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NEW YORK UNIVERSITY
INSTITUTE OF MATHEMATICAL SCIENCES
LIBRARY
t* Washington Place, New York 3, N. Y. I M M - N Y U 3 1



JULY 1962




NEW YORK UNIVERSITY
COURANT INSTITUTE OF
MATHEMATICAL SCIENCES



A Note on the Integral Equation of the
First Kind with a Cauchy Kernel

A. S. PETERS



PREPARED UNDER
CONTRACT NO. NONR-285(06)
WITH THE
OFFICE OF NAVAL RESEARCH



IMM-NYU 301
July 1962



New York University
Courant Institute of Mathematical Sciences



A NOTE ON THE INTEGRAL EQUATION OF THE
FIRST KIND WITH A CAUCHY KERNEL

A. S. Peters



This report represents results obtained at the
Courant Institute of Mathematical Sciences,
New York University, with the Office of Naval
Research, Contract No. Nonr-285(06) .
Reproduction in whole or in part permitted for
any purpose of the United States Government.



There are many physical problems which can be reduced to the
solution of the equation

r b $(t)dt
(1) J t _ T = F(t) , a < x < b

a

where F(x) is a prescribed function; and it has been recognized
for well over thirty years that this equation is of central
importance for the theory of singular integral equations. Several
methods have been devised to find the solution of (1) in closed
form. These methods can be classified according to the fundamental
mathematical ideas which are used to obtain the solution:

1. Methods which involve the use of techniques which have
their roots in the theory of analytic functions of a complex
variable, [l], [2], [3].

2. Methods which resort to the Fourier theory of expansions
in terms of orthonormal functions. [4], [5].

3. Methods which employ the Hardy-Poincare-Bertrand formula.

[13. [31.

4. Methods which depend on transform theory without the use
of function theory techniques.

The author has not seen the fourth method in the literature
but many readers will be able to reproduce this method with the
clue that after the transformation t = (b +a!)/(£ +1),
t = (b+ax)/(x+l) which transforms the interval (a,b) to (00, 0);
then (without the use of the Mellin inversion formula, and without
the use of function theory techniques) equation (1) can be reduced
to the solution of






i^iHl



which under appropriate assumptions possesses only the trivial
solution i/(i) = 0.

We are concerned here with the presentation of another
method which leads to a new formula for the solution of (1). The
method is in some sense more elementary than any of the methods
mentioned above, and it consequently may have some pedagogical
value. We proceed to show that (1) can be reduced to the solution
of Abel's integral equation. The reduction is an example of the
application of a known idea (probably under exploited) which is
described below.

A simple translation shows that (1) can be replaced by

1

(2) f 4|iU|i = f(x) , < x < 1



without loss of generality. We assume for simplicity that (£) may have a singularity
like in |a-||, or l/|ct-£| 7 where y < 1. Furthermore, we suppose
that the prescribed function f(x) is a member of the class of
functions to which 4>(£) belongs. Let us prepare the equation (2)
by writing it in the form



(5)

and then

(4)

where



f ifmsi . xf(x) + c



/"tTS*



Lildj



[(/?) 2 -(/3^) 2 ] yx"



yx f(x) +



•x



c = [mm



Integration of (4) gives

1 _ x

(5) - f lnl ^-^ 1 Jl 4>(€)d4 = T/" f(A)dA + 2c/^ .

J V? + ^'

The kernel of this equation can be expressed as

i



(6)



-in



/T-/x = j o
7I + /X 1



J J /T^ 7 /3TTr






d e



4 > x .



If this representation is substituted in (5) we find



A C,

(7) fs?bU)f



d, D.C. (1)

Bjxk.ij University
Graduate Division of Applied

Mathematics
Providence 12, Rhode Island
(1)

California Institute of

Technology
Hydrodynamics Laboratory
Pasadina ij., Calif.
Attn: Professor M.S. Plesset (1)
Professor V.A. Vanoni (1)



Distribution List (Cont.)



N-iii



Mr. C.A. Gongwer
Aerojet General Corporation
6352 N. Irwindale Avenue
Azusa, Calif.



(1)



Professor M,L. Albertson
Department of Civil engineering
Colorado A. + M. College
Fort Collins, Colorado (1)

Professor G. Birkhoff
Department of Mathematics
Harvard University
Cambridge 38, Mass. (1)

Massachusetts I n stitute of

Technology
Department of Naval architecture
Cambridge 39, Mass. (1)

Dr. R.R. Revelle

Scripps Institute of Oceanography

La Jolla, California (1)

Stanford University
Applied Mathematics and
Statistics Laboratory
Stanford, California



Professor J.W. Johnson
Fluid Mechanics Laboratory
University of California
Berkeley Jx, California

Professor H.A. Einstein
Department of Engineering
University of California
Berkeley I4., Calif.

Dean K.E. Schoenherr
College of Engineering
University of Notre Dame
Notre Dame, Indiana

Director

Woods Hole Oceanographic

Institute
Woods Hole, Mass.

Hydraulics Laboratory
Michigan State College
Eas-£ Lansing, Michigan
Attn: Professor H.R. Henry



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Via: Air Force Liaison Office

The RAND Corporation

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Santa Monica, Calif.

Attn: Library (1)

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NROTC and Naval Administrative

Unit
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Laboratory
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OCT 2



'Q62

DATE DUE



FEB 1 * ' fc -'








a&B 4 «








■AY 23 S3








JMf








a*-








|UQ27


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Online LibraryArthur S PetersA note on the integral equation of the first kind with a Cauchy kernel → online text (page 1 of 1)