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TitleL. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich-A Collection of Problems on Complex Analysis-Dover Publications (1991)
TagsPower Series Continuous Function Integral Complex Number Laplace Transform
File Size18.3 MB
Total Pages437
Table of Contents
                            Cover
S Title
A Collection of Problems on COMPLEX ANALYSIS
Copyright © 1965 by Pergamon Press
	ISBN 0486669130
	QA331. 7. V6513 1991 515´.9´.076-dc20
CONTENTS
FOREWORD
CHAPTER I  COMPLEX NUMBERS AND FUNCTIONS OF A COMPLEX VARIABLE
	§1. Complex numbers
	§2. Elementary transcendental functions
	§3. Functions of a complex variable
	§4. Analytic and harmonic functions
CHAPTER II  CONFORMAL MAPPINGS CONNECTED WITH ELEMENTARY FUNCTIONS
	§ 1. Linear functions
	§ 2. Supplementary questions of the theory of linear transformations
	§ 3. Rational and algebraic functions
	§ 4. Elementary transcendental functions
	§ 5. Boundaries of univalency, convexity and starlikeness
CHAPTER Ill  SUPPLEMENTARY GEOMETRICAL QUESTIONS. GENERALISED ANALYTIC FUNCTIONS
	§ 1. Some properties of domains and their boundaries. Mappings of domains
	§ 2. Quasi-conformal mappings. Generalised analytic functions
CHAPTER IV  INTEGRALS AND POWER SERIES
	§ 1. The integration of functions of a complex variable
	§ 2. Cauchy's integral theorem
	§ 3. Cauchy's integral formula
	§ 4. Numerical series
	§ 5. Power series
	§ 6. The Taylor series
	§ 7. Some applications of Cauchy's integral formula and power series
CHAPTER V  LAURENT SERIES, SINGULARITIES OF SINGLE-VALUED FUNCTIONS. INTEGRAL FUNCTIONS
	§ 1. Laurent series
	§ 2. Singular points of single-valued analytic functions
	§ 3. Integral functionst
CHAPTER VI  VARIOUS SERIES OF FUNCTIONS. PARAMETRIC INTEGRALS. INFINITE PRODUCTS
	§ 1. Series of functions
	§ 2. Dirichlet seriest
	§ 3. Parametric integrals
	§ 4. Infinite products
CHAPTER Vll  RESIDUES AND THEIR APPLICATIONS
	§ 1. The calculus of residues
	§ 2. The evaluation of integrals
	§ 3. The distribution of zeros. The inversion of series
	§ 4. Partial fraction and infinite product expansions. The summation of series
CHAPTER VIII   INTEGRALS OF CAUCHY TYPE. THE INTEGRAL FORMULAE OF POISSON AND SCHWARZ. SINGULAR INTEGRALS
	§ 1. Integrals of Cauchy type
	§ 2. Some integral relations and double integrals
	§ 3. Dirichlet´s integral, harmonic functions, the logarithmic potential and Green´s function
	§ 4. Poisson's integral, Schwarz's formula, harmonic measure
	§ 5. Some singular integrals
CHAPTER IX  ANALYTIC CONTINUATION. SINGULARITIES OF MANY-VALUED CHARACTER. RIEMANN SURFACES
	§ 1. Analytic continuation
	§ 2. Singularities of many-valued character. Riemann surfaces
	§ 3. Some classes of analytic functions with non-isolated singularities
CHAPTER X  CONFORMAL MAPPINGS (CONTINUATION)
	§ 1. The Schwarz-Christoffel formula
	§ 2. Conformal mappings involving the use of elliptic functions
CHAPTER XI  APPLICATIONS TO MECHANICS AND PHYSICS
	§ 1. Applications to hydrodynamics
	§ 2. Applications to electrostatics
	§ 3. Applications to the plane problem of heat conduction
ANSWERS AND SOLUTIONS
	CHAPTER I
	CHAPTER II
	CHAPTER III
	CHAPTER IV
	CHAPTER V
	CHAPTER VI
	CHAPTER VII
	CHAPTER VIII
	CHAPTER IX
	CHAPTER X
	CHAPTER XI
                        
Document Text Contents
Page 1

A COLLECTION
OF PROBLEMS ON

COMPLEX ANALYSIS
-

L. I. Volkovyskii, G. L. Lunts,
. and I.G. Aramanovich

Page 2

A Collection of Problems
on

Complex Analysis

Page 218

SINGULARITIES OF MANY-VALUED CHABAOTEB 207

1309. Let

Construct the Riemann surfaces of the functions:
(1) w = y(f(z));
(2) w = Logf(z);
(3) w =Log 1(;)+Logf{:z)·

HINT. First prove that the function /(z) is sohlioht in the disk lzl < I
and has the oirole fzf = I as its natural boundary.

§ 3. Some classes of analytic functions with non-isolated singu-
laritiest

Let E be a closed set of points with connected complement !J in the extended
z-plane. Let AB, AD and AO be classes of single-valued analytic functions
in !J, respectively: bounded (the class AB), having a bounded Dirichlet integral
with respect to !J (the class AD), continuous in the extended z-plane (the
class AO). If the set Eis in some respect small it can be expected that from
the analyticity of the function in !J and the additional limitations connected
with the class of function, it would follow that the function in E is analytic.
Then the function is a constant in virtue of Liouville's theorem. In this case
the set E is said to be a nul aet of the class of functions considered.

The classes of nul sets corresponding to the given classes of functions are
denoted by NAB• N AD• N AC• and the class of their complements in the
domain !J by 0 AB• 0 AD• 0 AC•

A set Eis said to be .AB-removable in the domain G, containing E, if from
the analyticity and boundedness in G-E there follows analyticity everywhere
in G (if G is identical with the extended z-plane, then AB-removableness in
Gisequivalent to the fact that EeNAB)· AD-removableness and .AO-remova·
bleness of a set E in the domain G are similarly defined (in the case of AO·
removableness, functions are considered of the class AO in G, that is, analytic
in G-E and continuous everywhere in G). For brevity F denotes any of the
classes AB, AD, AO, if it is a question of some property common to them.

1310. Prove that the set E with connected complement !J, which
is the set of singularities of a single-valued function analytic in Q,
is closed.

1311. Prove that the isolated singular points of functions of the
classes .AO, .AB, .AD are removable and the sets of singularities
of the functions of these classes are perfect.

t For this section see L. Alu.FOBS, A. BEtJBLING, (1950), Conformal invariants
and function theoretic null-sets, Acta Math. vol. 83, 100-129.

Page 219

208 PROBLEMS ON COMPLEX .ANALYSIS

HINT. For /e.AD consider a.Bf?-+0 the Dirichlet integral f J l/'l8 rdrd•
C1<fzi<C11

(a singularity at zero), starting from the Laurent expansion

""
.f(z) = .J: Cnzn •

-00

1312. Let G be an arbitrary domain in the extended z-plane
and Ek(k = I, 2, ... , q) bounded closed sets with connected com-
plements Dk, located in G, no two of which have points in common:

(i#=j).

Prove that the function /(z), single-valued and analytic in G-E
q

E = LJ E", can be represented uniquely in the form
k=l

q

f(z) = ,P(z)+ 21J11i(z),
k=l

where ,P(z) is analytic in G, 1Jlk(z) is analytic in D" and 1J11i(oo) = 0.
In particular, if G is the extended z-plane, then

q

J(z) =f(oo)+ L1Pk(z)
k=l

HINT. Separate the sets Et in (} by contours "" (la = I, 2, ..• , q) and in
the domain cut off from the variable domain Gn, converging to G, by the con-
tours Yk• represent /(z) by Cauchy's formula..

REMARK, 'Pt(z) has on Ek the same singularities a.s /(z) and is the analogue
of the principal pa.rt of the Laurent expansion in the neighbourhood of an
isolated singularity.

1313. Prove that if Ee NF, then E is .F-removable in any domain
G => E (removableness of nul sets).

HINT, Make use of the result of the preceding problem.

1314. Prove that if Eke NF (k = I, 2, ... , q) and E1 n EJ = 0
q

(i =F j), then E = LJ Ek c: NF (the property of analyticity of nul
k=l

sets).
1315. Prove that if E c: NF• then E does not contain interior

points.

Page 436

ANSWERS AND SOLUTIONS 425

n
1497. t1(z) =Im f(z)- J; oc1cw1c(z), where f(z) conformally maps D onto

k-1
the plane with horizontal outs, and

I pi if f (z) = Z-iJ + ... • a #: oo,
piz + ... , if a = oo.

n
~ 1 J og(C, z)

1498. t1(z) = 2gg(z, a)+ L.J IXk°'lc(z), w1c(z) = -2k ---a,;-da·
k=l r1c

1499. If /(a) = oo, then the field is formed by the dipole (a; p) where
I' is determined from the expansion off (z) close to the point a;

'( ) 1--1!!.__ + ... , if a #: oo, z = z-a
piz+ ••. , if a = oo.

1500. (1) If /(a) == 0, f(b) = oo, then the field is formed by the point
charges (a; 2q), (b; -2q) the flux of the vector intensity through each boundary
contour being equal to irero;

(2) If /(a) = 0, then the field is formed by the point charge (a, 2q),
the flux of the vector intensity through a boundary contour, corresponding to
a circle, in the direction of the outward normal to the domain D being equal
to 4:.irq, end through every other contour being equal to zero;

(3) The field is regular everywhere. The flux of the vector intensity
through boundary contours, transformed into circles, in the direction of the
outward normal to Dis equal to ±4:.irq (+for the contour, transformed into
the outer circle), end through every other contour it is equal to zero.

1501. See problems 1487 and 1489.
n-1

1502. (1) t1(Z) = J; or:lc°'1c(z)+c, where the or:1c a.re uniquely determined from
k=l

n-1
the system J; Plkott = 2qi (i = 1, 2, ... , n-1) (see problem 1104) e.nd c is

k=l
an arbitrary real number. The problem is equivalent to the construction of
the :Bow in D, streamlining the boundary contours I'1c with circulations 4:.irqrc
(k = 1, 2, ... , n), if oo e D, and by the circulations 4:.irq1c (k = 1, 2, •.. , n - 1),
-4nqn, if oo e D (I'n is the external contour).

(2) t1(z) = t10 (z)-2qg(z, a), where t10 (z) is determined as in part (1),
form the charges of the layer 2q1c+2qt, where

!lt = _ _!_ f og(C,a) dB.
2:.ir on

r1c

uoa. "(z) = 2q1 log ~ +c, if !l = 0, and
r1

t1(z) = 2(q1-.Aq)log"jZj'"-2qg(z,a)+c,

Page 437

426 PROBLEMS ON COMPLEX ANALYSIS

8 (log z+log a)
l 1 2:.iri

where A - og a g(z, a) = lzlA log ------1
- loge' 8 (logz-loga)

l 2:.iri

for.,,.= (loge)/:.iri, where e < a < 1, if q:;tO (in the notation of problem 1439

the Green's function g(z, a) = Im 4i ( : log 111) for I'= 2:.ir and I'1 = -2:.ir;

the latter from the condition 'P = 0 on the boundary of the ring).

15H. The source (a; q) is transformed into the source (a*; -q), where

a• is the point symmetrfoal to a. The function u = -2q log -11
1 +c, where

:ii (111, all
f (111, a) conformally maps the domain D onto the unit disk (here and in what
follows the coefticient of heat conductivity Tc is assumed to be equal to 1).

1505. u - -2q log 1 z-ii I +c. :ii 111-a
1506. u = 2~ log1;;~:)1 +c.

sin~+isinh :.irk
q 2a 2a

1507. u = 2_ log +c. •• . m 'sinh nh
s1nra-• 2a

1508. u = :Tc log I:~!~~: j +c, e = sn [ ! (111 + ib), u], where Tc is de-
. d ..... h l. K 2b termme ..-om t e re at1on K = a.

1509. (1) The Green's function g(z, a) of the domain D can be considered
as the temperature created in D by the heat source (a; 2:.ir) when the tempera-
ture on the boundary is equal to zero;

n

(2) u(z) = -2q g(z, a)+ 2 u1;w1:(111), where w1(111) is the harmonic
:ii k-1

measure of I'1;.

q U1-U1 lzl
1510, u = -2 g(z, a)+ 1 ( I ) log- +uv where g(z, a) is the Green :ii og r1 ,., f't

function (see the answer to problem 1508).

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