Elements of the Theory of Functions of a Complex Variable

308 Pages·2010·13.901 MB·English

by  Dr. H. Durege

ELEMENTS
OP THE
THEORY OF FUNCTIONS
COMPLEX VARIABLE
WITH
ESPECIAL REFERENCE TO THE METHODS OF RIEMANN
BY
DR. H. DUREGE
LATE PROFESSOR IN THE UNIVERSITY OF PBAGTJE
AUTHORIZED TRANSLATIONFROM THEFOURTH GEJtMAtfEDITION'
BY
GEORGE EGBERT FISHER, M.A., PH.D.
ASSISTANTPROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PENNSYLVANIA
AND
ISAAC J. SCHWATT, PH.D.
INSTRUCTOR IN MATHEMATICS IN THE UNIVERSITY OF PENNSYLVANIA
PHILADELPHIA
G. E. F1SHEB AND I. J. SCHWATT
1896
OOPTKIGHT,1896,BY
.E.PISHESANDI.J.SCHWATT.
HorSuootilwi$
3.S.Cushmg&Co. Berwick&Smith
NorwoodMass.U.SA.
TRANSLATORS' NOTE.
WE desire to express our indebtedness to Professor James
McMahon, Cornell University, for Ms kindness in reading the
translation, and for his valuable suggestions. Also to Messrs.
&
J. S. Gushing Co., Norwood, Mass., for the typographical
excellence of the book.
G. E. F.
I. J. S.
UNIVERSITY OF PENNSYLVANIA,
PHILADELPHIA, December, 1895.

FROM THE AUTHOR'S PREFACE TO THE THIRD
EDITION.
NUMEROUS additions have been made to Section IX., which,
treats of multiply connected surfaces. If Riemann's funda-
mental proposition on these surfaces be enunciated in such
a form that merely simply connected pieces are formed by
both modes of resolution, as is ordinarily, and was also in
49, the case, then it must be supplemented for further
applications. Such supplementary matter was given in 52
in the classification of surfaces, and in 53, V. But if we
express the fundamental proposition in the form in which
Riemann originally established it, in which merely simply
connected pieces are formed by only one mode of resolution,
while the pieces resulting from the other mode of resolution
may or may not be simply connected, then all difficulties are
obviated, and the conclusions follow immediately without
requiring further expedients.
This was shown in a supplementary note at the end of
the book.
AUTHOR'S PREFACE TO THE FOURTH EDITION.
IN the present new edition only slight changes are made,
consisting of brief additions, more numerous examples, differ-
ent modes of expression, and the like.
In reference to the above extract from the preface of the
preceding edition, I have asked myself the question, whether
I should not from the beginning adopt the original Riemann
enunciation of the fundamental proposition instead of that
which is given in 49. Nevertheless, I have finally adhered
to the previous arrangement, because I think that in this way
the difference between the two enunciations is made more
prominent, and the advantages of the Riemann enunciation
are more distinctly emphasized.
H. DUREGE.
PRAGUE,April,1893.
CONTENTS.
PAGE
INTRODUCTION 1
SECTION
I.
GEOMETRIC REPRESENTATION OF IMAGINARY QUANTITIES.
1. Acomplexquantityx+ty orr(cos -4- isin<) is represented
byapointinaplane, ofwhichxand yaretherectangular,
r and <p the polar co-ordinates. Astraight line is deter-
mined in length and direction by a complex quantity.
Direction-coefficient .... 12
2. Construction ofthefirstfouralgebraic operations 15
1. Addition .... 15
2. Subtraction. Transference ofthe origin 16
3. Multiplication 18
4. Division. Applications 20
3. Complexvariables. Theycandescribedifferentpathsbetween
twopoints. Direction ofincreasingangles 22
SECTION
II.
FUNCTIONS OF A COMPLEX VARIABLE IN GENERAL.
4. The correspondence of thevalues of thevariable and of the
function is the most essential characteristicof afunction 25
5. Conditionsunderwhichw=u4-ivisafunctionofz=x+iy 27
6. Thederivative isindependentofdz 29
dz
7. Iftobe afunction of ,the systemof ^-pointsissimilarin its
infinitesimal elements to the system of ^-points. Con-
formalrepresentation. Conformation 33
Vlll CONTENTS.
SECTION III.
MULTIFORM FUNCTIONS.
PAG&
8. The value of a multiform function depends upon, the path
describedbythevariable. Branch-points 37
9. Two paths lead to differentvalues of a function only when
they encloseabranch-point 43
10. Examples:
z V^.
(1) vS, (2) O~1)VS, (3) ^-=2, (4) -\'f5f+
Cyclicalinterchange offunction-values 47
11. Introduction of Riemann surfaces which cover the plane
w-fold. Branch-cuts 55
12. Proof that this modeofrepresentation of%-valuedfunctions
isconformal 61
13. Continuous and discontinuous passage overthe branch-cuts.
Simple branch-pointsandwinding-pointsofhigherorders 68
14. Theinfinitevalue ofz. Surfacesclosedatinfinity (Eiemann
spherical surfaces). Proof that theformation of a Kie-
mann spherical surface for agiven algebraicfunction is
alwayspossible. Examplesofdifferentarrangementsof
branch-cuts 71
15. Everyrationalfunction ofwandzislike-branchedwithw . 80
/
SECTION IV.
INTEGRALS WITH COMPLEX VARIABLES.
16. Definitionofanintegral. Dependence of the sameuponthe
pathofintegration 80
17. The surface integral JfJf\(O2X_^_IJ)}dxdy is equal to the
linear integral \(Pdx+ Qdy) extended over the boun-
dary. Positive boundary-direction 84
18. IfPdx+Qdy be acomplete differential, then f(Pdx+Qdy*),
takenalongtheboundary ofasurfacein whichPand Q
arefiniteandcontinuous, iszero. Also, \f(?)dz 0, if
this integral be taken along the boundary of a surface
in which/(#) is finite and continuous. Importance of
simplyconnectedsurfaces 90
19. Thevalueofaboundaryintegraldoesnotchangewhenpieces
inwhich/(#) isfiniteandcontinuousareaddedtoorsub-