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{{Infobox scientist
| name = Carlo Severini
| image = 
| image_size =
| alt =
| caption =
| birth_date = 10 March 1872
| birth_place = [[Arcevia]] ([[Ancona]])
| death_date = {{death date and age|1951|5|11|1872|4|10|df=yes}}
| death_place = [[Pesaro]]
| residence =
| citizenship =
| nationality = [[Italy|Italian]]
| fields = [[Real analysis]]
| workplaces = [[Università di Bologna]]<br>[[University of Catania]]<br>[[University of Genova]]
| alma_mater = [[Università di Bologna]]
| doctoral_advisor = [[Salvatore Pincherle]]
| academic_advisors =
| doctoral_students =
| notable_students =
| known_for = [[Severini-Egorov theorem]]
| author_abbrev_bot =
| author_abbrev_zoo =
| influences =
| influenced =
| awards =
| signature = 
| signature_alt =
| footnotes =
}}

'''Carlo Severini''' (10 March 1872 – 11 May 1951) was an [[Italians|Italian]] [[mathematician]]: he was born in [[Arcevia]] ([[Province of Ancona]]) and died in [[Pesaro]]. Severini, independently from [[Dmitri Egorov|Dmitri Fyodorovich Egorov]], proved and published earlier a proof of the theorem now known as [[Egorov's theorem]].

==Biography==
He graduated in [[Mathematics]] from the [[University of Bologna]] on November 30, 1897:<ref>According to the summary of his student file available from the {{Harvtxt|Archivio Storico dell'Università di Bologna|2004}} (an electronic version of the [[archive]]s of the [[University of Bologna]]).</ref><ref>The content of this section is based on references {{Harv|Tricomi|1962}} and {{Harv|Straneo|1952}}: this last one also refers that he was married and had several children, however without giving any other detail.</ref> the title of his "[[Laurea]]" [[thesis]] was "''Sulla rappresentazione analitica delle funzioni arbitrarie di variabili reali''".<ref>An [[English language|English]] translation reads as "On the Analytic Representation of Arbitrary Functions of Real variables"; despite the similarities in the title and the same year of publication, the biographical sources do not say if the paper {{Harv|Severini|1897}} is somewhat related to his thesis.</ref> After obtaining his [[academic degree|degree]], he worked in [[Bologna]] as an assistant to the chair of [[Salvatore Pincherle]] until 1900.<ref>The [[#{{harvid|Università di Bologna|1898|p=170}}|1897–1898 yearbook of the university]] already lists him between the [[assistant professor]]s.</ref> From 1900 to 1906, he was a senior high school teacher, first teaching in the [[Institute of Technology]] of [[La Spezia]] and then in the [[lyceum]]s of [[Foggia]] and of [[Turin]];<ref name="Str1" >According to {{Harvtxt|Straneo|1952|p=98}}.</ref> then, in 1906 he became full professor of [[Infinitesimal Calculus]] at the [[University of Catania]]. He worked in [[Catania]] until 1918, then he went to the [[University of Genova]], where he stayed until his retirement in 1942.<ref name="Str1"/>

==Work==
He authored more than 60 papers, mainly in the areas of [[real analysis]], [[approximation theory]] and [[partial differential equation]]s, according to {{Harvtxt|Tricomi|1962}}. His main contributions belong to the following fields of [[mathematics]]:<ref>Only his most known results are described in the following sections: {{Harvtxt|Straneo|1952}} reviews his research in greater detail.</ref>

===Approximation theory===
In this field, Severini proved a generalized version of the [[Weierstrass approximation theorem]]. Precisely, he extended the original result of [[Karl Weierstrass]] to the class of [[Bounded function|bounded]] [[locally integrable function]]s, which is a class including particular [[discontinuous function]]s as members.<ref>According to {{Harvtxt|Straneo|1952}}, the result is given in various papers, source {{Harv|Severini|1897}} perhaps being the most accessible of them.</ref>

===Measure theory and integration===
Severini proved [[Egorov's theorem]] one year earlier than [[Dmitri Egorov]]<ref>Egorov's proof is given in the paper {{Harv|Egorov|1911}}.</ref> in the paper {{Harv|Severini|1910}}, whose main theme is however [[Sequence (mathematics)|sequences]] of [[orthogonal functions]] and their properties.<ref>Also, according to {{Harvtxt|Straneo|1952|p=101}}, Severini, while acknowledging his own priority in the publication of the result, was unwilling to disclose it publicly: it was [[Leonida Tonelli]] who, in the note {{Harv|Tonelli|1924}}, credited him the priority for the first time.</ref>

===Partial differential equations===
Severini proved an [[existence theorem]] for the [[Cauchy problem]] for the [[Non-linear partial differential equation|non linear]] [[hyperbolic partial differential equation]] of first order

:<math>\left\{
\begin{array}{lc}
\frac{\partial u}{\partial x}=f\left(x,y,u,\frac{\partial u}{\partial y}\right) & (x,y)\in\mathbb{R}^+\times[a,b]\\
u(0,y)=U(y) & y\in[a,b]\Subset\mathbb{R}
\end{array}\right.,
</math>

assuming that the Cauchy data <math>U</math> (defined in the [[bounded interval]] <math>[a,b]</math>) and that the [[Function (mathematics)|function]] <math>f</math> has [[Lipschitz continuity|Lipschitz continuous]] first order [[partial derivative]]s,<ref>This means that f belongs to the [[Differentiability class|class]] <math>C^{(1,1)}</math>.</ref> jointly with the obvious requirement that the [[Set (mathematics)|set]] <math>\scriptstyle \{(x,y,z,p)=(0,y,U(y),U^\prime(y));y\in[a,b]\}</math> is contained in the [[Domain (mathematics)|domain]] of <math>f</math>.<ref>For more details about his researches in this field, see {{Harv|Cinquini-Cibrario|Cinquini|1964}} and the references cited therein</ref>

===Real analysis and unfinished works===
According to {{Harvtxt|Straneo|1952|p=99}}, he worked also on the foundations of the theory of [[real function]]s.<ref>{{Harvtxt|Straneo|1952|p=99}} lists Severini's researches on this field under as "''Fondamenti dell'analisi infinitesimale (Foundations of infinitesimal analysis)''": however, the topics covered range from the theory of integration to [[absolutely continuous function]]s and to operations on series of real functions.</ref> Severini also left an unpublished and unfinished [[treatise]] on the theory of [[real function]]s, whose title was planned to be "''Fondamenti dell'analisi nel campo reale e i suoi sviluppi"''.<ref>"''Foundations of Analysis on the Real Field and its Developments''": again according to {{Harvtxt|Straneo|1952|p=101}}, the treatise would have included his later original results and covered all the fundamental topics required for the study of [[functional analysis]] on the [[real number|real field]].</ref>

==Selected publications==
*{{Citation
| first = Carlo
| last = Severini
| author-link =
| title = Sulla rappresentazione analitica delle funzioni reali discontinue di variabile reale
| language = Italian
| journal = Atti della Reale Accademia delle Scienze di Torino.
| volume = 33
| origyear = 1897-1898
| year = 1897
| url = http://www.archive.org/stream/attidellaraccad04unkngoog#page/n1120/mode/1up
| pages = 1002–1023
| id =
| jfm = 29.0354.02
}}. In this paper (whose title reads as "''On the analytic representation of discontinuous real functions of a real variable''" in the English translation) he proves an extension of Weierstrass approximation theorem to a class of functions including among its members particular discontinuous ones.
*{{Citation
| last = Severini
| first = C.
| author-link =
| title = Sulle successioni di funzioni ortogonali
| language = Italian
| journal = Atti dell'[http://www3.unict.it/gioenia/ Accademia Gioenia]
| series = serie 5<sup>a</sup>,
| volume = 3
| issue = 5
| pages = Memoria XIII, 1–7
| year = 1910
| id =
| jfm = 41.0475.04
}}. "''On sequences of orthogonal functions''" contains his most known result, i.e. the Severini–Egorov theorem.

==See also==
*[[Hyperbolic partial differential equation]]
*[[Orthogonal functions]]
*[[Severini-Egorov theorem]]
*[[Weierstrass approximation theorem]]

==Notes==
{{Reflist|30em}}

==Biographical references==
*{{Citation
| author = Archivio Storico dell'Università di Bologna
| author-link = Università di Bologna
| title = Carlo Severini
| journal = [http://www.archiviostorico.unibo.it/Template/listStudenti.asp?IDFolder=143&LN=EN Fascicoli degli studenti]
| series = Fascicolo della Facoltà di Scienze Fisiche Matematiche Naturali n°
| volume = 2843
| origyear = 1897
| year = 2004
| language = Italian
| url = http://www.archiviostorico.unibo.it/template/listStudenti.asp?Testo=Carlo&IDNazione=&TestoCognome=Severini&IDProvincia=&Numero=&LuogoDiNascita=&Note=&IDFacolta=&Tipologia=&DI=&DF=&TitoloTesi=&AnnoAccademico=&filtra=Avvia+ricerca&IDFolder=143&LN=IT&Or=&start=true
| accessdate = March 1, 2011
}}. A very short summary of the student file of Carlo Severini, giving however useful information about his [[laurea]].
*{{Citation
| last = Straneo
| first = Paolo
| author-link =
| title = Carlo Severini
| language = Italian
| journal = [[Bollettino della Unione Matematica Italiana]]
| series = Serie 3,
| volume = 7
| issue = 3
| pages = 98–101
| year = 1952
| url = http://www.bdim.eu/item?id=BUMI_1952_3_7_1_87_0
| id =
| mr = 50531
}}, available from the [http://www.bdim.eu/ Biblioteca Digitale Italiana di Matematica].
*{{Citation
| last = Tonelli
| first = Leonida
| author-link = Leonida Tonelli
| title = Su una proposizione fondamentale dell'analisi
| language = Italian
| journal = [[Bollettino della Unione Matematica Italiana]]
| series = Serie 2,
| volume = 3
| issue =
| pages = 103–104
| year = 1924
| url =
| id =
| zbl = 50.0192.01
}}. In this paper, Tonelli credits Severini for the first proof of Severini–Egorov theorem. An English translation of the title reads as:–"''On a fundamental proposition of analysis''".
*{{Citation
| first = F. G.
| last = Tricomi
| author-link = Francesco Tricomi
| contribution = Carlo Severini
| contribution-url = http://www.dm.unito.it/sism/m_italiani/biografie/tricomi/severini.html
| title = Matematici italiani del primo secolo dello stato unitario
| language = Italian
| journal = Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali
| series = series IV
| volume = I
| year = 1962
| pages = 120
| place = Torino
| id =
| zbl = 0132.24405
}}. "''Italian mathematicians of the first century of the unitary state''" is an important historical memoir giving brief biographies of the Italian mathematicians who worked and lived between 1861 and 1961. Its content is available from the website of the [http://www.dm.unito.it/sism/ Società Italiana di Storia delle Matematiche].
*{{Citation
| author = Università di Bologna
| author-link = Università di Bologna
| contribution = Facoltà di Scienze Fisiche, Matematiche e Naturali. Assistenti
| contribution-url = http://amshistorica.unibo.it/archivio/000193/000039/000172.jpg
| title = Annuario della Regia Università di Bologna
| place = Bologna
| publisher = Premiato Stabilimento Tipografico Succ. Monti
| year = 1898
| page = 170
| pages =
| language = Italian
| url = http://amshistorica.unibo.it/193
| doi =
| id =
}}.

==References==
*{{Citation
| last = Cinquini-Cibrario
| first = M.
| author-link = Maria Cinquini-Cibrario
| last2 = Cinquini
| first2 = S.
| author2-link= Silvio Cinquini
| title = Equazioni a derivate parziali di tipo iperbolico
| language = Italian
| place = Roma
| publisher = Edizioni Cremonese
| year = 1964
| series = Monografie matematiche del Consiglio Nazionale delle Ricerche
| volume = 12
| pages = VIII+552
| id =
| mr = 0203199
| zbl = 0145.35404
}}. "''Partial differential equations of hyperbolic type''" is a monograph surveying the theory of [[hyperbolic equation]]s up to its [[state of the art]] in the early 1960s, published by the [[Consiglio Nazionale delle Ricerche]].
*{{Citation
| last = Egoroff
| first = D. Th.
| authorlink = Dmitri Egorov
| title = Sur les suites des fonctions mesurables
| journal = [[Comptes rendus de l'Académie_des_sciences#1835-1965|Comptes rendus hebdomadaires des séances de l'Académie des sciences]]
| language = French
| volume = 152
| pages = 244–246
| year = 1911
| url = http://gallica.bnf.fr/ark:/12148/bpt6k3105c.image.f244
| id =
| jfm = 42.0423.01
}}, available at [[Gallica]].

==External links==
*{{Citation
| last = Guerraggio
| first = Angelo
| author-link =
| last2 = Nastasi
| first2 = Pietro
| author2-link =
| last3 = Tricomi
| first3 = Francesco
| author3-link = Francesco Tricomi
| title = Carlo Severini (1872 – 1951)
| language = [[Italian language|Italian]]
| date =
| year = 2008-2010
| url = http://mathematica.sns.it/autori/1239/
| accessdate = March 2, 2011
}}. Available from the [http://mathematica.sns.it/ Edizione Nazionale Mathematica Italiana].

{{Persondata 
| NAME =Severini, Carlo
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Italian mathematician
| DATE OF BIRTH =10 March 1872
| PLACE OF BIRTH =[[Arcevia]] ([[Ancona]])
| DATE OF DEATH =11 May 1951
| PLACE OF DEATH =[[Pesaro]]
}}
{{DEFAULTSORT:Severini, Carlo}}
[[Category:1872 births]]
[[Category:1951 deaths]]
[[Category:People from Arcevia]]
[[Category:19th-century Italian mathematicians]]
[[Category:20th-century Italian mathematicians]]
[[Category:Mathematical analysts]]

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