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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!)/(Â£ +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 (Â£) may have a singularity

like in |a-||, or l/|ct-Â£| 7 where y < 1. Furthermore, we suppose

that the prescribed function f(x) is a member of the class of

functions to which 4>(Â£) 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) +

â€¢x

c = [mm

Integration of (4) gives

1 _ x

(5) - f lnl ^-^ 1 Jl 4>(â‚¬)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-Â£ Lansing, Michigan

Attn: Professor H.R. Henry

(1)

(1)

(1)

(1)

(1)

(1)

Director, USAE Project RAND

Via: Air Force Liaison Office

The RAND Corporation

1700 Main Street

Santa Monica, Calif.

Attn: Library (1)

Commanding Officer

NROTC and Naval Administrative

Unit

Massachusetts Institute ofTech,

Camdridge 39, Massachusetts , ,x

Commanding Officer and Director

U.S. Navy Mine and Defense

Laboratory

Panama City, Florida (I)

OCT 2

'Q62

DATE DUE

FEB 1 * ' fc -'

a&B 4 Â«

â– AY 23 S3

JMf

a*-

|UQ27

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!)/(Â£ +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 (Â£) may have a singularity

like in |a-||, or l/|ct-Â£| 7 where y < 1. Furthermore, we suppose

that the prescribed function f(x) is a member of the class of

functions to which 4>(Â£) 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) +

â€¢x

c = [mm

Integration of (4) gives

1 _ x

(5) - f lnl ^-^ 1 Jl 4>(â‚¬)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-Â£ Lansing, Michigan

Attn: Professor H.R. Henry

(1)

(1)

(1)

(1)

(1)

(1)

Director, USAE Project RAND

Via: Air Force Liaison Office

The RAND Corporation

1700 Main Street

Santa Monica, Calif.

Attn: Library (1)

Commanding Officer

NROTC and Naval Administrative

Unit

Massachusetts Institute ofTech,

Camdridge 39, Massachusetts , ,x

Commanding Officer and Director

U.S. Navy Mine and Defense

Laboratory

Panama City, Florida (I)

OCT 2

'Q62

DATE DUE

FEB 1 * ' fc -'

a&B 4 Â«

â– AY 23 S3

JMf

a*-

|UQ27

1

Online Library → Arthur S Peters → A note on the integral equation of the first kind with a Cauchy kernel → online text (page 1 of 1)