Kummer's function

In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer.

Kummer's function is defined by

\Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.

The duplication formula is

\Lambda _{n}(z)+\Lambda _{n}(-z)=2^{1-n}\Lambda _{n}(-z^{2})

Compare this to the duplication formula for the polylogarithm:

\operatorname {Li} _{n}(z)+\operatorname {Li} _{n}(-z)=2^{1-n}\operatorname {Li} _{n}(z^{2}).

An explicit link to the polylogarithm is given by

\operatorname {Li} _{n}(z)=\operatorname {Li} _{n}(1)\;\;+\;\;\sum _{k=1}^{n-1}(-1)^{k-1}\;{\frac {\log ^{k}|z|}{k!}}\;\operatorname {Li} _{n-k}(z)\;\;+\;\;{\frac {(-1)^{n-1}}{(n-1)!}}\;\left[\Lambda _{n}(-1)-\Lambda _{n}(-z)\right].

References

Lewin, Leonard, ed. (1991), Structural Properties of Polylogarithms, Providence, RI: American Mathematical Society, ISBN 0-8218-4532-2.

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

Read other articles:

ADCY10

ADCY10Struktur yang tersediaPDBPencarian Ortolog: PDBe RCSB Daftar kode id PDB4CLK, 4CLL, 4CLP, 4CLS, 4CLT, 4CLU, 4CLW, 4CLY, 4CLZ, 4CM0, 4CM2, 4OYA, 4OYB, 4OYI, 4OYM, 4OYO, 4OYP, 4OYW, 4OYX, 4OYZ, 4OZ2, 4OZ3, 4UST, 4USU, 4USV, 4USW, 5D0RPengidentifikasiAliasADCY10, HCA2, SAC, SACI, Sacy, hsAC, HEL-S-7a, adenylate cyclase 10 (soluble), adenylate cyclase 10, soluble, adenylate cyclase 10ID eksternalOMIM: 605205 MGI: 2660854 HomoloGene: 10188 GeneCards: ADCY10 Lokasi gen (Tikus)Kr.Kromosom...

Another Course to College

Pilot school in the United StatesAnother Course to CollegeFormer home of Another Course to College (ACC), 1976 – 1989Address612 Metropolitan AvHyde Park, MassachusettsUnited StatesInformationTypePilotEstablished1976 (1976)School districtBoston Public SchoolsCEEB code220336NCES School ID250279000698[2]Head of SchoolMichele PellamGrades9–12aEnrollment237 (2015-16)[1]Campus typeUrbanWebsitehttp://accbps.org Another Course to College (ACC) is a Public Pilot School located...

KJNI-LP

Radio station in Lake Elsinore, CaliforniaKJNI-LPLake Elsinore, CaliforniaFrequency101.7 MHzProgrammingFormatChildren's radioOwnershipOwnerJennifer Smart FoundationHistoryFirst air dateNovember 12, 2014; 9 years ago (2014-11-12)Technical information[1]Licensing authorityFCCFacility ID193349ClassL1ERP100 wattsHAAT−14 meters (−46 ft)Transmitter coordinates33°39′29.20″N 117°15′25.80″W / 33.6581111°N 117.2571667°W / 33.6...

Ghislarengo

Ghislarengo commune di Italia Tempat Negara berdaulatItaliaRegion di ItaliaPiedmontProvinsi di ItaliaProvinsi Vercelli NegaraItalia Ibu kotaGhislarengo PendudukTotal805 (2023 )GeografiLuas wilayah12,5 km² [convert: unit tak dikenal]Ketinggian206 m Berbatasan denganArborio Carpignano Sesia Rovasenda Sillavengo Lenta Informasi tambahanKode pos13030 Zona waktuUTC+1 UTC+2 Kode telepon0161 ID ISTAT002062 Kode kadaster ItaliaE007 Lain-lainSitus webLaman resmi Ghislarengo adalah komune y...

14th Street (Manhattan)

West-east street in Manhattan, New York For other uses, see 14th Street. 14th Street14th Street looking as seen from the east at Fifth AvenueMaintained byNYCDOTLength2.0 mi (3.2 km)[1]LocationManhattan, New York CityPostal code10014, 10011, 10003, 10009Coordinates40°44′09″N 73°59′34″W / 40.7357°N 73.9929°W / 40.7357; -73.9929West end NY 9A (11th Avenue) in Chelsea / MeatpackingEast endAvenue C in East Village / Stu...

Perang Vietnam

Perang Vietnam(Chiến tranh Việt Nam)Bagian dari Peperangan Indochina dan Perang DinginSearah jarum jam dari kiri atas: Helikopter Huey AS memasukkan pasukan ARVN Vietnam Selatan, 1970. * Pasukan Vietnam Utara sedang beraksi c. 1966. * Marinir AS sedang menggunakan Pelontar api, 1967. * Jenderal Vietnam Selatan, Nguyễn Ngọc Loan dengan cepat mengeksekusi perwira Viet Cong, Nguyễn Văn Lém selama Serangan Tet berlangsung. * Dua pesawat Douglas A-4C Skyhawk terbang melewati kapa...

Обири, Хеллен

У этого термина существуют и другие значения, см. Обири (значения). Хеллен Обириангл. Hellen Obiri Общая информация Полное имя Hellen Onsando Obiri Дата и место рождения 13 декабря 1989(1989-12-13)[1] (34 года)Кисии[d], Кения Гражданство Кения IAAF 14424921 Международные медали Лёгкая атлетика О�...

Wat Mai Suwannaphumaham

Tampak luar Wat Mai Suwannaphumaham Patung Budha yang ada di Vat Mai Suwannaphumaham Wat Mai Suwannaphumaham (bahasa Lao: ວັດໃໝ່ສຸວັນນະພຸມມາຣາມ) atau sering disebut dengan Wat Mai (Biara Baru) adalah salah satu kuil terbesar dan terindah yang ada di Luang Prabang, tepatnya pada lokasi pusat. Kuil ini berada di depan pasar malam dan di samping istana kerajaan.[1] Sejarah Wat Mai dibangun pada akhir abad XVII oleh raja Anourout, kemudian dip...

الحجل (مناخة)

قرية الحجل - قرية - تقسيم إداري البلد اليمن المحافظة محافظة صنعاء المديرية مديرية مناخة العزلة عزلة بني برة السكان التعداد السكاني 2004 السكان 210 • الذكور 105 • الإناث 105 • عدد الأسر 34 • عدد المساكن 30 معلومات أخرى التوقيت توقيت اليمن (+3 غرينيتش) ت�...

Battlefield

BattlefieldAliranPenembak orang pertamaPengembangUtamaDICE (2002–) Lainnya Easy Studios (2009, 2011)Neowiz Games (2010)Visceral Games (2015)Criterion Games (2019, 2021–)Ripple Effects Studio (2021–)Industrial Toys (2021–2023) PenerbitElectronic ArtsPembuatJohan PerssonPelantarMicrosoft Windows, OS X, PlayStation 2, Xbox, PlayStation 3, Xbox 360, PlayStation 4, Xbox One, PlayStation 5, Xbox Series X/S, Android, iOSTerbitan pertamaBattlefield 194210 September 2002Terbitan terakhirBattle...

Otto H. Kahn House

Building in Manhattan, New York United States historic placeOtto H. Kahn HouseU.S. National Register of Historic PlacesNew York State Register of Historic PlacesNew York City Landmark No. 0675 Location1 East 91st Street, Manhattan, New YorkCoordinates40°47′05″N 73°57′28″W / 40.78472°N 73.95778°W / 40.78472; -73.95778Built1914–1918ArchitectJ. Armstrong Stenhouse; C. P. H. GilbertNRHP reference No.06000821NYSRHP No.06101.000248NYCL...

فراخاتي

فراخاتي تقسيم إداري البلد اليونان [1] خصائص جغرافية إحداثيات 37°57′21″N 22°48′10″E / 37.9557°N 22.8028°E / 37.9557; 22.8028 الارتفاع 20 متر السكان التعداد السكاني 3603 (resident population of Greece) (2021)2656 (resident population of Greece) (2001)2224 (resident population of Greece) (1991)3338 (resident population of Greece) (2011)1629 (de fa...

WITL-FM

Country music radio station in Lansing, Michigan WITL-FMLansing, MichiganBroadcast areaLansing, MichiganFrequency100.7 MHz (HD Radio) [1]Branding100.7 WITLProgrammingFormatCountrySubchannelsHD2: WONU simulcastHD3: Air1AffiliationsCompass Media NetworksWestwood OneOwnershipOwnerTownsquare Media(Townsquare License, LLC)Sister stationsWFMK, WJIM, WJIM-FM, WMMQ, WVFNHistoryFirst air date1961 (as WMRT-FM)Former call signsWYFE (1964–1966)WMRT (1961–1964)Call sign meaningWhITtLeTechnical...

Folium vermis

Folium (vermis)Upper surface of the cerebellum.IdentifiersNeuroNames675TA98A14.1.07.206TA25828FMA83889Anatomical terms of neuroanatomy[edit on Wikidata] The folium vermis is a short, narrow, concealed band at the posterior extremity of the vermis, consisting apparently of a single folium, but in reality marked on its upper and under surfaces by secondary fissures. Laterally, it expands in either hemisphere into a considerable lobule, the superior semilunar lobule (lobulus semilunaris supe...

Turret Peak (Colorado)

Turret PeakTurret Peak (left) and Pigeon Peak (behind, right)Highest pointElevation13,841 ft (4,219 m)[1][2]Prominence735 ft (224 m)[2]Parent peakPigeon Peak[2]Isolation0.50 mi (0.80 km)[2]Coordinates37°37′39″N 107°38′23″W / 37.627499°N 107.6397833°W / 37.627499; -107.6397833[3]GeographyTurret PeakColorado LocationLa Plata County, Colorado, U.S.[3]Parent rangeSan Juan...

The Last Princess (film)

Untuk kegunaan lain, lihat The Last Princess. The Last PrincessPoster rilis teatrikalNama lainHangul덕혜옹주 Hanja德惠翁主 Alih Aksara yang DisempurnakanDeokhye-ongjuArtiPrincess Deokhye SutradaraHur Jin-hoProduserKim Jung-hwaSkenario Hur Jin-ho Lee Han-eol Seo Yoo-min BerdasarkanPrincess Deokhyeoleh Kwon Bi-youngPemeran Son Ye-jin Park Hae-il PerusahaanproduksiHofilmDistributorLotte EntertainmentTanggal rilis 3 Agustus 2016 (2016-08-03) (Korea Selatan) Durasi127 menitN...

Suisse aux Jeux olympiques d'hiver de 1948

Suisse aux Jeux olympiques d'hiver de 1948 Code CIO SUI Comité Comité olympique Suisse Lieu Saint-Moritz Participation 5e aux Jeux d'hiver Athlètes 70 (59 hommes et 11 femmes) dans 9 sports Porte-drapeau Felix Endrich(bobsleigh, ouverture)Karl Molitor(ski alpin, clôture) MédaillesRang : 3e Or3 Arg.4 Bron.3 Total10 Historique Jeux olympiques d'été 1896 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012 ...

Royal Street

Cet article est une ébauche concernant la route et la Louisiane. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Royal Street. La Royal Street (« Rue Royale », en espagnol : Calle Real) est une rue de La Nouvelle-Orléans en Louisiane, aux États-Unis. Elle traverse le Vieux carré français de La Nouvelle-Orléans et en est la deuxième voie la plus connue après Bourbon Street. Personnalités...

دوري لوكسمبورغ الوطني 2010–11

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (نوفمبر 2019) دوري لوكسمبورغ الوطني 2010–11 تفاصيل الموسم دوري لوكسمبورغ الوطني النسخة 97 البلد لوكسمبورغ التا�...

Let all mortal flesh keep silence

Greek early Christian hymn text Let all mortal flesh keep silencePrayers during the Canon of the Mass during the Mass of the Faithful.EnglishLet all mortal flesh keep silenceNative nameGreek: Σιγησάτω πᾶσα σάρξ βροτείαGenreHymnBased onHabakkuk 2:20 and Zechariah 2:13Meter8.7.8.7.8.7MelodyPicardy Let all mortal flesh keep silence (Greek: Σιγησάτω πᾶσα σάρξ βροτεία), also known as Let all mortal flesh keep silent, is an ancient cha...