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MATHEMATIC.q.

_

REVUD D'ANAI]YSE NUMÉRIQUE ET DE THÉORIE DD I]'APPROXIMATION

L'ANALYSE

NUMÉRIQUE

ET

I,A.

T'IIÉORIE ÐE

L'APPROXIMATION Tome 12, No

2,

1983, PP. 113-123

GENERAI, SOI,UTION OF THE ARCTANGËNT

FUNCTIONAL EQUATION

by

BORISL,{V CRSTICI, IOAN MUNTEAN a¡d NECUI,AE VORNICESCU

(Timiçoara)

(Cluj-NaPoca)

Abstraet

We

shall

show

that

evet'y soiution

/: R* R of the functional

equa-

tion

f(*) + fU) : Í

frY*r) for all

n,

! eR with'ry <

1,

which is bounded or measurable on an

interval

of positive length, possesses

a finite derivative at the point x:0 and has the

f.orm

f(x):"f'(O)

.

.

arctan

x, x e R.

Nonmeasurable

solutions of this

equatiori

are

exhi-

bited.

The

general solutions

for functional

equations consid.ered.

by I.

Sta-

mate

and.

N.

Ghircoiagiu

and by H.

Kiesewetter

are

derived.

from

our results und.er weaker hypotheses

on

unknown functions.

1.

Introduction

There are many methods

to

define what

is

commonly callecl arctangent

function. The

methods

of

Euclid.ean geometry

first

introduce

the

direct trigonometric functions sine, cosine

and tangent by

quotients

of

lengths

of

some adequate straight

line

segments, and, then

the

arctangent function

f, by

inversion

of the restriction to the - interval l-:, J 2' + Tlof

2L

thetan-

gent function. The

method

of definite integrals

generates

the

values of 2 - L'a¡alyse nuroérlque et la théo¡le de I'approxlmation Tome 12, nr,2, 1983

(2)

I74 B CRSTICI, I MUNTEAN and N VollNIctìScU

the

arctangent

f,

bY

a,

ARCTÀNGENT FUNCTION.AL EQUATION 115

3

further, when the condition

%y

<7 in (1.2) is

replaced

H. Kiesewetter t8l

showed

that the null function f(x) :

the

single continuous

solution of the functional

equation

by

xy # _7,

0, ø eR

is

r

(%

t -9--

1e1

ø eR

J 0

4+Q

l+t'

1+

(see

[6], p.389, and [4]).'l'he

method

of

recuirent-sequences- proposed

ü-A -rr"'i*ià'[7]

arrd'subseqrrct-ttlv dcvcloped

in [9'],.pp' 33-36'

an¿

IiO], pp. 20-'27,

introdtrces

thc

arctangcnt functron

lt

DY

Í"(*) :lim

2"'2,,(x)

lot x eR"

(1.4) Í(*) +l(y):f ii:";) for all x, ! = R with *! +.\.'

2, would

allow

the

reduction

of

the onal ecluation

for the

function

ng the difficulties

concerning

t

equation, : we - shall ; directlY

/: Il--* R of the functional

èqqa.

l. / is ,;tt;*tt

regularitv

conditions:

2.f is R,

'

3./is :

I

4.

"f

ís

5.

/ is stlr,

possesses

u -the'foim

(1.5) Í(*):f'(0)'arctan x, x

e.R.

As an application of these results,

the

general solutiorrs for, functional equations córisidereil

by I. Stamate and-N. Ghircoiaçiu ll2l'

a_n_d try

H'.

Kiesewetter

[B]

are -d.erived under general hypotheses

of

boundedness

or measurability of

unknown functions'

2. Dilfercntiable

solutions

and

continuous solutions

Let

us

first

remark

that for a function/:R-"R sptisfying

(1'2) we have

(2,1) , Í.(0):0

:

'(Pal

x: Y:0 into (l'2)),

and

(2.2) Í(-*) : -Í(x) for x eR

(ptrty: -xinto

(1.2) anduse

(2.1));from

(1.2)

withy -

% we also obtain

(2.3) 2f(x):f(."n i tot J\ ' " lt-xr1 lxl<7. r: ,. '

,;

under the differentiability

condition of

/

on R,

the

following theorem

is

essentially

due to G. H. Hard.y [6], p. 360:

:

THEoRDM

2,1. IÍ /: R* R is

þoint

x :

he functio

show

th e

R:

For

R

having btain

where

zo@): x

anð' 2,1

t:

',(x) f.ot

n2O i

þ,@))

other

methods

to

define

the

arctangent

function are

reviewed

in

[10],

pp. 2-5.

The

number

n,

defined

in thc

Euciidean geonr.etry.as

the

quotient

,r, oiirr"

-iàngth

of any

circle

of

positive radius

by its

cliameter, appears

inio

later metîods

th;;;gh ih" i*tio1""

Tz

:

4

f "(I)

and' Ts

: 2'¡mf

t(x) ' respectiveiy.

' It

iu

be expected.,

of

corlfse,

that the

enumerated methods afe no-

thire

else

but diff;;;;-;t*eedingå to

introduce orle and'

the

same func- tion,-

that is the following

equlities

(1.1)

.f'Lx)

: Írlx) : f"i!x) for all ø e

R'

are

true,

which then im

lly

ær

:

7!2 :.æ3. Bach oJ

the

functions

f ,' f ,

and

/, rr1irfi". the

same conditional

functional

equation

(L2) Í(*)

-r

f(y) : f (:+r) for all

x,

! e R with xv < f

lscc 16l.

pp' 390-391,

for

f,,

anð'

[7] for/'),

the strorrg conc]'itiorr

of

diffc-

ientia¡iiiiv on R and the

equalitics

(1.3) /i(o) : f|p):,fi(o) :

1'

We

intend.

to

derive

(1.1

and some weaker regularity

conditions on the

func

Deno'te

by arctan l,' l.'3\d /'

and

by

æ thc

"orrJupäáingä,t-t"r' rn i" it'zl

haé been considcred

by w, Alt

12) even

in a

more general

setting, but"without taking

into account

of its condîtioo"l

"turncîer.

When

thã condition xy <

1

fu (i'2)

is

replaced

by xy>ï;th"-r"""tiãit f@f J

arctan

x' x e:R'

satisfics ano-

ther functional

equation :

Í(*) +Í(y):f(i"r) f æ'sign x for

alr

x, i e Il with xv >

7"

(3)

Using (2.2), (2.1) and

the continuity

of.

f al 0 we

get

tim

f

(k)

: Í(x) -

tim

f(h) : f(*).

h+x è+0

To

prove

the differentiability

of

f at x :

0, choose a positive number ø such

that

ø

( l,

integrate (1.2)

with

respect

to y in [0, 1]

and obtain

{2.4) Í(*):ø+lr(}fi)ø,

I

xet-ø,øt

where å

is a

constant.

ett"l tn"

change

of

variable

z:!-lY, ye[0, l],

we

have L-try

,=f,, f,lc[- o,f,] ""a | { xz , l¡,

o

md

(2.4) becomes

ARCTANGENT FUNCTIONAL EQUATION 777

3. Boundetl

solutions

and

monotone solutions

THEoREM 3.1.

If /: R--*R is a

bound,ed'

function on

am interaal' of þositiue Lengtk and. satisfies tke

functional

equation (1.2), tken

f is

d'ifferen-

tiabl,e

at x :0

and, (1.5) hold,s.

Proof.

I.et xo eF,

a

¡ 0 and

suppose

that

there

is an Z > 0

such

that lf(fl < ¿ for

all

! =

f-xo

- a, %o*

ø1. Choose ó

¡

0

with blxol 1 |

and.

å(1

I

x'r)

<

ø(1

- blxoù. If x e l-b, bl, then xxs

<,

blxol < |

and. for

J-

xiro

l-t*o

we have

l!-xol:l*l

and,

so

(1.2) yields

(3.1) tf@)t:l-tø,1+r(itu_,."11* trt,.l

I

+ tÍu)t < M, xe l-b,bl,

where

U:lf(xo)l+L.

Now,

we-assert

that/is

continuous

at x':0.

Supposing the contrary,

there exists

an

E

¡ 0

such

that for

each integer

n )- I

we can

indicate

a

number ,,=l- +, +1 with lÍ@,)l > ", rt follows from (3.1) that

there is a subsequence (denoted

in

the same manner) of

the

seilueflce (x,),>r and. there

is a

number c such

that

I

cl > e

and,

(3.2) fi*,: o and tlaf@,) :

c.

By induction we associate

with

each

integer k ) 0 a

sequênce

(x!),>,, having the

properties

(3.3) li* *I:0 and ¡rnf(xï) :2b'c'

Namely,

lork:0

we

put %o: I

and'

xf;: fi,for all n 21,

and we see

that the equalities (3.3) revert to

(3.2).

Admitting that the

sequence

(x!),r,r with the

properties (3.3)

is

constructed-,

we first

determine an

index np¡1 so large

as l*l,l <

1

for all n 2

np+t and

then

we

put

*I+':-2!=

'

fornÞtp+t.

I -(xÍ\

Then

lim *!,*t :0

antl

Í(*1*\ :

2f (x!,) (see (2.3)), hence

limf(x!+r) :2

.

2'

c

:2þ+r

. c.

116 B. CRSTtrCI, ¡1. MUNTEAN and N. VoRNICESCU 5

l*z

4

-*l* +

h)

< l,

whence

by

(2.2), (1.2),

(2,1) and the differentiability

of.

f at 0 we

derive

lim :lim!.¡( u

-.1

: t .tim/0-10):l'Q)

h+o h+o It -\l+*(x!h)) t+*, Þ+o h llr2

Thus,

/

is differentiable

at

each

point r e R

and

.f'(x) :

{':',,,

hencethereis ac eRsuch that f(x):Í'(0).arctan xlc, ø eR;The

equalities

(2.1) and

arctan

0:0

yielð.

f(ù):,f'(0).arctan x, x eF..

fn the

case

of

continuous solutions we have:

TTTEoREM 2.2.

IÍf

:

R-* Ris

ø continuousfunction atthe

þoint x:0

ønd

søtisfigt

jlrt- functional

equation (1.2), then

f is

diJferentløbl,e

øt

tkis

þoint

ønd, (1.5) hold.s.

Proof

.

We sha1l

first

transfer

the continuity

of.

f

fuom

x :

O

to

any

point x eR. For all h e R sufficiently

close

to ø

we

have -xh 1l

and,

by

(1.2) we obtain

Íþc) -f Í(-

h)

: t

f

(* - hI.

lt+

rn)

r*rã a

l-xxo ó

t+nâ l-blxol <a.

lþ):bl(tlx,) __JþL

(l I xz)z ¿,

1-*

a

I

,xel-a,ø1.

The

theorem

on differentiation of

integrals

with

re6pect

to a

parameter ensures

the

differentiation of

f at x :

0 and. Theorem

2.1 applies. \

(4)

118 B. CRSTICI, L MUNTEAN and N. VORNICESCU

l*tr,

ø ll2

6 I ARCTANGENT FUNCTIONAL EQUATION 119

Now

choose

an integer h

>-

0

such

that

2h

. e

)-

M + l. It

follows

from

(3.3)

that

there exists

tx

>

nr so large

that

|

*:l < b

anð, lÍ@X) I

>

> 2r.ltl - | >

2re

- I

whence,

in virtue of

(3.1),

we arrive at the

con-

tradiction ,M )- lf(x:)

l

>

2"e

- | >

1l1. Consequently,

/ must be

conti-

nuous

át x:0 andlTh"ot"- 2.2

applies..

, #,

#,if i

íi,

"v,

o

{;ir'.::,: ;# *T ; :' i;f:

isfies

the functionø|, equøtion (7.2) length,

0 e I,

suclt.

tkøt xf(x) >

0

for_al,l-2.=I (or xf(x) 40for

ø1,1,

x eI), thenf is

d,ifferentiable

at x:0

ønd (1.5) lrclds.

.

Proof. ^U:_t-rS

-(2.2) we may admit that I

has

the form I :

10,

al,

where ø

>

0.

we

shall prove

that if xf(x) )

0

for

all

x e r, then/is i"ót""- singcn L Let x, z e /with x 12. Theny:ftr>0, y 42, xy <l

an! f(ù ).0,

hence

Í(,) : r (i=r) :.r(x) + Í(v)

>

Í(*).

" Simila,ily,if^xf(x) (.0 for

a77

x e

1,

then/is

decreasing on _L

In

both cases Corollary

3.2

applies.

3.\.If /:R-" R is ø

d.ifferentiøbte

(or

continuous) func-

,in

R^and

_søiisfj.es the functionøI equøtion (1.2), tken

f

is

x :0

and. (1.5) holds.

4.

Measurable solutions and nonmeasurablo solutions

' I,etþ, {, x eRwithþ*q, Iþl <qandqlxl<

1. Thehomographic

runcrion k: h,: lþ, ql-*130, #;1,

d.efined

by

(4.r)

h(v)::Yt!elþ,q1,

r-fiy

is

strictly

increasing-and continuous together

with

íts inverse. Consequently,

maps^open (closed) sets_into o_pe4 (clõsed, ¡espectively) sets and. thä ima[é

by h of an

open

interval lø, blCLþ, gl,

ø

<

ó,

is ihe

open

interval

e

(4.2) h(la, b[) : l[,where 6-!J-" and,d.- rtb.r

Moreover, '-:' l-bx

(4.3) (b

-

o)'

T,EMMA

4.1.

Tkeimøge by h' of any nxe(Ìs1,t/øbl,e setÐ

Clþ' ql is

measu'

vøble ønd. satisfies

(4.4)

mes

h(E) , ,+--'

(1

-l

qlxl)" rnes -Ð'

Proof.

we

use

the following well-known

characLetization

of

rneasu-

rable sets

in R (cf. [11], pp. 73-76):

a set

M ínP. is

measurable

if

and

onlyiffor each; > 0therôèxist

all openset G

f

ancl

a closedset]i f fl

such

that F CluI f G and nes

(G

\F) <

".

I,et

e

¡ 0.

Since

E is

measurable, there exist an open set G

6 lþ' ql

and a

closed

set Ff IlsuchthatFCECGand (4.5) mes(G\P) < \-qVl)'-'e.

l*x'

The

open det

G

..

F

can be represented as union

of a

countable farnily of

mutually

disjoirrt open intervals : G

\ F :

LJ {1' :

h > l}

(cf

.

[11],

p.

54),:

and so

the

measure

of

G

\.. F is

given b1'

(4.6)

mes (G

\ F) : 'f) -". r,.

The

irnages G,

: h(G)

anó,

¡;, - h(F).Ïton"n

and closed, respectively, and

verify Frçh(E) f

G..

Irrorn

(4.3), (4.6) and (4.5) we derive

(4.7)

mes (G1

'.. F,) <

111e,j /z(G ... -{i)

<

mes

(U {k(Ir):Ä > 1})

<

*f *", h(Ir) <,;itr-".)),,'"0 1*: (,'+#4y 'mcs(G .F) <

u,

i.e. the

measurability

of h(E) is

proved.

Ihe inequality $.\

can

be

dedrlced

froin a known result (cf'

[11],

pp.228-229) .

Hcwever',

for the

sake

of

completeness

we

present here ã-direct proof

of

(4.4). The open set G, can also be

written

as

union of

a countablõ

famil¡' of rnutually disjoint

open

intervals:

G,

: U {"f,:

h,

> l},

where

f r: fc,

d*1. Denote

by ar: ø

and.

br:

b

the

numbers obtained

from

tñË

lasi

irvo'èqualities

(4.2)

lor

c

: cr

and d,

: d,

Cleally,

,f, :

: k(Lr), where Ln: lør, b*[.

Moreover,

(4.8) GCU{Lr:h>t}.

Ind.eed, supposingthe contrary, there exists

an y e Gwitlny ø l) {Lr: lt

>-

)

1). For the numb"r

z: h(y) e

k(G)

: Grit

must exist an inte.ge¡ k'

->

7

ru"h thut , = Jr:

h(Lr), heîce there

is

an

y' e Lr wltl¡

z

: h(y')'

Now,

the injectivity of h'leads to the contradiction y : !' =

Lr.

Using

(4.3)

and

(4.8)

we

obtain

-9- I å

mes.E

mes G,

:

Jt F9s./¡ >

(1 +

qwry þi

mes

L, >'

11

¡

qln1y'

(r-q

(5)

720

which together

with

G,

:

h(E)

U

(G'

\

¿(E))

Ch(E) U (G'\F')

anð' (4.7)

yield

rrres

h(E) >

mes G1

-

rnes (G,

\FJ > ---:! i- -

e

Since o

is an artritrary

positive number, (4.4)

is

proved'

TrrEoRÞM 4.2.

IÍ /:R-'R is a'

measurable

fønction

otu

øn

interual'

of

þositiae lcngth and' søtisfies tloe

functional

equati,on

(l'2),

tken

f is

d'iffe-

ientiøble

at x :0 and

(1.5) hold's.

Proof. Our

argurnent

is

based

on the

Banach's

method [3] for

the

integration of

Cauchy

functional

equation.

the

interval

lxo

- a,

1(o

* ø],

where

'!ir'1,""î'"åTî"jo:'#"0il:'i:it1

is a continuo's runction s: lx,-., îT*þ*tljtår:,iJ; 118-11e)

there

(4.9) mesl/.i,

where

H:{x =lxo- d, no*øl:Í(x)+

SØ)\.

Let

E

be a positive

number. Since

g is uniformly

continuous,

there is

a

8>0with

(4.10) le@)-

S(?r)

l< efor tt,a elx¡-ttr, %otø1, lu-al <

Ð.

Pttt þ : xo- |,

ø

: xo* !,

and,

(4.11) 4 : min

^'-^'ìz("0

{=-f- ¡

a7' z¡t

, * -L.

+,1@o*

o)l' --t

Z¡t

+

ø')

I'

I

Let x eR with lxl<r¡. Clearly, lþl <q, þ+q

anð.

qlxl<1.

Accor-

ding to

(4.11)

the function h: k*

defined

in (a.l) fulfils

k(þ)

>

no

-

a

anð, k:(q)

1

xo

I

ø, hence

(4.12) h(lþ,

qD

C lro -

&, xo

-f

ø1.

By

(4.9), the measurable set

E:lþ, qt\H

satisfies mesE

> a'- i :1o' 44 In virtue of Lemma 4.1

we have

(4.r3)

mes

h(E) ' > -Ïelq-=+:+¡l-". (ltqlx\'' Ílsn)' 3 B2

4

!

since qr¡

< ls -

1.

The

inequalities

in

(4.9)

and

(4.13) show

that the

ser k'(E)

\ I/

is

nonvoid, hence

by $J2)

there is a number

z e

h(E)

Ck(lþ,

ql)

C lx, - a,

xo

I al

g ARCTANGENT FUNCTIONÄL ESUATION 727

with

z 4 H,

anð.so there exists an

! eE such ttat lÏy: h(y):

z'

Therefore,

! ê H ""a !_ - 4 H.

Tlne

last relations can be written in the

form

(4.r4) f(t): eu) and rl:=,): t(i:, -,,ì :

e(,)

By

(a.11) we obtain

lz - yl :

I

xl lLt: < n'!f-t: <2nl + q') (

8'

l-xY L-qn

hence

from

(1.2), (4.14) and (4.10)

we

derive

tÍ@)-/(0) t: tÍ@)t:l¡í-", -Í(v) l: ttt') -su)t<u

and.

the continuity

o1.

f at x : 0 is

proved'

RÞMARr( 4.3.- G.

"Hamel t5]

clonstructed.

a

d.iscontinuous solution

g: R-" R of the

Cauchy

functional

eqnation

(4.15)

g(u

+

u)

:

g(u)

+

g(ü)

for all

ot',

a eH"

We shall prove

that the function/:R-'R

defined

þv Í@): g(arctanx)

is an

unbäunded. and. nonmeasutuËl" solution

of

(1.2)

on any interval

of

positivc

length.

----in"fun"ction/satisfies

(1.2) since,

if x, y e

Rand.

xy <7,from

(4.15)

we

derive

(4,16) Í(*) + Í(y) :

g(atctan

x) |

g(arctaî

!) :

:

g(arctan

x I

arctan

!) :

B(arctan

i+r\ : f (U, å)

Now,

/

is neither bounded noï rueasufable on any ínhetvalof positive.]9ngth,

i"ã"ää, -*pposing

the contrary,

(4.16) together_with .Theorems

3.1

and

¿.2

*"ittlitãptvihe continuity

of

f

air

x:0. B'lrt g(y):f(tany), y e

=l-;,å[,.ogwouldbecontinuousaty:0andtheobtainedcon-

tradiction

achieves

the proof of our

assertion'

5.

Some applieations

Und.er

the differentiability

condition of

/

on

R the

following corollary

has been proved

by I.

Stamate and

N.

Ghircoiaçiu ^[12].:

-- ðó"oir,,lnv 5.i.

Let

f, g, h:I\-,R

be giuen functi,ons ult'iclt, satisfy

the

cond,itional,

functionøl

eqøat'ion

(5.1) Í(*) + eU) : r(=*) for

øtt' %,

! eR' uitk xv 1r'

B. CRSTICI, ,I. MUNTEAN and N. VORNICESCU 8

(6)

722 B. CRSTJCI, '¡. MUNTEAN and N. VORNICESCU 11 ARCTANGENT FUNCTTONAL EQUATION 723 10

,

then the

erl

by

the tod.

h(x):

d then

with

y: 0,

we find

(s.2)

h(z)

: f(0) + gþ)

and, h(z)

: Í(z) l- s(0) for alt z e

R,

whence

(5.3)

e(z)

:f(z) +

s(0)

- f(0) for alt z =

R.

Define the functior.q.jl*R by ç(*)

=f(x) _f(0)

and.

remark that,

according

to

(s.2) and (5.8), the eqiratiò'i

ls.ij tãr."í ìrí" iåì,,

ç(x) l- ç(y): r(=+rl for all

x,

! eR with xy <

t.

Now, from Theorems 3.1 and 4.2

it

forlows

that

g is differentiable

at x :

o

13d.it is given bv qV): p'(O).arcta' *, * ='F., ì;-flr"

conclusion of Corollary

5.I is

immediáte.

' '

'

s on

R the

following

result

has been bound.-ed.

(or

meøsotrøbte) function sfies the cond.itional functionat equø-

rems

3.1

and.4.2it

follows

rhat

f,, u,hilit"f,"nit':i ;':àÏati",'l

"ii::.

ssed.

by

^(1.5)..

From

(1.4)

and

(2.2)

we get 2f(JD _

1)

: flt), zf(^lî ¡

*

1)

: Í(-r): -/(1)

and

/(1) +f(J,_\:fdr*1) : _f,f:J,

[6] If arily, G. 11., A course of þure watkemøti.cs, Fourth Eclition, Cambriclge University [7] I{urw

l8l I(iese

Math.-Natur. Reihe ; 12t' 417'421, (1965).

[9] Maak,w., Di.fferential - und, Integtalrechnung,4.Ãtflage,vanclenhoeck uncl Ruprecht, Göttingen, 1969.

t10] Muntean i., Elementøry trør¿scendental functions (Romanian), Universitatea ,,Babeç-Bolyai", Cluj-NaPoca, 1982.

t11l Natansoa,'I. P., Theory of fiutctions of a real aariabl'¿, Seconcl Etlition (Russian),

Gos. Izdat. Tehn.-'I'eoret. Lit., Moscow, 1957.

t12l St am.ate, f. ancl Ghircoiagir, ñ., Fuøctional' equalions defining trigonotn-etri.c ftr,ncl'iotcs (Romanian), Bul. $tiin!, fnst. Politehn. Cluj, 10' 57-60, (1967)'

Rece¡ved 1s.VI.1983

Føaultatea d,¿ malemaliad.

Uniuersitøteø B abeç- B olyøi

Str. Kogãl,niceønu

I

3400

Ctruj-Naþocø

whence ,f(1)

:0. Using

(1.5)

with

ø

: 0,

we

obtain

0

:/(1) : Í'(0)

.

arctan l:f'(0).; ,

hence

Í(x) :0 for all z e

R.

RÞFERENCES

tl I A c z éI, !., V nagleichungen ultd, ihre Anuaod.tmgen,, Birkhäuser

Verlag, iget.

L2l Alt, W., Üter ciner yeellen Veyänd,erlichcn, uelche cin røtio¡tal¿s Ad,d,itionstheorew besi,tzen, Deutsche Math., 5, l_12. llé40)

[3] Banach, s., t2 Søz t'équation fonc!,ionneue ¡ø fil:iø l'¡üí,'äo¿"-. Math., 1, [4] Eberlei eletnenlary trønscendental, functions, Amer. Math. Monthly

81 ). '

[5] Ilamel, ie Zahlen unil die unstetige l-ösungen.d,ey Funhüionalgl,eichung

f@

I

y)

-

Í(t¡) + f(y), Marh. Ann., G0, 4sö-462, (ie05).

Referințe

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