Solution to Problem 72-11: Limit of an integral

Solution to Problem 72-11: Limit of an integral

Citation for published version (APA):

Lossers, O. P. (1973). Solution to Problem 72-11: Limit of an integral. SIAM Review, 15(2), 390-391. https://doi.org/10.1137/1015046

DOI:

10.1137/1015046

Document status and date: Published: 01/01/1973

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390 PROBLEMS AND SOLUTIONS

By [1],

it is sufficient to construct a 2m+x 2m+

1(__

1,1) matrix W for which det W- 2(’/1)2". Let A denote the 2 x 2 matrix and define W= (R)"

+IA,

the (m+ 1)th Kronecker power of A. W satisfies detW

(det

A)(’/x)2- _2(./

2-which provides a desired matrix.

Problem 72-11,Limit

_of

anIntegral,by N. MULLINEUXandJ. R. REED(University of

Aston,

Birmingham,England).

Determine

! lim log

Isin

t-

13/21

dt

-o sin

_13/2

sin

The problem arose in the numerical inversion of singular integral equations

such asthose which occur inaerodynamics and problems associated with plane transmission lines.

Editorial note. Most solvers

(see

laterdiscussion) noted that the integral as

given above does not exist since for fixed e, 0, the integrand behaves like

-

log

{(13/2)sin

13/2}

as 0. The error is a typographical one since in the original proposal the term

_Isin

t/2l

appears as

_Isin

(t

13/2)1.

In the following solution ofO. P. LOSSERS (Technological University of Eindhoven, Eindhoven,

the

Netherlands)

the latter expression is used. Replacing 13 by 213 we get

log Isin(t 13)1 dt

,2

sin13 sin ₀

dt

log

Icos

cot13.sin

tl

sin

logcost---

+

10gll cot13.tan

tl

sn sn

Since (logcost)/sin

t/2

+

O(t

3)

(t

0), we have that the first term is

O(132),

(13

0).

By

substitution ofv cot13.tan we get

fcote.tan2e

log

_l1

_vl

I lim O(132

--e--,0 _d0

(1

+

/)2tan2

_13)-1/2

21ogll_/)ldv=

v 4

In the solutions by N. M. BLACHMAN (Sylvania Electronics Systems), L. KHATCHATOORIANTZ (California State College, Los Angeles), S. L. PAVERI-FONTANA(University of California,Berkeley), Z. C. MOTTELERandO. G.RUEHR

(Michigan Technological University),J. W. RIESE(Kimberly-ClarkCorporation),

J.D.TALMAN(University of

Western

Ontario,London,Canada)andP. TH. L. M.

VANWOERKOM (National

Aerospace

Laboratories,Amsterdam,theNetherlands), it was remarked that the integral as originally stated does not exist. Various

alternative statementsof the problem weremade. We donot listthem all.

Most

PROBLEMS AND SOLUTIONS 391

the stated integral. The solution then proceeds much like that given above and leads to the same answer.

In

the solutions by

A.

G. KONHEIM (IBM Research Center),

A.

J. STRECOK (ArgonneNationalLaboratory)andI. FARKAS(University ofToronto,

Toronto,

Canada), the existence ofthe integral was ignored. They proceededinafashionmuchakin tothat in the above solution and alsoarrivedat

the samevalue,

-7t2/4.

Also, solvedby the proposers.

[Y.

L.

L.]

Problem 72-12, Construction

_of

.a

Maximal(0, 1)Determinant, by J. R. VENTURA, JR. (Naval Underwater

Systems

Center).

Itisknown

[1]

thatif n 2 1 forsomepositive integer m, then the

deter-minant ofthe n n matrix

_V.

having a maximal value over all n n matrices havingO’sand l’sas entriesisgivenby

f(n)

_IEI

2-"(n

+

1)("

+

Construct a

(2n

+

1)

x

(2n

+

1)

maximal (0,

1)

determinant.

REFERENCE

Ill J.COnN,Onthevalue_ofdeterminants,Proc.Amer.Math.Soc.,14(1963),pp.581-588.

Solution by the proposer.

Let

U,

be the n n matrixconsisting of all l’s. Let V2,+1 be given by

-0...0 1...1

1-E

0 Then,

IV2n+ 11

(2n

+

2)

"+1

22n

+

F(2n

+

1).

Solution by P. J. NIKOLAI (Aerospace Research Laboratories,

Wright-Patterson AFB).

Also solved by C. C.

Rousseau

(MemphisStateUniversity).

ERRATUM FOR JANUARY 1973 PROBLEM SECTION