This monograph deals with the problems of mathematical physics which are
improperly posed in the sense of Hadamard. The first part covers various
approaches to the formulation of improperly posed problems. These
approaches are illustrated by the example of the classical improperly
posed Cauchy problem for the Laplace equation. The second part deals
with a number of problems of analytic continuations of analytic and
harmonic functions. The third part is concerned with the investigation
of the so-called inverse problems for differential equations in which it
is required to determine a dif- ferential equation from a certain family
of its solutions. Novosibirsk June, 1967 M. M. LAVRENTIEV Table of
Contents Chapter I Formu1ation of some Improperly Posed Problems of
Mathematic: al Physics § 1 Improperly Posed Problems in Metric Spaces. .
. . . . . . . § 2 A Probability Approach to Improperly Posed Problems. .
. 8 Chapter II Analytic Continuation § 1 Analytic Continuation of a
Function of One Complex Variable from a Part of the Boundary of the
Region of Regularity . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 13 § 2 The Cauchy Problem for the Laplace Equation . . . . . . .
18 § 3 Determination of an Analytic Function from its Values on a Set
Inside the Domain of Regularity. . . . . . . . . . . . . 22 § 4 Analytic
Continuation of a Function of Two Real Variables 32 § 5 Analytic
Continuation of Harmonic Functions from a Circle. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 38 § 6 Analytic Continuation of
Harmonic Function with Cylin drical Symmetry . . . . . . . . . . . . . .
. . . . . . . . . . . 42 Chapter III Inverse Problems for Differential
Equations § 1 The Inverse Problem for a Newtonian Potential . . . . . .
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