Representation of some classes of quaternionic hyperholomorphic functions
In the algebra of complex quaternions H(C) we consider the left- and right-ψ-hyperholomorphic functions, and left-Λ — ψ-hyperholomorphic functions. We justify the transition in left- and right-ψ-hyperholomorphic functions to a simpler basis i.e., to the Cartan basis. Using Cartan's basis we...
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Інститут математики НАН України
2025
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| author | Kuzmenko, Tetiana Shpakivskyi, Vitalii Кузьменко, Тетяна Шпаківський, Віталій |
| author_facet | Kuzmenko, Tetiana Shpakivskyi, Vitalii Кузьменко, Тетяна Шпаківський, Віталій |
| author_institution_txt_mv | [
{
"author": "Тетяна Кузьменко",
"institution": "Житомирський військовий інститут імені С. П. Корольова"
},
{
"author": "Віталій Шпаківський",
"institution": "Інститут математики НАН України"
}
] |
| author_sort | Kuzmenko, Tetiana |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2026-02-24T12:22:41Z |
| description | In the algebra of complex quaternions H(C) we consider the left- and right-ψ-hyperholomorphic functions, and left-Λ — ψ-hyperholomorphic functions. We justify the transition in left- and right-ψ-hyperholomorphic functions to a simpler basis i.e., to the Cartan basis. Using Cartan's basis we find the solution of Cauchy-Fueter equation. By the same method we find representations of left- and right-ψ-hyperholomorphic functions, and representation of left-Λ — ψ-hyperholomorphic functions. |
| doi_str_mv | 10.3842/trim.v21n1.543 |
| first_indexed | 2026-08-04T01:09:09Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Ií-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè (2024) ò. 21, �1, 102�120
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ
êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ
ôóíêöié
Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
Abstract. In the algebra of complex quaternions HpCq we consider the
left� and right�ψ�hyperholomorphic functions, and left�Λ ´ ψ�hyperholo-
morphic functions. We justify the transition in left� and right�ψ�hyper-
holomorphic functions to a simpler basis i.e., to the Cartan basis. Using
Cartan's basis we �nd the solution of Cauchy�Fueter equation. By the same
method we �nd representations of left� and right�ψ�hyperholomorphic
functions, and representation of left�Λ ´ ψ�hyperholomorphic functions.
Àíîòàöiÿ.  àëãåáði êîìïëåêñíèõ êâàòåðíiîíiâ HpCq ðîçãëÿíóòî ëiâî�
i ïðàâî�ψ�ãiïåðãîëîìîðôíi ôóíêöi¨, à òàêîæ ëiâî�Λ´ψ-ãiïåðãîëîìîðô-
íi ôóíêöi¨. Îá ðóíòîâàíî ïåðåõiä ó ëiâî� i ïðàâî�ψ�ãiïåðãîëîìîðôíèõ
ôóíêöiÿõ äî ïðîñòiøîãî áàçèñó, à ñàìå äî áàçèñó Êàðòàíà. Âèêîðè-
ñòîâóþ÷è áàçèñ Êàðòàíà, çíàéäåíî ðîçâ'ÿçîê ðiâíÿííÿ Êîøi�Ôóåòåðà.
Òàêèì æå ìåòîäîì çíàéäåíî ïðåäñòàâëåííÿ ëiâî� i ïðàâî�ψ�ãiïåðãîëî-
ìîðôíèõ ôóíêöié, à òàêîæ ïðåäñòàâëåííÿ ëiâî�Λ ´ ψ�ãiïåðãîëîìîðô-
íèõ ôóíêöié.
1. Âñòóï
Îñíîâíèì îá'¹êòîì äîñëiäæåííÿ ¹ ìíîæèíà, ÿêó çàçâè÷àé íàçèâàþòü
ìíîæèíîþ êîìïëåêñíèõ êâàòåðíiîíiâ i ÿêà òðàäèöiéíî ïîçíà÷à¹òüñÿ ÷å-
ðåç HpCq. Öå àñîöiàòèâíà íåêîìóòàòèâíà àëãåáðà íàä ïîëåì êîìïëå-
êñíèõ ÷èñåë, ïîðîäæåíà åëåìåíòàìè 1, I, J , K òàêèìè, ùî âèêîíóþòüñÿ
íàñòóïíi ïðàâèëà ìíîæåííÿ:
I2 “ J2 “ K2 “ IJK “ ´1,
IJ “ ´JI “ K, JK “ ´KJ “ I, KI “ ´IK “ J,
i êîìïëåêñíà óÿâíà îäèíèöÿ i êîìóòó¹ ç I, J,K. Äëÿ àëãåáðè HpCq òàêîæ
âèêîðèñòîâó¹òüñÿ íàçâà � àëãåáðà áiêâàòåðíiîíiâ.
This work was supported by grants from the Simons Foundation (1290607, V.S.S.)
2020 Mathematics Subject Classi�cation: 30G35; 32A10
Êëþ÷îâi ñëîâà: êîìïëåêñíi êâàòåðíiîíè, áàçèñ Êàðòàíà, ëiâî� i ïðàâî�ψ�
ãiïåðãîëîìîðôíà ôóíêöiÿ, âàãîâèé îïåðàòîð Äiðàêà, ðiâíÿííÿ òèïó Êîøi�Ôóåòåðà,
ëiâî�Λ ´ ψ�ãiïåðãîëîìîðôíà ôóíêöiÿ
DOI : https://doi.org/10.3842/trim.v21n1.543
102
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 103
Ðîçãëÿíåìî â HpCq iíøèé áàçèñ te1, e2, e3, e4u, ÿêèé ¹ áàçèñîì Êàðòà-
íà [7], ðîçêëàä åëåìåíòiâ ÿêîãî â áàçèñi Ãàìiëüòîíà ì๠âèãëÿä:
e1 “ 1
2
p1 ` iIq, e2 “ 1
2
p1 ´ iIq, (1.1)
äå i � êîìïëåêñíà óÿâíà îäèíèöÿ.
Òàáëèöÿ ìíîæåííÿ â áàçèñi Êàðòàíà ïîäà¹òüñÿ ó âèãëÿäi
¨ e1 e2 e3 e4
e1 e1 0 e3 0
e2 0 e2 0 e4
e3 0 e3 0 e1
e4 e4 0 e2 0
. (1.2)
Ïðè öüîìó îäèíèöÿ àëãåáðè ì๠ðîçêëàä 1 “ e1 ` e2 .
Âiäìiòèìî, ùî ïiäàëãåáðà ç áàçèñîì te1, e2u ¹ àëãåáðîþ áiêîìïëåêñíèõ
÷èñåë BC àáî àëãåáðîþ êîìóòàòèâíèõ êâàòåðíiîíiâ Ñåãðå (äèâ., íàïðè-
êëàä, [3, 16]).
Ñïðàâåäëèâi òàêîæ ðiâíîñòi:
1 “ e1 ` e2 , I “ ´ie1 ` ie2 , J “ ´ie3 ´ ie4 , K “ e4 ´ e3 . (1.3)
Î÷åâèäíî, ùî ôîðìóëè (1.1) i (1.3) ¹ ôîðìóëàìè ïåðåõîäó âiä áàçèñó
Ãàìiëüòîíà äî áàçèñó Êàðòàíà i íàâïàêè.
Ðàçîì ç áàçèñîì Ãàìiëüòîíà i Êàðòàíà ðîçãëÿíåìî òàêîæ áàçèñ Ïàóëi.
Âiäîìî, ùî êîìïëåêñíi êâàòåðíiîíè ìîæóòü áóòè ïðåäñòàâëåíi ÷åðåç
ìàòðèöi Ïàóëi:
σ0 :“
ˆ
1 0
0 1
˙
, σ1 :“
ˆ
0 1
1 0
˙
, σ2 :“
ˆ
0 ´i
i 0
˙
, σ3 :“
ˆ
1 0
0 ´1
˙
.
 öüîìó áàçèñi òàáëèöÿ ìíîæåííÿ íàáóâ๠âèãëÿäó:
σ21 “ σ22 “ σ23 “ σ0 , σ1σ2σ3 “ iσ0 ,
σ1σ2 “ ´σ2σ1 “ iσ3 , σ2σ3 “ ´σ3σ2 “ iσ1 , σ1σ3 “ ´σ3σ1 “ iσ2 .
Ïðè öüîìó ôîðìóëè ïåðåõîäó âiä áàçèñó Êàðòàíà äî áàçèñó Ïàóëi
ìàþòü âèãëÿä:
e1 “ ´1
2
`
σ0 ´ σ3
˘
, e2 “ 1
2
`
σ0 ` σ3
˘
,
e3 “ 1
2
`´σ2 ´ iσ1
˘
, e4 “ 1
2
`´σ2 ` iσ1
˘
.
(1.4)
104 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
2. Êëàñè ãiïåðãîëîìîðôíèõ ôóíêöié
Íåõàé ψ1 , ψ2 , ψ3 , ψ4 � ôiêñîâàíi åëåìåíòè àëãåáðè HpCq ç íàñòóïíèì
ðîçêëàäîì â áàçèñi Êàðòàíà:
ψ1 :“
4ÿ
s“1
αses , αs P C, ψ2 :“
4ÿ
s“1
βses , βs P C,
(2.1)
ψ3 :“
4ÿ
s“1
γses , γs P C, ψ4 :“
4ÿ
s“1
δses , δs P C.
Ðîçãëÿíåìî çìiííó z “ z1e1 ` z2e2 ` z3e3 ` z4e4 , zs P C, s “ 1, 2, 3, 4 i
ôóíêöiþ
fpzq “
4ÿ
s“1
fspz1, z2, z3, z4qes , fs : Ω Ñ HpCq,
äå Ω � îáëàñòü â C4. Íåõàé êîìïîíåíòè fs, s “ 1, 2, 3, 4, � ãîëîìîðôíi
ôóíêöi¨ ÷îòèðüîõ êîìïëåêñíèõ çìiííèõ z1, z2, z3, z4 â Ω.
Ðîçãëÿíåìî îïåðàòîðè
ψDrf spzq :“ ψ1
Bf
Bz1 ` ψ2
Bf
Bz2 ` ψ3
Bf
Bz3 ` ψ4
Bf
Bz4 , (2.2)
Dψrf spzq :“ Bf
Bz1 ψ1 ` Bf
Bz2 ψ2 ` Bf
Bz3 ψ3 ` Bf
Bz4 ψ4 . (2.3)
Îçíà÷åííÿ 2.1. Ôóíêöiÿ f : Ω Ñ HpCq, Ω Ă C4, íàçèâà¹òüñÿ ëiâî�ψ�
ãiïåðãîëîìîðôíîþ (àáî ïðàâî�ψ�ãiïåðãîëîìîðôíîþ), ÿêùî êîìïîíåíòè
fs ¹ ãîëîìîðôíèìè ôóíêöiÿìè ÷îòèðüîõ êîìïëåêñíèõ çìiííèõ z1, z2,
z3, z4 â Ω i f çàäîâîëüíÿ¹ ðiâíÿííÿ
ψDrf spzq “ 0. (2.4)
(àáî Dψrf spzq “ 0.)
Êëàñ ψ�ãiïåðãîëîìîðôíèõ ôóíêöié â àëãåáði äiéñíèõ êâàòåðíiîíiâ
âïåðøå áóëî ââåäåíî Ì. Øàïiðî òà Í. Âàñèëåâñüêèì â ðîáîòàõ [21,22].
Òàêèé êëàñ ôóíêöié çàöiêàâèâ áàãàòüîõ äîñëiäíèêiâ. Çîêðåìà, Ê. Ãþð-
ëåáåê òà éîãî ó÷åíü Õ. Ì. Íãó¹í ïðèäiëÿþòü îñîáëèâó óâàãó çàñòîñó-
âàííþ ψ�ãiïåðãîëîìîðôíèõ ôóíêöié (äèâ., íàïðèêëàä, ñòàòòi [5, 11, 12]
òà äèñåðòàöiþ Ã. Ì. Íãó¹íà [17]). Çàóâàæèìî, ùî îïåðàòîðè (2.2) i (2.3)
òàêîæ íàçèâàþòü âàãîâèìè îïåðàòîðàìè Äiðàêà. Àíàëiç i çàñòîñóâàííÿ
òàêèõ îïåðàòîðiâ âèâ÷àþòüñÿ â ñòàòòÿõ [23,24].
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 105
Àêòèâíî äîñëiäæóþòüñÿ ðiçíi óçàãàëüíåííÿ ψ�ãiïåðãîëîìîðôíèõ ôóí-
êöié. Îñòàííiì ÷àñîì ñòàëè öiêàâèìè óçàãàëüíåííÿ íà âèïàäîê äðîáîâèõ
ïîõiäíèõ (íàïðèêëàä, ðîáîòè [9, 10]).
Òàêîæ ïî÷àëè ðîçãëÿäàòè îïåðàòîðè áiëüø çàãàëüíîãî âèãëÿäó íiæ
(2.2). À ñàìå, ó ñòàòòi [8] äîñëiäæåíî îïåðàòîð âèãëÿäó
ψ
ΛDrf s :“ Λf ` ψ1
Bf
Bz1 ` ψ2
Bf
Bz2 ` ψ3
Bf
Bz3 ` ψ4
Bf
Bz4 , Λ P HpCq. (2.5)
Îçíà÷åííÿ 2.2. Ôóíêöiÿ f : Ω Ñ HpCq, Ω Ă C4, íàçèâà¹òüñÿ ëiâî�
Λ ´ ψ�ãiïåðãîëîìîðôíîþ, ÿêùî êîìïîíåíòè fs ¹ ãîëîìîðôíèìè ôóíê-
öiÿìè ÷îòèðüîõ êîìïëåêñíèõ çìiííèõ z1, z2, z3, z4 â Ω i f çàäîâîëüíÿ¹
ðiâíÿííÿ
ψ
ΛDrf spzq “ 0. (2.6)
 ðîáîòi [2] ðîçâèíóòî òåîðiþ òàê çâàíèõ pϕ, ψq�ãiïåðãîëîìîðôíèõ
ôóíêöié. Çàâäÿêè ìàòðè÷íîìó ïiäõîäó, äëÿ òàêèõ ôóíêöié óçàãàëüíåíî
ôîðìóëó Áîðåëÿ�Ïîìïåþ òà âñòàíîâëåíî ôîðìóëè Ïëåìåëÿ�Ñîõîöüêî-
ãî. Äîñëiäæåííÿ [2] áóëî ïðîäîâæåíî â ñòàòòÿõ [1, 18�20].
Ðàçîì ç òèì çàëèøà¹òüñÿ âiäêðèòîþ ïðîáëåìà ïðåäñòàâëåííÿ (àáî
îïèñó â ÿâíîìó âèãëÿäi) ψ�ãiïåðãîëîìîðôíèõ i ëiâî�Λ ´ ψ�ãiïåðãîëî-
ìîðôíèõ ôóíêöié. Öÿ ðîáîòà ïðèñâÿ÷åíà âèâ÷åííþ ñàìå öüîãî ïèòàííÿ.
2.1. Ïðèêëàäè. Ñïî÷àòêó ðîçãëÿíåìî ïðèêëàäè ëiâî� i ïðàâî�ψ�ãi-
ïåðãîëîìîðôíèõ ôóíêöié.
Ïðèêëàä 2.1. Ðîçãëÿíåìî îáëàñòü Ω Ă C2 » BC, çìiííó ζ “ z1e1`z2e2
i ôóíêöiþ f : Ω Ñ HpCq âèãëÿäó
f “
4ÿ
s“1
fspz1, z2qes , fs : Ω Ñ C.
Öå ñëiä ðîçóìiòè íàñòóïíèì ÷èíîì. Ìè îòîòîæíþ¹ìî C2 ç BC. Ïiñëÿ
öüîãî ìíîæèíà Ω â BC ñò๠ïiäìíîæèíîþ â HpCq, à íå â C2. Äàëi ìè
ðîçãëÿäà¹ìî äåÿêi îá'¹êòè ÿê òàêi, ùî çíàõîäÿòüñÿ â HpCq. Çîêðåìà,
ìíîæèíà Ω çíàõîäèòüñÿ â HpCq. Òîìó ìè ïðàöþ¹ìî ç ôóíêöiÿìè, ÿêi
âèçíà÷åíi i ïðèéìàþòü çíà÷åííÿ â HpCq. Òîáòî, ζ ìiñòèòüñÿ â îáëàñòi ç
HpCq: ìè âñå âêëàäà¹ìî â HpCq.
Âðàõîâóþ÷è òàêi ïðèïóùåííÿ, ââåäåìî íàñòóïíi îçíà÷åííÿ.
Ôóíêöiÿ f : Ω Ñ HpCq, Ω Ă BC, íàçèâà¹òüñÿ ïðàâî�BC�ãiïåðãîëî-
ìîðôíîþ, ÿêùî iñíó¹ åëåìåíòè àëãåáðè HpCq f 1
rpζq òàêèé, ùî
lim
εÑ0
fpζ ` εhq ´ fpζq
ε
“ h ¨ f 1
rpζq @h P BC. (2.7)
106 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
Ôóíêöiÿ f : Ω Ñ HpCq, Ω Ă BC, íàçèâà¹òüñÿ ëiâî�BC�ãiïåðãîëîìîðô-
íîþ, ÿêùî iñíó¹ åëåìåíò àëãåáðè HpCq f 1
l pζq òàêèé, ùî
lim
εÑ0
fpζ ` εhq ´ fpζq
ε
“ f 1
l pζq ¨ h @h P BC. (2.8)
Óìîâà (2.7) îçíà÷à¹, ùî
Bf
Bz1 “ e1f
1
rpζq ïðè h “ e1 (2.9)
i
Bf
Bz2 “ e2f
1
rpζq ïðè h “ e2. (2.10)
Ç ðiâíîñòåé (2.9) i (2.10) âèïëèâ๠àíàëîã óìîâ Êîøi�Ðiìàíà
e2
Bf
Bz1 “ e1
Bf
Bz2 . (2.11)
Àíàëîãi÷íî, ç ðiâíîñòi (2.8) âèïëèâà¹
Bf
Bz1 e2 “ Bf
Bz2 e1. (2.12)
Îòæå, ïðàâî� i ëiâî�BC�ãiïåðãîëîìîðôíà ôóíêöiÿ ¹ óçàãàëüíåííÿì
òåîði¨ ãîëîìîðôíèõ ôóíêöié â àëãåáði BC (äèâ., íàïðèêëàä, [3, 16]).
Ëåãêî ïîáà÷èòè, ùî ìíîæèíà ïðàâî�BC�ãiïåðãîëîìîðôíèõ ôóíêöié i
ëiâî�BC�ãiïåðãîëîìîðôíèõ ôóíêöié ¹ ïiäìíîæèíîþ ëiâî�ψ�ãiïåðãîëî-
ìîðôíèõ i ïðàâî�ψ�ãiïåðãîëîìîðôíèõ ôóíêöié, âiäïîâiäíî. Ñïðàâäi,
äëÿ ζ “ z1e1 ` z2e2 ðiâíiñòü (2.11) íàáóâ๠âèãëÿäó (2.4) ïðè ψ1 “ e2,
ψ2 “ ´e1 , ψ3 “ ψ4 “ 0. Àíàëîãi÷íî, ïðàâî�BC�ãiïåðãîëîìîðôíi ôóíê-
öi¨ ¹ ïiäìíîæèíîþ ïðàâî�ψ�ãiïåðãîëîìîðôíèõ ôóíêöié.
Ùå îäèí ïðèêëàä âiäîáðàæåíü ç îáëàñòi â R3 â àëãåáðó HpCq, ÿêèé ¹
÷àñòèííèì âèïàäêîì ëiâî� i ïðàâî�ψ�ãiïåðãîëîìîðôíèõ ôóíêöié, ðîç-
ãëÿíóòî â ðîáîòàõ [13,14].
 ðiâíîñòi (2.4) ïîêëàäåìî ψ1 “ 1, ψ2 “ I, ψ3 “ J, ψ4 “ K. Ó öüîìó
âèïàäêó
α1 “ α2 “ 1, α3 “ α4 “ 0, β1 “ ´i, β2 “ i, β3 “ β4 “ 0,
γ1 “ γ2 “ 0, γ3 “ ´i, γ4 “ ´i, δ1 “ δ2 “ 0, δ3 “ ´1, δ4 “ 1.
Òîäi ðiâíiñòü (2.4) íàáóâ๠âèãëÿäó
Bf
Bz1 ` I
Bf
Bz2 ` J
Bf
Bz3 `K
Bf
Bz4 “ 0
� äîáðå âiäîìå ðiâíÿííÿ òèïó Êîøi�Ôóåòåðà (äèâ., íàïðèêëàä, [6, 15]).
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 107
2.2. Îñíîâíà âëàñòèâiñòü ëiâî� i ïðàâî�ψ�ãiïåðãîëîìîðôíèõ
ôóíêöié.
Òåîðåìà 2.1. Íåõàé ôóíêöiÿ f ëiâî�ψ�ãiïåðãîëîìîðôíà (àáî ïðàâî�ψ�
ãiïåðãîëîìîðôíà) â äåÿêîìó áàçèñi àëãåáðè HpCq. Òîäi â iíøîìó áàçèñi
àëãåáðè HpCq iñíó¹ íàáið ôóíêöié Ψ :“ pΨ1,Ψ2,Ψ3,Ψ4q, Ψs P HpCq, s “
1, 2, 3, 4, òàêèõ, ùî ôóíêöiÿ f ¹ ëiâî�Ψ�ãiïåðãîëîìîðôíîþ (àáî ïðàâî�
Ψ�ãiïåðãîëîìîðôíîþ).
Äîâåäåííÿ. Äîâåäåìî öþ òåîðåìó ó âèïàäêó ëiâî�ψ�ãiïåðãîëîìîðôíèõ
ôóíêöié. Íåõàé te1, e2, e3, e4u � áàçèñ Êàðòàíà â HpCq i ti1, i2, i3, i4u �
iíøèé áàçèñ â HpCq. Öå îçíà÷à¹, ùî
e1 “ k1i1 ` k2i2 ` k3i3 ` k4i4,
e2 “ m1i1 `m2i2 `m3i3 `m4i4,
e3 “ n1i1 ` n2i2 ` n3i3 ` n4i4,
e4 “ r1i1 ` r2i2 ` r3i3 ` r4i4,
äå ki,mi, ni, ri ïðè i “ 1, 2, 3, 4, � êîìïëåêñíi ÷èñëà.
Ðîçãëÿíåìî ðiâíiñòü
ψDrf sptq :“ ψ1
Bf
Bt1 ` ψ2
Bf
Bt2 ` ψ3
Bf
Bt3 ` ψ4
Bf
Bt4 “ 0, (2.13)
äå t :“ t1e1 ` t2e2 ` t3e3 ` t4e4, t1, t2, t3, t4 P C. Ó çìiííié t ïåðåéäåìî
äî áàçèñó ti1, i2, i3, i4u. Òîäi
t “ i1pt1k1 ` t2m1 ` t3n1 ` t4r1q ` i2pt1k2 ` t2m2 ` t3n2 ` t4r2q
`i3pt1k3 ` t2m3 ` t3n3 ` t4r3q ` i4pt1k4 ` t2m4 ` t3n4 ` t4r4q.
Ïîêëàäåìî
z1 :“ t1k1 ` t2m1 ` t3n1 ` t4r1,
z2 :“ t1k2 ` t2m2 ` t3n2 ` t4r2,
z3 :“ t1k3 ` t2m3 ` t3n3 ` t4r3,
z4 :“ t1k4 ` t2m4 ` t3n4 ` t4r4.
(2.14)
Ç ðiâíîñòåé (2.14) îòðèìà¹ìî
Bf
Bt1 “ k1
Bf
Bz1 ` k2
Bf
Bz2 ` k3
Bf
Bz3 ` k4
Bf
Bz4 ,
Bf
Bt2 “ m1
Bf
Bz1 `m2
Bf
Bz2 `m3
Bf
Bz3 `m4
Bf
Bz4 ,
Bf
Bt3 “ n1
Bf
Bz1 ` n2
Bf
Bz2 ` n3
Bf
Bz3 ` n4
Bf
Bz4 ,
Bf
Bt4 “ r1
Bf
Bz1 ` r2
Bf
Bz2 ` r3
Bf
Bz3 ` r4
Bf
Bz4 .
108 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
Òîäi ðiâíiñòü (2.13) ðiâíîñèëüíà íàñòóïíié ðiâíîñòi:
ψDrf sptq “ pψ1k1`ψ2m1`ψ3n1`ψ4r1q Bf
Bz1 `pψ1k2`ψ2m2`ψ3n2`ψ4r2q Bf
Bz2
`pψ1k3`ψ2m3`ψ3n3`ψ4r3q Bf
Bz3 `pψ1k4`ψ2m4`ψ3n4`ψ4r4q Bf
Bz4 . (2.15)
Âèêîðèñòîâóþ÷è ïîçíà÷åííÿ (2.1), ìà¹ìî:
ψ1 “
4ÿ
s“1
αses “
4ÿ
s“1
ispα1ks ` α2ms ` α3ns ` α4rsq,
ψ2 “
4ÿ
s“1
βses “
4ÿ
s“1
ispβ1ks ` β2ms ` β3ns ` β4rsq,
ψ3 “
4ÿ
s“1
γses “
4ÿ
s“1
ispγ1ks ` γ2ms ` γ3ns ` γ4rsq,
ψ4 “
4ÿ
s“1
δses “
4ÿ
s“1
ispδ1ks ` δ2ms ` δ3ns ` δ4rsq,
Ç (2.15) îòðèìà¹ìî
ψDrf sptq “
4ÿ
s“1
is
”
pα1ks`α2ms`α3ns`α4rsqk1`pβ1ks`β2ms`β3ns`β4rsqm1
`pγ1ks ` γ2ms ` γ3ns ` γ4rsqn1 ` pδ1ks ` δ2ms ` δ3ns ` δ4rsqr1
ı Bf
Bz1
`
4ÿ
s“1
is
”
pα1ks ` α2ms ` α3ns ` α4rsqk2 ` pβ1ks ` β2ms ` β3ns ` β4rsqm2
`pγ1ks ` γ2ms ` γ3ns ` γ4rsqn2 ` pδ1ks ` δ2ms ` δ3ns ` δ4rsqr2
ı Bf
Bz2
`
4ÿ
s“1
is
”
pα1ks ` α2ms ` α3ns ` α4rsqk3 ` pβ1ks ` β2ms ` β3ns ` β4rsqm3
`pγ1ks ` γ2ms ` γ3ns ` γ4rsqn3 ` pδ1ks ` δ2ms ` δ3ns ` δ4rsqr3
ı Bf
Bz3
`
4ÿ
s“1
is
”
pα1ks ` α2ms ` α3ns ` α4rsqk4 ` pβ1ks ` β2ms ` β3ns ` β4rsqm4
`pγ1ks ` γ2ms ` γ3ns ` γ4rsqn4 ` pδ1ks ` δ2ms ` δ3ns ` δ4rsqr4
ı Bf
Bz4
“: Ψ1
Bf
Bz1 ` Ψ2
Bf
Bz2 ` Ψ3
Bf
Bz3 ` Ψ4
Bf
Bz4 “ 0.
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 109
□
Çàóâàæåííÿ 2.1. Ç öi¹¨ òåîðåìè âèïëèâà¹, ùî íàäàëi äîñòàòíüî ðîç-
ãëÿäàòè êîíñòàíòè ψ i ôóíêöiþ f â íàéïðîñòiøîìó áàçèñi, òîáòî â áàçèñi
Êàðòàíà.
Çàóâàæåííÿ 2.2. Âiäîìî, ùî â àëãåáðàõ Êëiôôîðäà ðiâíîñòi
Bf
Bt0 ` I
Bf
Bt1 ` J
Bf
Bt2 `K
Bf
Bt3 “ 0
i ψDrf sptq “ 0 ñïiâïàäàþòü ç òî÷íiñòþ äî îðòîãîíàëüíîãî ïåðåòâîðåííÿ.
Çàçíà÷èìî, ùî òåîðåìà 2.1 ¹ ñàìå öèì òâåðäæåííÿì, àëå ñôîðìóëüî-
âàíå â iíøèõ òåðìiíàõ.
3. Çàñòîñóâàííÿ äî ðîçâ'ÿçàííÿ ðiâíÿííÿ òèïó
Êîøi�Ôóåòåðà
Òåïåð âñòàíîâèìî çâ'ÿçîê ìiæ ðîçâ'ÿçêàìè ðiâíÿííÿ
Drf sptq :“ Bf
Bt0 ` I
Bf
Bt1 ` J
Bf
Bt2 `K
Bf
Bt3 “ 0, (3.1)
äå t :“ t0 ` t1I` t2J` t3K, t0, t1, t2, t3 P C, i ðîçâ'ÿçêàìè ðiâíÿííÿ (2.4).
Ç öi¹þ ìåòîþ â çìiííié t ïåðåéäåìî äî áàçèñó Êàðòàíà. Òîäi
t “ t0pe1 ` e2q ` t1p´ie1 ` ie2q ` t2p´ie3 ´ ie4q ` t3pe4 ´ e3q
“ pt0 ´ it1qe1 ` pt0 ` it1qe2 ` p´it2 ´ t3qe3 ` p´it2 ` t3qe4.
Ïîçíà÷èìî ÷åðåç
z1 :“ t0 ´ it1, z2 :“ t0 ` it1, z3 :“ ´it2 ´ t3, z4 :“ ´it2 ` t3. (3.2)
Ç ðiâíîñòåé (3.2) îòðèìà¹ìî
Bf
Bt0 “ Bf
Bz1 ` Bf
Bz2 ,
Bf
Bt1 “ ´i Bf
Bz1 ` i
Bf
Bz2 ,
Bf
Bt2 “ ´i Bf
Bz3 ´ i
Bf
Bz4 ,
Bf
Bt3 “ ´ Bf
Bz3 ` Bf
Bz4 .
Òîäi ðiâíÿííÿ (3.1) ðiâíîñèëüíå ðiâíÿííþ
Drf s “ Bf
Bz1 ` Bf
Bz2 ´ iI
Bf
Bz1 ` iI
Bf
Bz2 ´ iJ
Bf
Bz3 ´ iJ
Bf
Bz4 ´K
Bf
Bz3 `K
Bf
Bz4
“ p1 ´ iIq Bf
Bz1 ` p1 ` iIq Bf
Bz2 ` p´iJ ´Kq Bf
Bz3 ` p´iJ `Kq Bf
Bz4
“ 2
ˆ
e2
Bf
Bz1 ` e1
Bf
Bz2 ´ e4
Bf
Bz3 ´ e3
Bf
Bz4
˙
“ 0.
Òàêèì ÷èíîì äîâåäåíî òåîðåìó.
110 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
Òåîðåìà 3.1. Ôóíêöiÿ f çìiííî¨ t “ t0 ` t1I ` t2J ` t3K çàäîâîëüíÿ¹
ðiâíÿííÿ (3.1) òîäi i òiëüêè òîäi, êîëè ôóíêöiÿ f çìiííî¨ z “ z1e1 `
z2e2 ` z3e3 ` z4e4 çàäîâîëüíÿ¹ ðiâíÿííÿ
e2
Bf
Bz1 ` e1
Bf
Bz2 ´ e4
Bf
Bz3 ´ e3
Bf
Bz4 “ 0, (3.3)
äå z i t ïîâ'ÿçàíi ñïiââiäíîøåííÿìè (3.2).
Òåïåð ðîçâ'ÿæåìî ðiâíÿííÿ (3.3).
e2
Bf
Bz1 “ e2
ˆBf1
Bz1 e1 ` Bf2
Bz1 e2 ` Bf3
Bz1 e3 ` Bf4
Bz1 e4
˙
“ Bf2
Bz1 e2 ` Bf4
Bz1 e4 ,
e1
Bf
Bz2 “ Bf1
Bz2 e1 ` Bf3
Bz2 e3 ,
e4
Bf
Bz3 “ Bf1
Bz3 e4 ` Bf3
Bz3 e2 ,
e3
Bf
Bz4 “ Bf2
Bz4 e3 ` Bf4
Bz4 e1 .
Ðiâíÿííÿ (3.3) ðiâíîñèëüíå ñèñòåìi ðiâíÿíü
Bf1
Bz2 “ Bf4
Bz4 ,
Bf2
Bz1 “ Bf3
Bz3 ,
Bf3
Bz2 “ Bf2
Bz4 ,
Bf4
Bz1 “ Bf1
Bz3 .
Ìà¹ìî äâi íåçàëåæíi ñèñòåìè
Bf1
Bz2 “ Bf4
Bz4 ,
Bf1
Bz3 “ Bf4
Bz1 (3.4)
i Bf2
Bz1 “ Bf3
Bz3 ,
Bf2
Bz4 “ Bf3
Bz2 . (3.5)
Ðîçâ'ÿçêîì ñèñòåìè (3.4) â îäíîçâ'ÿçíié îáëàñòi Ω ¹ äîâiëüíà ãîëî-
ìîðôíà ôóíêöiÿ
f1 “ f1pz2, z3q
i
f4 “ z4
Bf1
Bz2 ` z1
Bf1
Bz3 .
Ðîçâ'ÿçêîì ñèñòåìè (3.5) â îäíîçâ'ÿçíié îáëàñòi Ω ¹ äîâiëüíà ãîëî-
ìîðôíà ôóíêöiÿ
f2 “ f2pz1, z4q
i
f3 “ z3
Bf2
Bz1 ` z2
Bf2
Bz4 .
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 111
Îòæå, ìà¹ìî ðîçâ'ÿçîê ðiâíÿííÿ (3.3):
fpzq “ f1pz2, z3qe1 ` f2pz1, z4qe2
`
ˆ
z3
Bf2
Bz1 ` z2
Bf2
Bz4
˙
e3 `
ˆ
z4
Bf1
Bz2 ` z1
Bf1
Bz3
˙
e4 . (3.6)
Òàêèì ÷èíîì, çãiäíî ç òåîðåìîþ 3.1 ìè îòðèìàëè
Òåîðåìà 3.2. Â îäíîçâ'ÿçíié îáëàñòi ôóíêöiÿ (3.6), äå z1, z2, z3, z4 çà-
äàíi ñïiââiäíîøåííÿìè (3.2), çàäîâîëüíÿ¹ ðiâíiñòü (3.1).
Òâåðäæåííÿ 3.1. Â îäíîçâ'ÿçíié îáëàñòi ôóíêöiÿ (3.6) çàäîâîëüíÿ¹
÷îòèðèâèìiðíå êîìïëåêñíå ðiâíÿííÿ Ëàïëàñà
∆C4f :“ B2f
Bt21
` B2f
Bt22
` B2f
Bt23
` B2f
Bt24
“ 0. (3.7)
Ïðî ðiâíÿííÿ (3.7) òà éîãî çâ'ÿçîê ç ðiâíÿííÿì Êîøi-Ôóåòåðà äèâ.
ó [15].
4. Ïðåäñòàâëåííÿ ëiâî�ψ�ãiïåðãîëîìîðôíèõ ôóíêöié ó
ñïåöiàëüíîìó âèïàäêó
Çíàéäåìî çàãàëüíèé ðîçâ'ÿçîê ðiâíÿííÿ (2.4) äëÿ ñïåöiàëüíîãî âèáî-
ðó ïàðàìåòðiâ ψ1, ψ2, ψ3 i ψ4. Ç öi¹þ ìåòîþ, ïåðåòâîðèìî ðiâíÿííÿ (2.4)
â ñèñòåìó ÷îòèðüîõ äèôåðåíöiàëüíèõ ðiâíÿíü â ÷àñòèííèõ ïîõiäíèõ. Òå-
ïåð ìà¹ìî
ψ1
Bf
Bz1 “ pα1e1 ` α2e2 ` α3e3 ` α4e4q
ˆBf1
Bz1 e1 ` Bf2
Bz1 e2 ` Bf3
Bz1 e3 ` Bf4
Bz1 e4
˙
“ Bf1
Bz1α1e1 ` Bf3
Bz1α1e3 ` Bf2
Bz1α2e2 ` Bf4
Bz1α2e4
`Bf2
Bz1α3e3 ` Bf4
Bz1α3e1 ` Bf1
Bz1α4e4 ` Bf3
Bz1α4e2
“ B
Bz1 pα1f1 ` α3f4q e1 ` B
Bz1 pα2f2 ` α4f3q e2
` B
Bz1 pα1f3 ` α3f2q e3 ` B
Bz1 pα2f4 ` α4f1q e4.
Àíàëîãi÷íî,
ψ2
Bf
Bz2 “ B
Bz2 pβ1f1 ` β3f4q e1 ` B
Bz2 pβ2f2 ` β4f3q e2
` B
Bz2 pβ1f3 ` β3f2q e3 ` B
Bz2 pβ2f4 ` β4f1q e4,
112 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
ψ3
Bf
Bz3 “ B
Bz3 pγ1f1 ` γ3f4q e1 ` B
Bz3 pγ2f2 ` γ4f3q e2
` B
Bz3 pγ1f3 ` γ3f2q e3 ` B
Bz3 pγ2f4 ` γ4f1q e4,
ψ4
Bf
Bz4 “ B
Bz4 pδ1f1 ` δ3f4q e1 ` B
Bz4 pδ2f2 ` δ4f3q e2
` B
Bz4 pδ1f3 ` δ3f2q e3 ` B
Bz4 pδ2f4 ` δ4f1q e4.
Òîäi ðiâíÿííÿ (2.4) ðiâíîñèëüíå ñèñòåìi
B
Bz1 pα1f1`α3f4q` B
Bz2 pβ1f1`β3f4q` B
Bz3 pγ1f1`γ3f4q` B
Bz4 pδ1f1`δ3f4q “ 0,
B
Bz1 pα2f2`α4f3q` B
Bz2 pβ2f2`β4f3q` B
Bz3 pγ2f2`γ4f3q` B
Bz4 pδ2f2`δ4f3q “ 0,
(4.1)
B
Bz1 pα1f3`α3f2q` B
Bz2 pβ1f3`β3f2q` B
Bz3 pγ1f3`γ3f2q` B
Bz4 pδ1f3`δ3f2q “ 0,
B
Bz1 pα2f4`α4f1q` B
Bz2 pβ2f4`β4f1q` B
Bz3 pγ2f4`γ4f1q` B
Bz4 pδ2f4`δ4f1q “ 0.
Òåîðåìà 4.1. Íåõàé
ψ1 “ α1e1 ` α2e2 ` α3e3 ` α4e4, α1α2 ‰ α3α4 ,
ψ2 “ λα1e1 ` µα2e2 ` µα3e3 ` λα4e4,
ψ3 “ θα1e1 ` ϑα2e2 ` ϑα3e3 ` θα4e4,
ψ4 “ να1e1 ` ηα2e2 ` ηα3e3 ` να4e4,
(4.2)
äå α1, α2, α3, α4, λ, µ, θ, ϑ, ν, η � äîâiëüíi êîìïëåêñíi ÷èñëà. Òîäi êîæíà
ëiâî�ψ�ãiïåðãîëîìîðôíà ôóíêöiÿ íàáóâ๠âèãëÿäó
fpzq “ f1prζ2, rζ3, rζ4qe1 ` f2pζ2, ζ3, ζ4qe2 ` f3prζ2, rζ3, rζ4qe3 ` f4pζ2, ζ3, ζ4qe4,
(4.3)
äå
rζ2 :“ λz1 ´ z2, rζ3 :“ θz1 ´ z3, rζ4 :“ νz1 ´ z4,
ζ2 :“ µz1 ´ z2, ζ3 :“ ϑz1 ´ z3, ζ4 :“ ηz1 ´ z4,
(4.4)
i f1, f2, f3, f4 � äîâiëüíi ãîëîìîðôíi ôóíêöi¨ ñâî¨õ òðüîõ àðãóìåíòiâ.
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 113
Äîâåäåííÿ. Äëÿ âèáðàíèõ ïàðàìåòðiâ (4.2) ïåðøå ðiâíÿííÿ ñèñòåìè (4.1)
íàáóâ๠âèãëÿäó
B
Bz1 pα1f1 ` α3f4q ` B
Bz2 pλα1f1 ` µα3f4q
` B
Bz3 pθα1f1 ` ϑα3f4q ` B
Bz4 pνα1f1 ` ηα3f4q “ 0. (4.5)
Àíàëîãi÷íî, äëÿ âèáðàíèõ ïàðàìåòðiâ (4.2) ÷åòâåðòå ðiâíÿííÿ ñèñòå-
ìè (4.1) íàáóâ๠âèãëÿäó
B
Bz1 pα4f1 ` α2f4q ` B
Bz2 pλα4f1 ` µα2f4q
` B
Bz3 pθα4f1 ` ϑα2f4q ` B
Bz4 pνα4f1 ` ηα2f4q “ 0. (4.6)
Ðîçãëÿíåìî ðiçíèöþ ìiæ ðiâíÿííÿì (4.5), äîìíîæåíèì íà α2, i ðiâ-
íÿííÿì (4.6), äîìíîæåíèì íà α3. Òîäi îòðèìà¹ìî ðiâíiñòü
B
Bz1
´
f1pα1α2 ´ α3α4q ` f4pα2α3 ´ α2α3q
¯
` B
Bz2
´
f1pλα1α2 ´ λα3α4q ` f4pµα2α3 ´ µα2α3q
¯
` B
Bz3
´
f1pθα1α2 ´ θα3α4q ` f4pϑα2α3 ´ ϑα2α3q
¯
` B
Bz4
´
f1pνα1α2 ´ να3α4q ` f4pηα2α3 ´ ηα2α3q
¯
“ 0.
Îòæå, ìà¹ìî
Bf1
Bz1 ` λ
Bf1
Bz2 ` θ
Bf1
Bz3 ` ν
Bf1
Bz4 “ 0. (4.7)
Äëÿ ðiâíÿííÿ (4.7) ðîçãëÿíåìî õàðàêòåðèñòè÷íå ðiâíÿííÿ
dz1
1
“ dz2
λ
“ dz3
θ
“ dz4
ν
. (4.8)
Ðîçâ'ÿçêàìè ñèñòåìè (4.8) ¹ iíòåãðàëè
c2 “ λz1 ´ z2, c3 “ θz1 ´ z3, c4 “ νz1 ´ z4.
Îòæå, çàãàëüíèé ðîçâ'ÿçîê ðiâíÿííÿ (4.7) ì๠âèãëÿä
f1 “ f1prζ2, rζ3, rζ4q,
äå rζ2, rζ3, rζ4 âèçíà÷åíi ðiâíîñòÿìè (4.4).
Âiäìiòèìî, ùî ïîëiíîìè (4.4) ¹ àíàëîãàìè äîáðå âiäîìèõ ïîëiíîìiâ
Ôóåòåðà [4].
Àíàëîãi÷íî ìîæíà îòðèìàòè ïðåäñòàâëåííÿ äëÿ êîìïîíåíò f2, f3, f4.
□
114 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
Îòæå, ôîðìóëà (4.3) ä๠ïðåäñòàâëåííÿ ëiâî�ψ�ãiïåðãîëîìîðôíî¨ ôóíê-
öi¨ çà óìîâè ñïåöiàëüíîãî âèáîðó ψ.
Çàóâàæåííÿ 4.1. Âèêîðèñòîâóþ÷è ôîðìóëè (1.4), ìîæíà çàïèñàòè ïðåä-
ñòàâëåííÿ (4.3) â áàçèñi Ïàóëi:
fpzq “
´
f1prζ2, rζ3, rζ4q ` f2pζ2, ζ3, ζ4q
¯
σ0`
´
if3prζ2, rζ3, rζ4q ´ if4pζ2, ζ3, ζ4q
¯
σ1
`
´
´f3prζ2, rζ3, rζ4q ´ f4pζ2, ζ3, ζ4q
¯
σ2 `
´
f2pζ2, ζ3, ζ4q ´ f1prζ2, rζ3, rζ4q
¯
σ3 .
5. Ïðåäñòàâëåííÿ ïðàâî�ψ�ãiïåðãîëîìîðôíèõ ôóíêöié ó
ñïåöiàëüíîìó âèïàäêó
 öüîìó ðîçäiëi áóäåìî øóêàòè çàãàëüíèé ðîçâ'ÿçîê ðiâíÿííÿ
Dψrf spzq “ Bf
Bz1 ψ1 ` Bf
Bz2 ψ2 ` Bf
Bz3 ψ3 ` Bf
Bz4 ψ4 “ 0 (5.1)
ïðè ñïåöiàëüíîìó âèáîði ïàðàìåòðiâ ψ1, ψ2, ψ3 i ψ4. Ç öi¹þ ìåòîþ ïåðå-
òâîðèìî ðiâíÿííÿ (5.1) â ñèñòåìó ÷îòèðüîõ äèôåðåíöiàëüíèõ ðiâíÿíü â
÷àñòèííèõ ïîõiäíèõ. Ìà¹ìî
Bf
Bz1 ψ1 “
ˆBf1
Bz1 e1 ` Bf2
Bz1 e2 ` Bf3
Bz1 e3 ` Bf4
Bz1 e4
˙
pα1e1 ` α2e2 ` α3e3 ` α4e4q
“ Bf1
Bz1α1e1 ` Bf1
Bz1α3e3 ` Bf2
Bz1α2e2 ` Bf2
Bz1α4e4
`Bf3
Bz1α2e3 ` Bf3
Bz1α4e1 ` Bf4
Bz1α1e4 ` Bf4
Bz1α3e2
“ B
Bz1 pα1f1 ` α4f3q e1 ` B
Bz1 pα2f2 ` α3f4q e2
` B
Bz1 pα3f1 ` α2f3q e3 ` B
Bz1 pα4f2 ` α1f4q e4.
Àíàëîãi÷íî,
Bf
Bz2 ψ2 “ B
Bz2 pβ1f1 ` β4f3q e1 ` B
Bz2 pβ2f2 ` β3f4q e2
` B
Bz2 pβ3f1 ` β2f3q e3 ` B
Bz2 pβ4f2 ` β1f4q e4,
Bf
Bz3 ψ3 “ B
Bz3 pγ1f1 ` γ4f3q e1 ` B
Bz3 pγ2f2 ` γ3f4q e2
` B
Bz3 pγ3f1 ` γ2f3q e3 ` B
Bz3 pγ4f2 ` γ1f4q e4,
Bf
Bz4 ψ4 “ B
Bz4 pδ1f1 ` δ4f3q e1 ` B
Bz4 pδ2f2 ` δ3f4q e2
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 115
` B
Bz4 pδ3f1 ` δ2f3q e3 ` B
Bz4 pδ4f2 ` δ1f4q e4.
Òîäi ðiâíÿííÿ (5.1) ðiâíîñèëüíå ñèñòåìi
B
Bz1 pα1f1`α4f3q` B
Bz2 pβ1f1`β4f3q` B
Bz3 pγ1f1`γ4f3q` B
Bz4 pδ1f1`δ4f3q “ 0,
B
Bz1 pα2f2`α3f4q` B
Bz2 pβ2f2`β3f4q` B
Bz3 pγ2f2`γ3f4q` B
Bz4 pδ2f2`δ3f4q “ 0,
(5.2)
B
Bz1 pα3f1`α2f3q` B
Bz2 pβ3f1`β2f3q` B
Bz3 pγ3f1`γ2f3q` B
Bz4 pδ3f1`δ2f3q “ 0,
B
Bz1 pα4f2`α1f4q` B
Bz2 pβ4f2`β1f4q` B
Bz3 pγ4f2`γ1f4q` B
Bz4 pδ4f2`δ1f4q “ 0.
Òåîðåìà 5.1. Íåõàé
ψ1 “ α1e1 ` α2e2 ` α3e3 ` α4e4, α1α2 ‰ α3α4 ,
ψ2 “ µα1e1 ` λα2e2 ` µα3e3 ` λα4e4,
ψ3 “ ϑα1e1 ` θα2e2 ` ϑα3e3 ` θα4e4,
ψ4 “ ηα1e1 ` να2e2 ` ηα3e3 ` να4e4,
(5.3)
äå α1, α2, α3, α4, λ, µ, θ, ϑ, ν, η � äîâiëüíi êîìïëåêñíi ÷èñëà. Òîäi êîæíà
ïðàâî�ψ�ãiïåðãîëîìîðôíà ôóíêöiÿ íàáóâ๠âèãëÿäó
fpzq “ f1pζ2, ζ3, ζ4qe1 ` f2prζ2, rζ3, rζ4qe2 ` f3prζ2, rζ3, rζ4qe3 ` f4pζ2, ζ3, ζ4qe4,
(5.4)
äå ζ2, ζ3, ζ4, rζ2, rζ3, rζ4 âèçíà÷åíi ðiâíîñòÿìè (4.4) i f1, f2, f3, f4 � äîâiëüíi
ãîëîìîðôíi ôóíêöi¨ òðüîõ âiäïîâiäíèõ àðãóìåíòiâ.
Äîâåäåííÿ. Äëÿ âèáðàíèõ ïàðàìåòðiâ (5.3) ïåðøå ðiâíÿííÿ ñèñòåìè (5.2)
íàáóâ๠âèãëÿäó
B
Bz1 pα1f1 ` α4f3q ` B
Bz2 pµα1f1 ` λα4f3q `
` B
Bz3 pϑα1f1 ` θα4f3q ` B
Bz4 pηα1f1 ` να4f3q “ 0. (5.5)
Àíàëîãi÷íî, äëÿ âèáðàíèõ ïàðàìåòðiâ (5.3) òðåò¹ ðiâíÿííÿ ñèñòåìè
(5.2) ì๠âèãëÿä
B
Bz1 pα3f1 ` α2f3q ` B
Bz2 pµα3f1 ` λα2f3q
` B
Bz3 pϑα3f1 ` θα2f3q ` B
Bz4 pηα3f1 ` να2f3q “ 0. (5.6)
116 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
Ðîçãëÿíåìî ðiçíèöþ ìiæ ðiâíÿííÿì (5.5), äîìíîæåíèì íà α2 i ðiâíÿí-
íÿì (5.6), äîìíîæåíèì íà α4. Òîäi îòðèìà¹ìî íàñòóïíó ðiâíiñòü
B
Bz1
´
f1pα1α2 ´ α3α4q ` f3pα2α4 ´ α2α4q
¯
` B
Bz2
´
µf1pα1α2 ´ α3α4q ` λf3pα2α4 ´ α2α4q
¯
` B
Bz3
´
ϑf1pα1α2 ´ α3α4q ` θf3pα2α4 ´ α2α4q
¯
` B
Bz4
´
ηf1pα1α2 ´ α3α4q ` νf3pα2α3 ´ α2α3q
¯
“ 0.
Îòæå, ìà¹ìî ðiâíÿííÿ
Bf1
Bz1 ` µ
Bf1
Bz2 ` ϑ
Bf1
Bz3 ` η
Bf1
Bz4 “ 0. (5.7)
Äëÿ ðiâíÿííÿ (5.7) ðîçãëÿíåìî õàðàêòåðèñòè÷íå ðiâíÿííÿ
dz1
1
“ dz2
µ
“ dz3
ϑ
“ dz4
η
. (5.8)
Ðîçâ'ÿçêàìè ñèñòåìè (5.8) ¹ iíòåãðàëè
c2 “ µz1 ´ z2, c3 “ ϑz1 ´ z3, c4 “ ηz1 ´ z4.
Òàêèì ÷èíîì, çàãàëüíèé ðîçâ'ÿçîê ðiâíÿííÿ (5.7) íàáóâ๠âèãëÿäó
f1 “ f1pζ2, ζ3, ζ4q,
äå ζ2, ζ3, ζ4 âèçíà÷åíi ðiâíîñòÿìè (4.4).
Àíàëîãi÷íî, îòðèìó¹ìî ðîçêëàäè êîìïîíåíò f2, f3, f4. □
Îòæå, ôîðìóëà (5.4) ä๠ïðåäñòàâëåííÿ ïðàâî�ψ�ãiïåðãîëîìîðôíî¨
ôóíêöi¨ ïðè ñïåöiàëüíîìó âèáîði ψ.
Ïîðiâíþþ÷è ïðåäñòàâëåííÿ (4.3) i (5.4), îòðèìà¹ìî íàñòóïíå òâåð-
äæåííÿ.
Òâåðäæåííÿ 5.1. Íåõàé âèêîíóþòüñÿ óìîâè òåîðåìè 4.1. Òîäi ôóíê-
öiÿ f ¹ îäíî÷àñíî ëiâî� i ïðàâî�ψ�ãiïåðãîëîìîðôíîþ, ÿêùî âîíà ïðè-
éì๠çíà÷åííÿ íà ìíîæèíi th3e3 ` h4e4 : h3, h4 P Cu.
Çàóâàæåííÿ 5.1. Âèêîðèñòîâóþ÷è ôîðìóëè (1.4) ìîæåìî çàïèñàòè
ïðåäñòàâëåííÿ (5.4) â áàçèñi Ïàóëi:
fpzq “
´
f1pζ2, ζ3, ζ4q ` f2prζ2, rζ3, rζ4q
¯
σ0`
´
if3prζ2, rζ3, rζ4q ´ if4pζ2, ζ3, ζ4q
¯
σ1
`
´
´f3prζ2, rζ3, rζ4q ´ f4pζ2, ζ3, ζ4q
¯
σ2 `
´
f2prζ2, rζ3, rζ4q ´ f1pζ2, ζ3, ζ4q
¯
σ3 .
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 117
6. Ïðåäñòàâëåííÿ ëiâî�Λ ´ ψ�ãiïåðãîëîìîðôíèõ ôóíêöié ó
ñïåöiàëüíîìó âèïàäêó
Ó öüîìó ðîçäiëi ðîçãëÿíåìî îïåðàòîð (2.5) i ðiâíÿííÿ (2.6). Òåïåð áó-
äåìî øóêàòè ïðåäñòàâëåííÿ ëiâî�Λ´ψ�ãiïåðãîëîìîðôíèõ ôóíêöié ïðè
ñïåöiàëüíîìó âèáîði ïàðàìåòðiâ ψ1, ψ2, ψ3, ψ4 i Λ. Ç öi¹þ ìåòîþ ïåðå-
òâîðèìî ðiâíÿííÿ (2.6) â ñèñòåìó ÷îòèðüîõ äèôåðåíöiàëüíèõ ðiâíÿíü â
÷àñòèííèõ ïîõiäíèõ. Ïîçíà÷èìî ÷åðåç
Λ :“ Λ1e1 ` Λ2e2 ` Λ3e3 ` Λ4e4. (6.1)
Âèêîðèñòîâóþ÷è ðåçóëüòàòè ðîçäiëó 4, îòðèìà¹ìî ðiâíÿííÿ (2.6) ðiâ-
íîñèëüíå íåîäíîðiäíié ñèñòåìè äèôåðåíöiàëüíèõ ðiâíÿíü â ÷àñòèííèõ
ïîõiäíèõ
B
Bz1 pα1f1 ` α3f4q ` B
Bz2 pβ1f1 ` β3f4q ` B
Bz3 pγ1f1 ` γ3f4q ` B
Bz4 pδ1f1 ` δ3f4q
“ ´Λ1f1 ´ Λ3f4
B
Bz1 pα2f2 ` α4f3q ` B
Bz2 pβ2f2 ` β4f3q ` B
Bz3 pγ2f2 ` γ4f3q ` B
Bz4 pδ2f2 ` δ4f3q
“ ´Λ2f2 ´ Λ4f3
B
Bz1 pα1f3 ` α3f2q ` B
Bz2 pβ1f3 ` β3f2q ` B
Bz3 pγ1f3 ` γ3f2q ` B
Bz4 pδ1f3 ` δ3f2q
“ ´Λ1f3 ´ Λ3f2
B
Bz1 pα2f4 ` α4f1q ` B
Bz2 pβ2f4 ` β4f1q ` B
Bz3 pγ2f4 ` γ4f1q ` B
Bz4 pδ2f4 ` δ4f1q
“ ´Λ4f1 ´ Λ2f4
(6.2)
Òåîðåìà 6.1. Íåõàé
ψ1 “ α1e1 ` α2e2 ` α3e3 ` α4e4, α1α2 ‰ α3α4 ,
ψ2 “ λα1e1 ` µα2e2 ` µα3e3 ` λα4e4,
ψ3 “ θα1e1 ` ϑα2e2 ` ϑα3e3 ` θα4e4,
ψ4 “ να1e1 ` ηα2e2 ` ηα3e3 ` να4e4,
(6.3)
äå α1, α2, α3, α4, λ, µ, θ, ϑ, ν, η � äîâiëüíi êîìïëåêñíi ÷èñëà. Êðiì òîãî,
íåõàé Λ ì๠âèãëÿä (6.1) òàêèé, ùî
α2Λ3 “ α3Λ2 , α1Λ4 “ α4Λ1 . (6.4)
Òîäi êîæíà ëiâî�Λ ´ ψ�ãiïåðãîëîìîôðíà ôóíêöiÿ íàáóâ๠âèãëÿäó
fpzq “ e1Φ1prζ2, rζ3, rζ4q ¨ exp
´α3Λ4 ´ α2Λ1
α1α2 ´ α3α4
z1
¯
`e2Φ2pζ2, ζ3, ζ4q ¨ exp
´α4Λ3 ´ α1Λ2
α1α2 ´ α3α4
z1
¯
(6.5)
`e3Φ3prζ2, rζ3, rζ4q ¨ exp
´α3Λ4 ´ α2Λ1
α1α2 ´ α3α4
z1
¯
118 Ò. Ñ. Êóçüìåíêî, Â. Ñ. Øïàêiâñüêèé
`e4Φ4pζ2, ζ3, ζ4q ¨ exp
´α4Λ3 ´ α1Λ2
α1α2 ´ α3α4
z1
¯
,
äå rζ2, rζ3, rζ4, ζ2, ζ3, ζ4 âèçíà÷åíi ñïiââiäíîøåííÿìè (4.4), à Φ1,Φ2,Φ3,Φ4 �
äîâiëüíi ãîëîìîðôíi ôóíêöi¨ òðüîõ êîìïëåêñíèõ çìiííèõ.
Äîâåäåííÿ. Äëÿ çàäàíèõ ïàðàìåòðiâ (6.3) ïåðøå ðiâíÿííÿ ñèñòåìè (6.2)
íàáóâ๠âèãëÿäó
B
Bz1 pα1f1 ` α3f4q ` B
Bz2 pλα1f1 ` µα3f4q `
` B
Bz3 pθα1f1 ` ϑα3f4q ` B
Bz4 pνα1f1 ` ηα3f4q “ ´Λ1f1 ´ Λ3f4. (6.6)
Àíàëîãi÷íî, äëÿ ïàðàìåòðiâ (6.3) ÷åòâåðòå ðiâíÿííÿ ñèñòåìè (6.2) íà-
áóâ๠âèãëÿäó
B
Bz1 pα4f1 ` α2f4q ` B
Bz2 pλα4f1 ` µα2f4q
` B
Bz3 pθα4f1 ` ϑα2f4q ` B
Bz4 pνα4f1 ` ηα2f4q “ ´Λ4f1 ´ Λ2f4. (6.7)
Ðîçãëÿíåìî ðiçíèöþ ìiæ ðiâíÿííÿì (6.6), äîìíîæåíèì íà α2, i ðiâ-
íÿííÿì (6.7), äîìíîæåíèì íà α3. Òîäi âðàõîâóþ÷è (6.4), îòðèìà¹ìî íà-
ñòóïíó ðiâíiñòü
B
Bz1
´
f1pα1α2 ´ α3α4q ` f4pα2α3 ´ α2α3q
¯
` B
Bz2
´
f1pλα1α2 ´ λα3α4q ` f4pµα2α3 ´ µα2α3q
¯
` B
Bz3
´
f1pθα1α2 ´ θα3α4q ` f4pϑα2α3 ´ ϑα2α3q
¯
` B
Bz4
´
f1pνα1α2 ´ να3α4q ` f4pηα2α3 ´ ηα2α3q
¯
“ pΛ4α3 ´ Λ1α2qf1.
Îòæå, ìà¹ìî ðiâíÿííÿ
Bf1
Bz1 ` λ
Bf1
Bz2 ` θ
Bf1
Bz3 ` ν
Bf1
Bz4 “ Λ4α3 ´ Λ1α2
α1α2 ´ α3α4
f1. (6.8)
Äëÿ ðiâíÿííÿ (6.8) ðîçãëÿíåìî õàðàêòåðèñòè÷íå ðiâíÿííÿ
dz1
1
“ dz2
λ
“ dz3
θ
“ dz4
ν
“ pα1α2 ´ α3α4q d f1
pΛ4α3 ´ Λ1α2qf1 . (6.9)
Ðîçâ'ÿçêàìè ñèñòåìè (6.9) ¹ iíòåãðàëè
c2 “ λz1 ´ z2, c3 “ θz1 ´ z3, c4 “ νz1 ´ z4,
c5 “ ln f1 ` α2Λ1 ´ α3Λ4
α1α2 ´ α3α4
z1.
Ïðåäñòàâëåííÿ äåÿêèõ êëàñiâ êâàòåðíiîííèõ ãiïåðãîëîìîðôíèõ ôóíêöié 119
Îòæå, çàãàëüíèé ðîçâ'ÿçîê ðiâíÿííÿ (6.8) ì๠âèãëÿä
f1 “ Φ1prζ2, rζ3, rζ4q ¨ exp
´α3Λ4 ´ α2Λ1
α1α2 ´ α3α4
z1
¯
,
äå rζ2, rζ3, rζ4 âèçíà÷åíi ðiâíîñòÿìè (4.4), à Φ1 � äîâiëüíà ãîëîìîðôíà
ôóíêöiÿ òðüîõ êîìïëåêñíèõ çìiííèõ.
Àíàëîãi÷íî ìîæíà îòðèìàòè ïðåäñòàâëåííÿ äëÿ êîìïîíåíò f2, f3, f4.
□
Îòæå, ôîðìóëà (6.5) ä๠ïðåäñòàâëåííÿ ëiâî�Λ´ψ�ãiïåðãîëîìîðôíî¨
ôóíêöi¨ ïðè ñïåöiàëüíîìó âèáîði Λ i ψ.
Çàóâàæåííÿ 6.1. Ç ðiâíîñòåé (2.2) i (2.5) âèïëèâà¹, ùî ïðè Λ “ 0
ìíîæèíà ëiâî�Λ ´ ψ�ãiïåðãîëîìîðôíèõ ôóíêöié ñïiâïàä๠ç ìíîæè-
íîþ ëiâî�ψ�ãiïåðãîëîìîðôíèõ ôóíêöié. Öå çàñâiä÷óþòü òåîðåìàìè 4.1
i 6.1, îñêiëüêè ïðåäñòàâëåííÿ (6.5) ñïiâïàä๠ç ïðåäñòàâëåííÿì (4.3) ïðè
Λ “ 0.
Ëiòåðàòóðà
[1] R. Abreu Blaya, J. Bory Reyes, A. Guzm�an½ U. K�ahler. On the ϕ�hyperderivative
of the ψ�cauchy-type integral in cli�ord analysis. Comput. Methods Funct. Theory,
17(1):101�119, 2017.
[2] R. Abreu Blaya, J. Bory Reyes, A. Guzm�an Ad�an½ U. Kaehler. On some structural
sets and a quaternionic pϕ, ψq�hyperholomorphic function theory. Mathematische
Nachrichten, 288(13):1451�1475, 2015.
[3] D. Alpay, M. E. Luna-Elizarraras, M. Shapiro½ D. C. Struppa. Basics of functi-
onal analysis with bicomplex scalars, and bicomplex schur analysis. Springer: Spri-
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Ò. Ñ. Êóçüìåíêî
Æèòîìèðñüêèé âiéñüêîâèé iíñòèòóò iìåíi Ñ. Ï. Êîðîëüîâà, ì. Æèòîìèð
Email: kuzmenko.ts15@gmail.com
ORCID: 0000-0001-9052-6230
Â. Ñ. Øïàêiâñüêèé
Iíñòèòóò ìàòåìàòèêè ÍÀÍ Óêðà¨íè, ì. Êè¨â
Email: shpakivskyi86d@gmail.com
ORCID: 0000-0003-4256-8975
|
| id | oai:trim.imath.kiev.ua:article-543 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:09:09Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/86/02b2595924c525bc2858eeff7ac75386.pdf |
| spelling | oai:trim.imath.kiev.ua:article-5432026-02-24T12:22:41Z Representation of some classes of quaternionic hyperholomorphic functions Представлення деяких класів кватерніонних гіперголоморфних функцій Kuzmenko, Tetiana Shpakivskyi, Vitalii Кузьменко, Тетяна Шпаківський, Віталій In the algebra of complex quaternions H(C) we consider the left- and right-ψ-hyperholomorphic functions, and left-Λ — ψ-hyperholomorphic functions. We justify the transition in left- and right-ψ-hyperholomorphic functions to a simpler basis i.e., to the Cartan basis. Using Cartan's basis we find the solution of Cauchy-Fueter equation. By the same method we find representations of left- and right-ψ-hyperholomorphic functions, and representation of left-Λ — ψ-hyperholomorphic functions. В алгебрі комплексних кватерніонів H(C) розглянуто ліво- і право-ψ-гіперголоморфні функції, а також ліво-Λ — ψ-гіперголоморфні функції. Обґрунтовано перехід у ліво- і право-ψ-гіперголоморфних функціях до простішого базису, а саме до базису Картана. Використовуючи базис Картана, знайдено розв’язок рівняння Коші—Фуетера. Таким же методом знайдено представлення ліво- і право-ψ-гіперголоморфних функцій, а також представлення ліво-Λ — ψ-гіперголоморфних функцій. Інститут математики НАН України 2025-08-18 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/543 10.3842/trim.v21n1.543 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 21 No. 1 (2024): Special Issue "Actual problems of modern mathematics: methods, models and applications"; 102-120 Сборник Трудов Института математики НАН Украины; Том 21 № 1 (2024): Спеціальний номер "Актуальні проблеми сучасної математики: методи, моделі та застосування" ; 102-120 Збірник Праць Інституту математики НАН України; Том 21 № 1 (2024): Спеціальний номер "Актуальні проблеми сучасної математики: методи, моделі та застосування" ; 102-120 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/543/526 Авторське право (c) 2024 Тетяна Кузьменко, Віталій Шпаківський http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Kuzmenko, Tetiana Shpakivskyi, Vitalii Кузьменко, Тетяна Шпаківський, Віталій Representation of some classes of quaternionic hyperholomorphic functions |
| title | Representation of some classes of quaternionic hyperholomorphic functions |
| title_alt | Представлення деяких класів кватерніонних гіперголоморфних функцій |
| title_full | Representation of some classes of quaternionic hyperholomorphic functions |
| title_fullStr | Representation of some classes of quaternionic hyperholomorphic functions |
| title_full_unstemmed | Representation of some classes of quaternionic hyperholomorphic functions |
| title_short | Representation of some classes of quaternionic hyperholomorphic functions |
| title_sort | representation of some classes of quaternionic hyperholomorphic functions |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/543 |
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