Introduction to the Operational Calculus

301 Pages·1967·16.222 MB·English

by  LOTHAR BERG (Eds.)

NORTH-HOLLAND SERIES ON
APPLIED MATHEMATICS
AND MECHANICS
EDITORS:
H.A.LAUWERIER
Institute of Applied Mathematics
University of Amsterdam
W.T.KOITER
Laboratory of Applied Mechanics
Technical University, Delft
VOLUME 2
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM
INTRODUCTION
TO THE
OPERATIONAL CALCULUS
BY
LOTHAR BERG
University of Halle
<m
c
1967
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM
©NORTH-HOLLAND PUBLISHING COM PANY-AMSTERDAM-1967.
No part of this book may be reproduced in any form by print, photoprint, microfilm
or any other means without written permission from the publisher
This book was originally published under the title:
EINFUHRUNG IN DIE OPERATORENRECHNUNG
by VEB Deutscher Verlag der Wissenschaften
Berlin 1962
Translation:
VENEDA PUBLISHING COMPANY LTD.
Oxford
Printed in Germany
FOREWORD TO THE FIRST EDITION
This book is the outcome of lectures delivered by the author over a period of
several years to engineering students of the Hochschule fur Elektrotechnik at Ilmenau,
and to mathematics and physics students at the Martin Luther University of Halle-
Wittenberg. During this period a revolution was taking place in the entire approach
to the operational calculus, which had previously always been based in one way or
another, following the superseding of Heaviside's interpretation, on the Laplace
integral. This revolution was the result of increasing familiarity with Jan Mikusinski's
fundamental work and was carried further by numerous authors.
My own lectures were under constant revision in the light of these developments,
the net result being the present book. While conceived as an introduction to the
realm of ideas of modern operational calculus, the book also enables the earlier
literature, based on the Laplace transformation, to be understood. In this it differs
essentially from Mikusinski's standard work "Operational Calculus", where the La­
place transformation is only mentioned at the end. In addition, the exposition is so
arranged that the various definitions of the Laplace transform to be found in the
literature (with or without preliminary factor) are equally accessible.
As in all the books of the present series*, the concern is with a branch of mathe­
matics that has a wide field of application in the natural sciences and technology.
In electrical and control engineering in particular, the operational calculus is virtually
indispensable. Yet in the technical literature the mathematical side of the subject is
either assumed to be familiar, or is only very unsatisfactorily developed. The present
book is therefore conceived as complementary to such literature. It starts with few
assumptions—though it is naturally expected that the reader will have had practice
in mathematical modes of thought. The middle section only demands a previous
knowledge of the usual material of a first-year course in mathematics (differential and
integral calculus, and differential equations), and it is only in the later sections that
some knowledge of theory of functions is required—though in these days this latter
will have been encountered by any electrical engineer during his studies. Some familiar
concepts and propositions are collected and explained in an appendix, so that the book
can also be used for self-study. (The individual sections of the appendix are indicated
by raised figures [*], [2], etc.) On the other hand, there are no exercises**, since the
reader can formulate these for himself on the basis of the worked examples, or can
* The Series meant here is called "Mathematik fur Naturwissenschaft uod Technik" published
by VEB Deutscher Verlag der Wissenschaften, Berlin—transl. ed.
** Please note that this English edition of the book does contain excercises which have been pre­
pared by the author at the special request of the editors of North-Holland Publishing Company.—
transl. ed.
v
vi
take them from a suitable compilation. To prevent breaking up the exposition, the
longer proofs are as a rule only briefly sketched, though in such a way that the reader
interested in mathematics can easily fill in the gaps.
A brief survey of the structure of the book will be found in the introduction. But
it should be mentioned right away that I found it unavoidable to present at the begin­
ning of the book, and not say in the appendix, those sections of algebra on which the
later theory is built up. At the present time, unfortunately, algebra does not form
part of the basic background of a student graduating from a technical college, in
spite of its impact on ever-widening fields of applied mathematics, physics and engi­
neering, and the fact that a graduate has to work for perhaps forty years with the
tools acquired during his student days. For this reason, the basic algebraic concepts are
explained at the relevant points as the subject of the book is developed.
More than one chapter is devoted to asymptotic properties, which have become
one of the most important present-day fields for research in analysis. Here, as at other
points in the latter part of the book, some themes are developed which are not to be
found in this form in any other book, and are still far from settled, so that it should
repay even the mathematician to study them. As distinct from the more detailed
treatment of the earlier part, at the close topics are only mentioned or indicated in
isolation. Similarly, the potentialities for applying the operational calculus, which
go beyond ordinary differential equations with constant coefficients, could in essence
only be explained with the aid of examples—which naturally allow the general
methods to be recognized. The reader who wishes to make a more fundamental study
of an individual topic must be referred either to more detailed accounts or to the
specialist literature, since the only aim of the present Introduction must be to acquaint
the reader with the basic material, to offer him some sort of survey and to interest
him in the trends of modern development.
I must first of all thank my teacher, Prof. Dr. L. Holzer (Rostock), with whom I
myself took my first steps in the operational calculus and asymptotic properties. Next I
must thank Prof. Dr. K. Bogel (Ilmenau), Prof. Dr. H. Grotzsch'(Halle), Dr. W. Jentsch
(Halle), Dr. H. Michel (Halle) and Dipl.-Math. F. Stopp (Halle) for numerous pieces
of advice and improvements stemming from their reading or revision of a version
of the manuscript, and Mrs. E. Wolf for careful copying of the manuscript. I am also
much indebted to the publisher and printer for their constant cooperation and
splendid work.
Halle/S., 1961 L.BERG
FOREWORD TO THE SECOND EDITION
After the first edition of the present book had sold out quite fast, there seemed no
point in attempting major modifications in the second edition. Corrections have simply
been made to the misprints and inaccuracies that have been noticed, while the treat­
ment has been made more precise and complete in a few places. Some additions have
also been made to the list of references. Let me take this opportunity to thank warmly
all those colleagues who have contributed, either by letter or orally, to the improve­
ment of the text. As regards the future I shall be more than happy to receive further
critical suggestions that might improve later editions, and also any off-prints in which
topics in operational calculus are discussed.
Of the short sections that are new to the present edition, special mention may be
made of the enlarged introduction, in which numerical methods are now mentioned
and more especially, of a supplement to Section 23, in which some extremely important
results published by Prof. Wunsch have been treated in a somewhat modified form.
These results are concerned with a generalization of the transfer factor (as compared
with the first edition) and of a consequent modification of the concept of perturbation
function for differential equations; further, with a clarification of the role of the
initial data and with a proof of the simplicity and usefulness of the operational cal­
culus even in the case of non-vanishing initial data; and finally, with a clearing up of
the misunderstandings that have arisen in these topics.
Another point worth making is that, where the matter has been mentioned in their
letters, mathematicians have recognized the importance for pure mathematics of
providing a justification for the algebraic foundations of the operational calculus,
but have regarded such justification as too difficult for engineers. Whereas engineers,
both in letters and reviews, have warmly welcomed the approach to the operational
calculus chosen here. The pedagogical advantage of this approach should really not
be overlooked—it provides an essential link between the fundamental concepts of
analysis and algebra, which is of great value to students.
Halle/S., 1964 L.BERG
FOREWORD TO THE ENGLISH EDITION
The English edition is a translation of the second edition of "Einfuhrung in die
Operatorenrechnung". By special request of the publisher the author has inserted
exercises with solutions.
vn
FOREWORD TO THE SECOND EDITION
After the first edition of the present book had sold out quite fast, there seemed no
point in attempting major modifications in the second edition. Corrections have simply
been made to the misprints and inaccuracies that have been noticed, while the treat­
ment has been made more precise and complete in a few places. Some additions have
also been made to the list of references. Let me take this opportunity to thank warmly
all those colleagues who have contributed, either by letter or orally, to the improve­
ment of the text. As regards the future I shall be more than happy to receive further
critical suggestions that might improve later editions, and also any off-prints in which
topics in operational calculus are discussed.
Of the short sections that are new to the present edition, special mention may be
made of the enlarged introduction, in which numerical methods are now mentioned
and more especially, of a supplement to Section 23, in which some extremely important
results published by Prof. Wunsch have been treated in a somewhat modified form.
These results are concerned with a generalization of the transfer factor (as compared
with the first edition) and of a consequent modification of the concept of perturbation
function for differential equations; further, with a clarification of the role of the
initial data and with a proof of the simplicity and usefulness of the operational cal­
culus even in the case of non-vanishing initial data; and finally, with a clearing up of
the misunderstandings that have arisen in these topics.
Another point worth making is that, where the matter has been mentioned in their
letters, mathematicians have recognized the importance for pure mathematics of
providing a justification for the algebraic foundations of the operational calculus,
but have regarded such justification as too difficult for engineers. Whereas engineers,
both in letters and reviews, have warmly welcomed the approach to the operational
calculus chosen here. The pedagogical advantage of this approach should really not
be overlooked—it provides an essential link between the fundamental concepts of
analysis and algebra, which is of great value to students.
Halle/S., 1964 L.BERG
FOREWORD TO THE ENGLISH EDITION
The English edition is a translation of the second edition of "Einfuhrung in die
Operatorenrechnung". By special request of the publisher the author has inserted
exercises with solutions.
vn
INTRODUCTION
1. General survey
The original point of the operational calculus was to replace, in the solution of
differential equations, the transcendental operation of differentiation by the algebraic
operation of multiplication—in much the same way as multiplication in ordinary cal­
culations is replaced by the even simpler operation of addition by using logarithms.
Oliver Heaviside was the first to develop the operational calculus as such, and to
exploit it in numerous applied problems. He introduced the simple formal differential
operator/? = d/dt and handled it (without justification) like a multiplication factor, in
accordance with
/>/(') = 4-/W^/'('). (i)
dt
As a consequence of the lack of justification, it was possible to arrive at false results;
for instance, in the case of the equation pc = 0, which, according to (1), holds for
every constant c, we cannot divide through by/? because c may be Φ 0. Functional
transformations were therefore brought in later for justifying the operational cal­
culus; of these, the best known is the Laplace transformation in the form
F{p) = &{/«)} - Γ <r»M dt. (2)
Precise limits can be set here to the validity of all formulae, i.e. we can write down
concrete assumptions, under which the proposed transformations are permissible.
The connection between the Laplace transform and the operational calculus is pro­
vided by the formula
pnrm = nfV)}+f(P), o)
with the aid of which essentially the same purpose is achieved as in (1): the derivative
f'(t) corresponds to a multiplication by /?. Of course this multiplication is not carried
out on the original function/(i) as in (1), but on the image function (2); it is now no
longer a mysterious operator multiplication, but a perfectly ordinary multiplication
of the variable p in the image space with the transform F(p) = &{f(i)}. A further
difference between equations (1) and (3) is the presence of the added term/(0). It is
only when/(0) = 0 that a differentiation is precisely mapped into a multiplication by
the Laplace transformation; when/(0) Φ 0 formula (3) appears at first sight to have
an imperfection. It will be seen, however, that this apparent imperfection is another
1
2 Introduction
reason why the operational calculus based on (2) is far superior to the old Heaviside
calculus. In fact, the appearance of the term/(0) in (3) enables initial value problems
with arbitrary initial values to be solved, whereas the Heaviside calculus only supplies
a solution when the initial values are zero. The superiority of the Laplace transforma­
tion also reveals itself in the fact that, as well as the property (3), it possesses numerous
other properties which will be dealt with in Chap. VI of this book. If the reader is
particularly interested in the Laplace transformation, he can start with this chapter,
apart from occasional back references. A detailed and comprehensive treatment of
the theory of the Laplace transformation, together with its applications, may be
found in the well known books by G.Doetsch, where the letter s is used instead of p.
These books may be expressly recommended for a deeper treatment.
But it must be realized that, for all its advantages, the Laplace transformation is
only an auxiliary aid in the operational calculus. It is the merit of J. Mikusinski to
have given an exact mathematical foundation to the operational calculus, which is
free from such aids. Starting from the convolution product
fg= [7('-r)g(T)dT (4)
he arrived at the operational calculus by a direct algebraic route. As with Heaviside,
multiplication is now again an operator multiplication, to be carried out direct with
the function f(t). But, as distinct from the Heaviside calculus, it is now well defined
and soundly based. And here again, the calculus based on (4) offers much more than
that based on (2), since it has a wider range of application, enables certain uniqueness
questions to be answered, and also supplies an exact basis for the Dirac delta function,
without having to call in the theory of distributions. Some rather superficial defects
of the product (4) can be avoided by defining multiplication in accordance with
fg = j{'At-T)g(r)dT (5)
instead of (4), as was first done by M. Rajewski, whose method we wish to follow. Not
surprisingly, the defects avoided are replaced by others, likewise superficial. Which
definition has the over-all advantage, time alone will tell. From the mathematical
stand-point the product (5) rather than (4) can be looked on as the average product of
the functions/and g, inasmuch as the integration is here counterbalanced by differ­
entiation. Also, in the case of the product (5) multiplication with a constant function
fit) = c is the same as ordinary multiplication with c. From the technical standpoint
there is the further advantage that the dimensions remain invariant with (5), whereas
in (4) time t always appears as an additional dimension in a product. Purely as regards
formulae, the calculus developed with the aid of (5) corresponds to the calculus based
on the transformation φ(ρ) = p&{f(t)}; this is described in the book by K. W. Wagner
cited at the end.
In control engineering it has been usual so far to make no distinction between the
original region and the transform region, even when using the Laplace transformation.
The Heaviside method 3
This is not strictly permissible, though it becomes so within the framework of the
operational calculus based on (4) or (5). Here again, the basis (5) is to be preferred,
since the dimensions are here invariant, so that there is no further obstacle to the
above-mentioned usage in control engineering, which is now firmly entrenched. The
only objection that could still be made against the calculus based on (4) or (5) is that
now, when p is again an operator, we no longer have at our disposal all the inter­
pretations, methods of calculation and asymptotic relations that stem essentially from
the conception of p as a complex variable. This disadvantage has been removed by
W. A. Ditkin, however, who, in a brief communication to the Academy of Sciences
of the USSR, showed in a purely algebraic manner that the differentiation operator s
(here p) that appears with Mikusinski can be identified with a complex variable.
According to this, the calculus based on the Laplace transformation is merely a
special case of the algebraic operational calculus. If we further replace the algebraic
reasoning of Ditkin by corresponding reasoning from asymptotic analysis, we arrive
in essence at the programme of this book, which was developed by the author as long
ago as 1959 in a lecture in Hannover to the Society for Applied Mathematics and
Mechanics.
Further details regarding the contents of this book—as for instance the operational
calculus for functions of a discrete variable and other operator methods—may be
gathered from the Contents and the introductions to the individual chapters. General
differential operators such as developed by M.A.Neumark and complex integral
operators such as those of St. Bergman are not dealt with here.
2. The Heaviside method
To obtain a first insight into the operational calculus, it is useful, even today, to
follow Heaviside's original method with the aid of a concrete example. We shall start
with the differential equation
y(t) + y>(t) = t2 (6)
and seek a function y(t) that satisfies this equation. Using the notation (1), we can
write (6) as
y + py = t2
or, if we treat/? as an algebraic quantity:
(1 +p)y = t2.
We can write down at once a formal solution of this equation, containing the new
operator (1 + p):
y = -±-t>. (7)
1 +P
Although we have now completed our solution of (6) for y from the purely formal
point of view, we are now faced with what is actually the main problem of the opera-