the region of convergence.
Verify that
by comparing the Taylor series expansions for sinh
z
and
ex.
Comment on the
convergence.
Prove
that
a
Laurent expansion
of
a function about a particular point is unique.
That is, show
if
m
m
n=-n.
n=-n,
then
gn
must equal
&
for ail
n.
Use the Cauchy integral formula.
Consider the function
(a)
Locate all the poles
of
&)
in
the -plane and identify their order.
(b)
Find the Taylor series expansion for
~(g)
about
z
=
0.
Identify the first three
coefficients
of
this series and indicate the region
of
convergence.
(c)
How
many different Laurent series for
~(z)
can be constructed about
z
=
O?
(d)
Find a series expansion around
z
=
0,
either Taylor
or
Laurent, that is a valid
representation of
~(z)
at
=
r.
Indicate the region of convergence for this
series.
Consider the function
(a)
Where
in
the complex Z-plane are the poles of
&)?
(b)
Determine the first three terms
for
the Taylor series expansion of
I&)
about
(c)
Identify the region of convergence for the Taylor series
of
part (b).
(d)
Determine the general expression
for
the
nrh
coefficient of the Taylor series
2
=
0.
-
expansion of part (b).
208
INTRODUCTION TO COMPLEX
VARIABLES
(e)
There is a Laurent series expansion for
I&)
about
z
=
0
in the region
1
<
lzl
-
<
2 that has the form
m
n=-m
Obtain the general expression for the
&.
24.
Consider the function
(a)
Can
this
function be expanded in a Taylor series about
g
=
O?
(b)
Find two different Laurent series expansions for
this
function, both expanded
about
g
=
0.
Indicate their regions of convergence.
(c)
Find the Taylor series for
this
function expanded about
z
=
1/2. Indicate
the region
of
convergence.
(d)
Find the Laurent series for
this
function expanded about
z
=
1/2. Indicate
the region
of
convergence.
(e)
What is the residue of
this
function at
z
=
O?
at
z
=
l? at
z
=
1/2?
(f)
What is the value of the integral
where
C
is a counterclockwise circle centered at the origin of radius 1/2?
of radius
1.5?
(g)
What
is the value of the integral in part
(f)
above if
C
is a clockwise circle
centered at
z
=
1
of radius
0.5?
of radius
1.5?
25.
Identify the locations and orders of the poles for the following functions. For
each of the functions evaluate the residue associated with each of its poles.
Z2
g2
+
1’
i.
~(g)
=
-
Z
iii.
w(z)
=
e.
sin
z
In
g
iv.
y(z)
=
-
g’
-
1’
EXERCISES
209
26.
Each of the following functions has
a
pole at
z
=
1.
In each case identify the
order of the pole and determine its residue.
1
22
-
22
+
1
1
ii.
___
-2-
1’
1.
--.
- -
I
1
iii.
-
In
g
1
-
cos
(z
-
1)
(z
-
113
.
iv.
v.
cot(g
-
1).
27.
Let
C
be a counterclockwise, circular contour
of
radius
2
centered at
z
=
0.
Evaluate the complex integrals:
(n
=
an integer).
28.
Evaluate the integral
29.
30.
(a)
where
C
is the clockwise, circular contour given
by
IzI
=
2.
(b)
where
C
is the clockwise, circular contour given
by
lgI
=
4.
Evaluate the complex integral
where
C
is the counterclockwise, circular contour given
by
lz
-
+
~/2)
=
1
Evaluate the complex integral
where
C
is the counterclockwise, circular contour
lg
-
11
=
2.
210
INTRODUCTION
TO
COMPLEX
VARIABLES
31.
Evaluate the complex integral
where
C
is the clockwise, circular contour
lz
-
24
=
1.
32.
Evaluate the complex integral
where
C
is the counterclockwise, circular contour
lgl
=
1.
33.
Consider the complex integral
sing
Adz
(g
-
.Ir/Nz_
+
.Ir/2)
and the two contours
C
=
C1
and
C
=
C2
shown below.
imag
-
z-plane
Let
Z,
be the integral around
C1
and
12
the
integral around
C2.
Show
by
contour
deformation that
12
-
11
=
-2i.
34.
Consider the function
(a)
Find all the poles
of
I&)
and identify their order.
(b)
Evaluate the integral
where
C
is a counterclockwise, circular contour given
by
lzl
=
1.
EXERCISES
211
(c)
Evaluate the integral of part
(b)
above if
C
is a counterclockwise, circular
contour given by
I?
-
TI
=
1.
35.
Consider the function
Find all the poles of
~(z)
and identify their order.
Evaluate the integral
where
C
is a counterclockwise, circular contour given by
IzI
-
=
1.
Evaluate the integral of part (b) above if
C
is
a counterclockwise, circular
contour given by
lz
-
274
=
1.
36.
Consider the integral
Locate
the
poles of the integrand and identify their order.
Evaluate the integral if
C
is a counterclockwise, circle of radius
1
centered
at
i.
z
~
=
0.
ii.
g
=
27r.
iii.
z
=
2m'.
37.
Consider the integral
where
n
is a positive integer, and
C
is the counterclockwise circle
lgl
=
1.
(a)
Find the Laurent series expansion about
z
=
0
for the integrand.
(b)
Use the residue theorem to evaluate the integral.
38.
Convert each
of
the following integrals to
an
equivalent one in the complex
plane. Identify all the poles
of
the complex integrands and their residues. Close
the contours and evaluate the integrals.
cos
x
x
sin
?I
ii.
1%
dx
___
x2
+
1'
212
INTRODUCTION TO COMPLEX VARIABLES
x2 cos (2x)
ex
v.
[Idx cosh
(m)
.
sinh(ax)
sinh
(
m)
--.rr<a<.rr.
vi.
1
dx
39.
Convert each of the following real integrals into a complex integral and evaluate
it using the residue theorem:
1
is
lnde
1
+
sin2
6‘
1
cos
e
cos
(38)
211
iii.
1
d05+4~os0’
40.
Show that the integral
where x is a pure real variable, can be evaluated by doing an integration
in
the
-
z-plane around the closed contour shown in the figure below.
Identify the similar closed contour in the z-plane that can be used to evaluate
the integral
where
m
is any positive integer.
EXERCISES
213
41.
Consider the integral
1
m
PPLrnk
(9
+
l)(x
+
1)’
where we must use the principal part because
of
the behavior of the integrand
at
x
=
-
I.
Convert this into an integral in the complex plane and determine an
appropriate contour of integration to evaluate this integral. Show that you obtain
the same answer regardless of which side of the
5
=
-
1
pole
you
go around.
42. Consider the integral
-
ixu
=
p
-9
where
x
and
u
are real variables.
(a)
Without performing the integration, show that
Z(u)
is a pure real function
(b)
Convert this integral into the complex plane and evaluate it using closure
of
u.
and the residue theorem.
(c)
Plot
Z(u)
vs. u.
(d)
The function
Z(u)
can
be
converted to a complex function
of
a complex
for
u, i.e.,
L(u)
--+
Z(4li).
Is
variable by substituting the complex variable
Lw
analytic?
43.
Use complex integral techniques to evaluate the integral
sin
(kx)
Ldk7-9
where
k
and
x
are both real variables.
44. Discuss the problems with trying to close the following integral in the complex
plane:
45.
Consider the semicircular contour shown below in the limit as
E
-+
0.
-€
,
imag
E
real
On this contour, evaluate
214
INTRODUCTION
TO COMPLEX
VARIABLES
i.
ldg-.
1
ii.
Ldz-
1
iii.
Ldz-
1
z
z-i'
-
z(g
-
i)'
46.
Use the results
of
the previous problem to evaluate
1
where
C
is the closed contour shown below in the limit
E
-+
0
and
R,
-+
w.
real
Based on
this
result, what is the principal part
of
the integral
where
x
is
a
real variable?
47.
Consider the function
(a)
Evaluate the integral
f
dzw(z),
C
where
C
is the counterclockwise, circular contour given
by
lzl
=
1
(b)
Evaluate the principal part
of
the integral
PPl:dx,.
cos
x
EXERCISES
215
18.
How does the function
y
=
g2
map the unit circle centered at the origin of the
49.
Using the mapping function
z-plane onto the y-plane?
(a)
how does the unit circle
1441
-
1
I
=
1 map onto the g-plane?
(b)
how does the line
y
=
1/2
map onto the z-plane?
and
u
in the y-plane get mapped onto the Z-plane:
50.
For each of the mapping functions below, indicate how the lines of constant
u
i.
~(z)
=
ec.
iii.
~(g)
=
lnz
-
1.
the origin
of
the E-plane for the following mapping functions?
51.
What part of the z-plane corresponds to the interior
of
the unit circle centered at
2-1
i.
y(z)
=
=
Zf
1'
z-i
ii.
w(z)
=
=
-
z+i'
52.
Consider the mapping function
(a)
Is
this mapping function analytic?
(b)
Show how this function maps the grid lines of the Z-plane onto the y-plane.
Specifically map, on a labeled drawing, the grid lines shown below.