# Francesco Giacomo Tricomi

### Quick Info

Born
5 May 1897
Naples, Italy
Died
21 November 1978
Turin, Italy

Summary
Francesco Tricomi studied differential equations which became very important in the theory of supersonic flight. He wrote a number of excellent books which were translated into several different languages.

### Biography

Francesco Tricomi's parents were Arturo Tricomi (1863-1928) and Corinna Di Lustro. Arturo Tricomi, born in Palermo on 15 September 1863, was an engineer and architect who became an extraordinary professor of Ornamental Design and Architecture at the University of Cagliari and in 1906 moved to the University of Naples where he taught ornamental design in the Faculty of Mathematical Sciences. As an architect, among his works were the Palazzo Santa Lucia and the Villino Berlingieri, both in Naples.

Francesco studied first at the local Technical Institute where he was taught mathematics by Alfredo Perna, an excellent teacher who had published papers such as L'immaginario i ed i numeri alternati i, j, k nello studio delle deformazioni infinitesime delle curve piane e delle curve storte (1898) and Le equazioni delle curve in coordinate complesse coniugate (1903) in the Rendiconti del Circolo Matematico di Palermo. In fact Perna became the ministerial supervisor and prepared the report on Teacher Training in Italy for the International Congress of Mathematicians in Zurich in 1932. At first Tricomi had not found mathematics interesting for, as a student, he was interested in practical science problems. When Perna showed him how trigonometry could be used to predict eclipses, he began to find mathematics more exciting but when he graduated from the Technical Institute at the age of sixteen, he decided that he wanted to study chemistry at university.

Tricomi studied first at the University of Bologna, where he enrolled in chemistry classes in 1913. He attended a few lectures by Federigo Enriques and quickly realised that he found mathematical physics much more to his liking than chemistry. After one year of chemistry at the University of Bologna, he decided that physics would suit him better so he spent 1914-15 taking physics courses. This move to physics, of course, meant that he needed to take further mathematics courses and, quite soon, he realised that it was mathematics where his true passion lay. He moved to the University of Naples in 1915, enrolling in the third year of the mathematics course there; his teachers included Roberto Marcolongo and Gabriele Torelli (1849-1931).

World War I had begun when Tricomi was still a student at Bologna. Italy had a defensive alliance with both Germany and Austro-Hungary but did not join them when the war began on 28 July 1914. Italy negotiated with both the Central Powers and with the Allies, signing a pact with the Allies in April 1915 and declaring war on Austro-Hungary in the following month. Fighting in the Alps was fierce but neither side gained the upper hand. In August 1916 Italy declared war on Germany and, around the same time, Tricomi, then at the University of Naples, was called up for military service. He was sent first to a training course at for students at the Military Academy of Turin where he trained and was appointed as an officer on 1 April 1917. He was then sent to the front at the Karst where the Italians were about to made a massive attack against the Austrians. This was nearly successful but, with victory in sight, they had to withdraw as their supply lines became too extended. Germany then sent troops to support the Austrians and the Italians, after being routed by the combined force, managed to halt the combined German-Austrian advance at Monte Grappa. Tricomi was involved in the fighting at Monte Grappa in December 1917. The Italian troops, however, fell back to defend Venice at the Piave river where they again successfully stopped the Austrian advance. Tricomi, always at the centre of hostilities, was involved in the fighting at Piave in June 1918.

Rather remarkably, despite all this involvement in fighting with the Italian army, Tricomi had been able to continue with his studies and he had been awarded a degree in mathematics from the University of Naples on 16 April 1918. World War I ended on 11 November 1918, but he remained in military service until 1920 when he was discharged, was decorated with two military crosses, and was able to return to the University of Naples. He undertook research under Gabriele Torelli but found the environment at Naples difficult. Always someone of independent thought, the environment at Naples at that time was too authoritarian for him and, with a strong recommendation from Ugo Amaldi, he was appointed as an assistant to Francesco Severi in the chair of analytical geometry at the University of Padua in 1921.

After a brief period in Padua he moved again, this time to the University of Rome in the spring of 1922. The reason for the move was simply that Severi had been appointed to the chair of algebraic and infinitesimal analysis at the University of Rome and he invited Tricomi to follow him. This move meant he arrived into a remarkably strong mathematical environment which included Vito Volterra, Guido Castelnuovo, Federigo Enriques, Tullio Levi-Civita and Giuseppe Bagnera. We have already mentioned Tricomi's very independent nature so, despite this collection of world-leading mathematicians, he was always going to work alone without having anyone tell him what to do. This did not mean, however, that he did not understand that an Institute required top class people and he firmly held this belief throughout his life. Soon after arriving in Rome, Tricomi had submitted his first paper for publication, namely Su di un'equazione integrale di prima specie which appeared in the Rendiconti del Circolo Matematico di Palermo in 1922. The paper begins:-
In a recent note, Prof Levi-Civita showed how the problem of determining the harmonic function close to an assigned function U - that is, of the harmonic function that in a given range (in three dimensions) is closest, on average, to U - can be reduced, on the one hand, to the inversion of a certain integral equation of the first kind with constant limits, and, on the other, to a biharmonic problem, that is to say to the determination of a biharmonic function whose surface values are known together with those of its normal derivative. For this second way the author then arrives at the resolution of the question considered by him.
In 1924 the University of Florence advertised a competition for the chair of Algebraic and Infinitesimal Analysis. Tricomi competed and was offered the chair which he accepted and took up the appointment in February 1925. Enrico D'Ovidio had retired from the University of Turin in 1922 and in November 1925 Tricomi was called to the University of Turin as an extraordinary professor of algebraic analysis and in charge of complementary mathematics. Erika Luciano writes [19]:-
Upon his arrival in the Turin faculty he immediately perceived some tensions in the teaching staff, determined by the presence of two groups: the 'Jewish' one, to which Gino Fano and Guido Fubini adhered, and the 'vectorist' one of Giuseppe Peano, Tommaso Boggio and Cesare Burali-Forti. The sympathies of Tricomi went without hesitation to the first group, also because of the solid friendship that bound him to Fubini and Alessandro Terracini. However, Tricomi was also strongly appreciated by Peano who, as early as 1925, proposed to him to exchange his teaching of complementary mathematics with that of infinitesimal calculus. He would formally become the owner of the latter, however, only from 1932.
In fact before he moved to Turin he had already published a paper in 1923 which was to become very famous. In this paper, Sulle equazioni lineari alle derivate parziali di 2° ordine di tipo misto , he studied the theory of partial differential equations of mixed type, in particular the equation
$yu_{xx} + u_{yy} = 0$,
now known as the 'Tricomi equation'. The equation became important in describing an object moving at supersonic speed. Of course there were no supersonic aircraft in 1923 but the equation was to play a major role in later studies of supersonic flight as shown by Theodore von Karman and the Austrian physicist Felix Frankl (1905-1961).

In 1931, Tricomi married Susanna Fomm (1904-1959) in Germany. They had no children and Susanna, who was "gifted with a meek and sweet-tempered temperament" suffered very poor health which was a constant worry to her husband.

Benito Mussolini had become Italian Prime Minister in 1922 and over the following years brought in laws which made Italy a fascist one-party dictatorship. Tricomi was a vigorous anti-Fascist so the political situation became more and more troublesome to him. One consequence was that it led to Tricomi giving all the support he could to people and organisations which fascism opposed. Two in particular we must mention. It made him a strong supporter of Jewish people and of the Waldensians. The first of these will need no explanation to our readers but the second of these might. The Waldensians were founded by Peter Waldo in the 1170s as a religious movement within the Catholic Church but declared heretical by the Catholic Church 40 years later. It is considered by many as a forerunner of the Reformation. By 1938 Italy began to promulgate racial laws requiring discrimination against Jews. Gino Fano was one of Tricomi's colleagues at Turin and, because of the racial laws, he was deprived of his chair in 1938. Tricomi did his utmost to assist Fano, getting books and journals for him so that he could continue his mathematics. Tricomi also helped his colleague Alessandro Terracini (1889-1968) publish the high school algebra textbook Algebra elementare ad uso dei licei (1940) under Tricomi's name. We note that in this book Terracini fully developed the theory of real numbers according to Dedekind's construction. Terracini also lost his chair in Turin in 1938 and, in the following year emigrated to Argentina.

Of course, Tricomi's actions in helping Jewish colleagues put him in considerable danger and the entry of Italy into World War II made his position even more difficult. The centre of the Waldensian church was in Torre Pellice, in the valleys about 45 km southwest of Turin and, in the autumn of 1942, he was forced to leave Turin and take refuge in Torre Pellice with his mother Corinna, his wife Susanna, and Susanna's sister. In September 1943 German troops entered the north of Italy and created the Republic of Salò, a puppet state they set up for Mussolini after Italy surrendered to the Allies on 8 September 1943. This put Tricomi and his family in extreme danger in Torre Pellice and they went into hiding, eventually being able to make their way to Rome. Once there, he lived clandestinely as a guest of the Waldensian, Paolo Bosio (1891-1959), Pastor of the Piazza Cavour Church in Rome. When there, Tricomi continued to help Jewish colleagues and friends, such as Guido Castelnuovo and Federigo Enriques both of whom had been forced into hiding but were running an illegal school instructing Jewish students who were suffering discrimination.

After Rome was freed by the Allies, Castelnuovo asked Tricomi to remain in Rome and help him with the reconstruction of mathematics in the university there. The Allies, however, quickly put a stop to this, appointing (no doubt for political reasons) a far less able person to be in charge of the reconstruction. Tricomi returned to Turin in 1944 where he resumed his university teaching.

In the late 1930s, Tricomi had begun to publish books, many of them based on lecture courses he was giving at Turin. After the war he published further editions of the books he had already published, and continued with further volumes. His books were widely praised and translated from Italian into English, French, German and Russian.

You can see the titles and lots more information about these books at THIS LINK.

Here are a couple of typical comments by reviewers. Arthur Erdélyi, reviewing Lezioni sulle funzioni ipergeometriche confluenti (1952) writes [7]:-
Tricomi's book is based on a course of lectures given at the University of Torino, and it is written with that gift for exposition for which its author is so justly famous.
Richard Bellman, reviewing Lezioni sulle equazioni a derivate parziali (1954), writes [1]:-
This book furnishes an excellent introduction to the rapidly expanding theory of partial differential equations, written in the author's usual lucid and interesting style.
The praise given in these and many other reviews certainly rated the books vastly superior to that suggested by the modest comment Tricomi wrote in the preface of one of them:-
Maybe I have not succeeded to make difficult things easy, but at least I have never made an easy subject difficult.
A few years after the end of the war, he was involved in the Bateman project. In 1946 Harry Bateman died and Arthur Erdélyi headed a team, which included Wilhelm Magnus and Tricomi, working at the California Institute of Technology to publish the vast range of material left by Bateman. The team produced three volumes of Higher Transcendental Functions and two volumes of Tables of Integral Transforms.

You can read part of Erdélyi's introduction, and some reviews of these books at THIS LINK.

Tricomi participated in the International Congress of Mathematicians at Cambridge, Massachusetts in September 1950 while in the United States. He gave the lecture On the Incomplete Gamma Function. He began as follows:-
In the preparation of a monograph on the confluent hypergeometric functions for the "Bateman Manuscript Project" I have had occasion to point out many probably new properties of the incomplete gamma function.
This was not the first International Congress of Mathematicians that Tricomi had attended, for he had been at Bologna in 1928 and Zurich in 1932 when he gave the lecture Periodische Lösungen einer Differentialgleichung erster Ordnung . He also attended later International Congresses of Mathematicians, at Amsterdam in 1954 when he gave the address Asymptotische Eigenschaften der konfluenten hypergeometrischen Funktionen and at Edinburgh, Scotland in 1958, when he gave the lecture Quo vadimus? . He began this lecture as follows:-
Exactly thirty years ago Gino Loria made a communication with the same title as the present at the International Congress of Mathematicians in Bologna. As in the case of Loria, this title wants to have the meaning of a cry of alarm in the face of the enormous, uncontrolled development of today's mathematical publications.
You can read a version of this lecture at THIS LINK.

In late 1950 Tricomi left the United States and returned to Turin to continue his remarkable research output. Tricomi's autobiography [32] lists 300 papers, while a further 46 are listed in [14]. These papers cover a vast range of subjects including singular integrals, differential and integral equations, pseudodifferential operators, functional transforms, special functions, probability theory and its applications to number theory. Erika Luciano writes [19]:-
Trichomi left a profound mark in the field of special functions and functional transformations in the classical sense. Starting from the study of the differential equation of confluent hypergeometric functions, he introduced an appropriate integral of this (called the Trichomi function) which together with the Kummer function allows one to obtain as particular cases many special functions, including those of Bessel, the polynomials of Laguerre, Hermite etc.

Equally numerous and important were his results on multidimensional singular integral equations containing the main value of an improper integral, and his works on asymptotic evaluations, on series developments, on the theory of transformations (by Bessel, Gauss, Hankel) with related applications. In the field of applied mathematics, the contributions of numerical computation and of probability and statistics stand out, including the first quantitative theory of the phenomenon of bacterial resistance.
Tricomi was an editor of Aequationes Mathematicae from the time the journal was founded in 1968 until his death. The editors described him as:-
A forthright man, outspoken opponent of dictatorships of all colours, of sloppiness, of abstraction for abstraction's sake, and of the 'publish or perish' syndrome, his interests went far beyond mathematical research. Here we mention only his interest in the teaching of mathematics and in expository work, which led him to write several excellent textbooks, eventually translated from their original Italian to English, French, German and Russian.
Tricomi had a deep interest in problems concerning the history of mathematics and he published many important articles on this topic. In fact, references to articles on Giovanni Plana, Guido Ascoli, Bernhard Riemann, Salvatore Pincherle, Jacques Hadamard, Tommaso Boggio, Giuseppe Veronese, Francesco Gerbaldi, Duilio Gigli, Cesare Arzelà, Giuseppe Basso, Giuseppe Vitali, Federigo Enriques, Roberto Marcolongo, Guido Fubini, Giovanni Ricci, Ernesto Pascal and Mauro Picone written by him appear in this Archive.

His influence on mathematics goes well beyond the impressive results of his research. His writings have made an important contribution towards the present development of science. As the editors of Aequationes Mathematicae write:-
... the problems ... which he has solved and the theories which he has initiated and others have continued to work on, will keep his name alive. His passing away is a great loss to the international mathematical community.
Gaetano Fichera writes [13]:-
Tricomi enjoyed the esteem and the deep respect for the entire Italian mathematical community. And truly sincere was the condolences for his death. Abroad, especially in the United States and the Soviet Union, he was considered a legendary figure, one of the greatest mathematicians of this century.
His wife died in her middle 50s in 1959 and, having no children, Tricomi spent his final years as a rather lonely figure after he retired in 1972. His final months are described in [13]:-
In the first days of July last year [1978], Francesco Tricomi left Turin at the wheel of his little Volkswagen. Driving in his unmistakable style, he crossed the Alps to reach Switzerland and spend his summer holidays there, as usual.

He had turned 81 on the previous 5 May, but his health was excellent, his mind was very clear, his physical appearance was excellent with his tall, elegant figure always erect and imposing. All of us had seen this during the last academic sessions, in June of last year.

After a peaceful vacation in Switzerland, he returned to Italy in September and resumed the life he had led in Turin since he retired. A solitary life, very methodical, according to his custom, but illuminated by long hours of reading and studying at his work table and interspersed with visits to the Turin Academy of Sciences, whose activity he scrupulously followed, and, by participation in some conferences of interest to him, at the Turin Mathematical Seminar. A few days ago he had attended a conference on relativity at the Turin Academy of Sciences and participated in a numerical analysis conference at the Mathematical Institute, when an illness, as sudden as a thunderbolt, struck him on the evening of 19 October, while he was in his studio looking for a publication in his highly ordered miscellany which he wished to consult. Transported to the hospital, the doctors immediately recognised the extreme severity of the disease. However, his very strong constitution lasted for over a month. He died in the early afternoon of 21 November, after long suffering endured with virile courage.
According to his wishes, after his death in a hospital in Turin from a circulatory disease, he was buried in the small cemetery of Torre Pellice.

### References (show)

1. R Bellman, Review: Lezioni sulle equazioni a derivate parziali, by F G Tricomi, Bull. Amer. Math. Soc. 61 (5) (1955), 87-88.
2. A H Black, Review: Lezioni di Analisi Matematica, by Francesco G Tricomi, Amer. Math. Monthly 50 (2) (1943), 118.
3. J C Burkill, Review: Differential Equations, by Francesco G Tricomi, The Mathematical Gazette 46 (358) (1962), 362-363.
4. J B Diaz, Review: Integral Equations, by Francesco G Tricomi, Quarterly of Applied Mathematics 17 (1) (1959), 66; 94.
5. J B Diaz, Review: Vorlesungen Über Orthogonalreihen by Francesco G Tricomi, Quarterly of Applied Mathematics 14 (4 (1957), 404.
6. G C Dong and M Y Chi, Influence of Tricomi's mathematical work in China, in Mixed type equations (Leipzig, 1986).
7. A Erdélyi, Review: Lezioni sulle funzioni ipergeometriche confluenti, by F G Tricomi, Bull. Amer. Math. Soc. 60 (1954), 185-189.
8. A Erdélyi, Review: Funzioni ipergeometriche confluenti, by F G Tricomi, Bull. Amer. Math. Soc. 61 (5) (1955), 456-460.
9. A Erdélyi, Review: Vorlesungen über Orthogonalreihen, by F G Tricomi, Bull. Amer. Math. Soc. 67 (5) (1961), 447-449.
10. A Ferri, Review: Aerodinamica Transonica by C Ferrari and F G Tricomi, Quarterly of Applied Mathematics 21 (4) (1964), 362.
11. G Fikera, Francesco Giacomo Tricomi (on the ninetieth anniversary of his birth) (Russian), Uspekhi Mat. Nauk 42 (3)(255)(1987), 203-211.
12. G Fichera, Francesco Giacomo Tricomi, in J M Rassias (ed.), Mathematical analysis (Leipzig, 1985), 6-31.
13. G Fichera, Francesco Giacomo Tricomi (Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 114 (1-2) (1980/81), 37-48.
14. G Fichera, Francesco Giacomo Tricomi (Italian), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 66 (5) (1979), 467-483.
15. G Fichera, Francesco Giacomo Tricomi (on his ninetieth birthday) (Russian), Uspekhi Mat. Nauk 42 (3)(255) (1987), 203-211.
16. A E Heins, Review: Integral equations, by F G Tricomi, Bull. Amer. Math. Soc. 64 (4) (1958), 197-198.
17. A E Heins, Review: Equazioni a derivate parziali, by F G Tricomi, Bull. Amer. Math. Soc. 65 (3) (1959), 169-170.
18. O Gurel, Review: Differential Equations, by Francesco G Tricomi, SIAM Review 4 (3) (1962), 269-270.
19. E Luciano, Tricomi, Francesco Giacomo, Dizionario Biografico degli Italiani 96 (2019).
20. E Luciano and L Rosso, L'archivio e la biblioteca di Francesco G Tricomi, Rivista di storia dell'Università di Torino VII (2018), 105-327.
21. S G Mikhlin, On Tricomi's works on integral equations, in J M Rassias (ed.), Mixed type equations (Leipzig, 1986), 195-204.
22. F J Murray, Review: Equazioni differenziali, by F G Tricomi, Bull. Amer. Math. Soc. 56 (2) (1950), 195-196.
23. I Opatowski, Review: Equazioni differenziali, by Francesco G Tricomi, Science New Series 109 (2827) (1949), 236.
24. W T Reid, Review: Equazioni differenziali (2nd ed.), by F G Tricomi, Bull. Amer. Math. Soc. 61 (4) (1955), 371-372.
25. G E Raynor, Review: Funzioni Analitiche, by F G Tricomi, Bull. Amer. Math. Soc. 44 (1) (9) (1938), 610-611.
26. G E Raynor, Review: Funzioni Ellittiche, by F G Tricomi, Bull. Amer. Math. Soc. 44 (1) (9) (1938), 610-611.
27. U Richard, Francesco Giacomo Tricomi (Italian), Boll. Un. Mat. Ital. A (6) 1 (1) (1982), 159-170.
28. F Skof, Francesco Giacomo Tricomi (Italian), in C S Roero (ed.), La facoltà di scienze matematiche fisiche naturali di Torino 1848-1998 (Torino, 1999), 598-602.
29. W Strodt, Review: Funzioni Analitiche (2nd ed.), by F G Tricomi, Bull. Amer. Math. Soc. 53 (7) (1947), 739-740.
30. C A Swanson, Review: Fonctions Hypergeometriques Confluentes, by Francesco G Tricomi, Amer. Math. Monthly 68 (1) (1961), 78.
31. F G Tricomi, Principles and maxims from a notebook of Francesco G Tricomi (1897-1978) (Italian), Archimede 31 (1-2) (1979), 15-16
32. F G Tricomi, La mia vita di matematico attraverso la cronistoria dei miei lavori (Bibliografia commentata 1916-1967) (Padova, 1967).
33. F G Tricomi, Ricordi di mezzo secolo di vita matematica torinese, Rend. Sem. Mat. Univ. e Politec. Torino 31 (1971/73), 31-43.
34. Tricomi, Francesco Giacomo, Enciclopedia Italiana III (Appendice) (1961).
35. C Truesdell, Review: Higher Transcendental Functions, by Harry Bateman, Arthur Erdelyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G Tricomi, Amer. Math. Monthly 61 (8) (1954), 576-578.
36. L C Young, Review: Integral Equations, by Francesco G Tricomi, Science New Series 127 (3313) (1958), 1494-1495.

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Written by J J O'Connor and E F Robertson
Last Update January 2021