Gösta Mittag-Leffler

How to use this course

Gösta Mittag-Leffler is known to every student of complex analysis for one theorem: a meromorphic function can be built with poles and principal parts prescribed at will. To historians he is better known as an organizer. He founded Acta Mathematica in 1882 and made it, within a decade, one of the leading journals in the world. He published Georg Cantor when Berlin would not, launched Henri Poincaré’s work on Fuchsian functions, arranged the King Oscar prize that produced Poincaré’s memoir on the three-body problem, and secured for Sofia Kovalevskaya the first full professorship of mathematics held by a woman in modern Europe. His villa at Djursholm became the Institut Mittag-Leffler, still a leading research institute.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–06 follow the theorem from Weierstrass’s products to its general form of 1884. Lessons 07–08 treat his later work on analytic continuation and summation. Lessons 09–10 follow two ideas descended from his theorem into twentieth-century mathematics.

Timeline

Stockholm and Uppsala, 1846–1873

Gösta Leffler was born in Stockholm on 16 March 1846. His father, Johan Olof Leffler, was a schoolteacher who became headmaster of a Stockholm school and sat in the Swedish parliament. His mother was Gustava Wilhelmina Mittag. Gösta was the eldest of four children, and all of them made names for themselves. His sister Anne Charlotte Leffler became a well-known writer, his brother Frits a linguist, and his brother Artur a civil engineer. At about twenty Gösta added his mother’s surname to his own, and became Mittag-Leffler.

He entered Uppsala University in 1865, supporting himself in part by private tutoring, and at first trained in actuarial work before settling on mathematics. In 1870 his father fell ill, and the family’s finances became tighter. In 1872 he took his doctorate at Uppsala with a thesis, written under Göran Dillner, on separating the roots of an analytic function of one variable, an application of the argument principle. His biographers have not rated it highly. He was made docent at Uppsala the same year.

Swedish mathematics was then provincial, and Mittag-Leffler knew it. In 1873 he won a travel scholarship, the Byzantine stipend, which allowed him three years abroad. He went first to Paris.

Paris and Berlin, 1873–1876

He arrived in Paris in October 1873 to learn from Charles Hermite. He found Hermite’s lectures hard to follow but formed a lasting friendship with him, and met Jean-Claude Bouquet, Charles Briot, Michel Chasles, Gaston Darboux and Joseph Liouville. Hermite told him, again and again, that the master of analysis was in Berlin, and that he should go to Weierstrass. After a stay in Göttingen, Mittag-Leffler went to Berlin.

Weierstrass’s lectures on the theory of analytic functions changed his mathematics for good. Weierstrass built function theory on power series and analytic continuation, with a rigour unknown elsewhere, and in 1876 he published his theorem on entire functions with prescribed zeros (Lesson 04). Mittag-Leffler set himself the matching problem for poles. He also came to know Leopold Kronecker, and he absorbed the Berlin standard of proof.

In Berlin he heard from Weierstrass about his former private student Sofia Kovalevskaya, then living in Russia and outside mathematics. In 1876 Mittag-Leffler visited her in St Petersburg and was deeply impressed. The friendship that began there would bring her to Stockholm seven years later.

Helsinki, 1877–1881

In 1877 Mittag-Leffler won the competition for the chair of mathematics at the University of Helsinki, then the Imperial Alexander University of the Grand Duchy of Finland, succeeding Lorenz Lindelöf. There he developed his theorem on meromorphic functions in a series of notes, and he trained his first doctoral students, among them Hjalmar Mellin, later known for the Mellin transform, who took his degree in Helsinki in 1882.

In Helsinki he also met Signe Lindfors, daughter of a wealthy Finnish family; they married in 1882. Her fortune would support his projects for the rest of his life.

Meanwhile Stockholm was building a new kind of institution. Stockholms högskola, a private university college supported by donors, opened in 1878 with the aim of research and public lectures, free of the old universities’ regulations. Mittag-Leffler was appointed its first professor of mathematics in 1881. He resigned from Helsinki and came home with two projects: an international journal and a research school in Stockholm.

Weierstrass products

For the general reader. A polynomial is determined, up to a constant, by its roots. Can we build a function with infinitely many prescribed zeros? Simply multiplying the factors \((1-z/a_n)\) usually fails, because the infinite product diverges. Weierstrass found convergence factors that tame the product without adding new zeros. Mittag-Leffler’s theorem is the analogue for poles, and its proof copies Weierstrass’s idea.

Elementary factors

For an integer \(p\ge0\) define

The exponential cancels the first \(p\) terms of \(\log(1-z)=-\sum_k z^k/k\), so \(E_p(z)\) is close to 1 when \(|z|\) is small.

Proof. Fix \(R>0\). For all but finitely many \(n\), \(|a_n|\ge2R\), and then for \(|z|\le R\) the lemma gives \(|1-E_{n-1}(z/a_n)|\le2^{-n}\). A product \(\prod(1+u_n)\) with \(\sum|u_n|\) uniformly convergent converges uniformly, and it vanishes only where a factor does. The last claim follows because the quotient of two entire functions with the same zeros is entire and zero-free, hence of the form \(e^g\) on the simply connected plane.

The Mittag-Leffler theorem in the plane

For the general reader. A rational function can be written as a sum of simple fractions, one for each pole. Mittag-Leffler showed that the same is true for functions with infinitely many poles, once each fraction is corrected by a polynomial so that the infinite sum converges. Any list of poles going off to infinity, with any prescribed singular behaviour at each, belongs to some function. Classical formulas such as the partial-fraction expansion of the cotangent become special cases.

A principal part at a point \(a\) is a polynomial in \(1/(z-a)\) without constant term, \(P\big(\tfrac{1}{z-a}\big)=\sum_{k=1}^{N}\frac{c_k}{(z-a)^k}\).

Proof. If some \(a_n=0\), keep that term without correction. For \(a_n\ne0\), the function \(P_n\big(\tfrac{1}{z-a_n}\big)\) is holomorphic in \(|z|<|a_n|\), so its Taylor series at 0 converges uniformly on \(|z|\le|a_n|/2\). Let \(T_n\) be a partial sum with \(\big|P_n\big(\tfrac{1}{z-a_n}\big)-T_n(z)\big|<2^{-n}\) there. On a disc \(|z|\le R\), all but finitely many \(n\) have \(|a_n|/2\ge R\), so the tail of the series is dominated by \(\sum2^{-n}\), and it converges uniformly to a holomorphic function. The finitely many remaining terms supply the poles. The difference of two solutions has no poles, so it is entire.

The cotangent

The function \(\pi\cot\pi z\) has simple poles of residue 1 at every integer. The principal parts \(\frac{1}{z-n}\) need only the constant correction \(T_n=-\frac1n\):

To prove equality, call the difference \(g\). It is entire, and differentiating gives \(g'(z)=-\frac{\pi^2}{\sin^2\pi z}+\sum_{n\in\mathbb Z}\frac{1}{(z-n)^2}\). This is entire, periodic with period 1, and bounded as \(|\operatorname{Im}z|\to\infty\) (both terms tend to 0), so by Liouville’s theorem it is a constant, which must be 0. Thus \(g\) is constant, and since \(g\) is odd, \(g=0\).

Arbitrary open sets and Cantor’s point sets

For the general reader. Mittag-Leffler wanted his theorem not only for the whole plane but for any region, where poles may pile up at the boundary instead of at infinity. To describe how a set of points can accumulate, he used the new language of Georg Cantor’s set theory, at a time when most mathematicians regarded it with suspicion. His 1884 memoir in Acta Mathematica was one of the first places where Cantor’s ideas did real work in analysis.

In Mittag-Leffler’s formulation, one starts with a set \(Q\) of isolated points in the extended plane and its set of limit points \(Q'\), which Cantor called the derived set. Any prescribed principal parts at the points of \(Q\) belong to a single-valued function that is meromorphic on the complement of \(Q'\). Taking \(Q'=\partial\Omega\) recovers the modern statement.

Proof sketch

Exhaust \(\Omega\) by compact sets \(K_1\subset K_2\subset\cdots\), each in the interior of the next, such that every bounded component of \(\mathbb C\setminus K_n\) contains a point outside \(\Omega\). Let \(Q_n\) be the sum of the principal parts at the finitely many points of \(A\cap(K_n\setminus K_{n-1})\). It is holomorphic near \(K_{n-1}\). By Carl Runge’s theorem of 1885, it can be approximated uniformly on \(K_{n-1}\), within \(2^{-n}\), by a rational function \(R_n\) whose poles lie outside \(\Omega\). Then \(f=\sum_n(Q_n-R_n)\) converges uniformly on each \(K_m\) after discarding finitely many terms, and has the right poles. When \(\Omega=\mathbb C\), the poles of \(R_n\) can be put at infinity, so \(R_n\) is a polynomial, as in Lesson 05.

Combined with Weierstrass’s product theorem for open sets, the result gives two classical corollaries: every meromorphic function on \(\Omega\) is a quotient of two holomorphic functions on \(\Omega\), and every open set is the natural domain of existence of some holomorphic function.

The star and analytic continuation

For the general reader. A power series converges only in a disc, whose edge is set by the nearest singularity, even though the function it represents may extend much further. Weierstrass extended functions by re-expanding series at new centres, chain after chain. Mittag-Leffler wanted a single formula, computed from the original Taylor coefficients, that converges to the function in as large a region as possible. The natural region is a star, and between 1899 and 1905 he gave such formulas in five notes in Acta.

The star

Let \(f(z)=\sum_{k\ge0}a_kz^k\) have positive radius of convergence. Say that a point \(z\) belongs to the star \(S_f\) if \(f\) can be continued analytically along the whole segment \([0,z]\), that is, along \(t\mapsto tz\), \(0\le t\le1\). On each ray from 0 the continuation stops at the first singular point met, so \(S_f\) is the plane with a closed ray removed behind each singularity visible from the origin. It is open and star-shaped about 0, contains the disc of convergence, and on it the continuation of \(f\) is single-valued.

Why the second formula works. For \(\alpha>0\), the factor \(1/\Gamma(1+\alpha k)\) decays faster than any geometric sequence, so the series defines an entire function of \(z\). Writing \(a_k=\frac{1}{2\pi i}\oint f(w)w^{-k-1}\,dw\) and summing under the integral,

As \(\alpha\to0^+\), \(E_\alpha(t)\) tends to \(\frac1{1-t}\) for \(t\) outside a sector around \([1,\infty)\) whose opening shrinks with \(\alpha\), and in that limit the integral is Cauchy’s formula for \(f(z)\). Since the kernel is large only when \(z/w\) lies near \([1,\infty)\), that is, when \(w\) lies near the segment \([0,z]\), the contour can be deformed far into the star while avoiding that segment. Making this precise needs the asymptotics of \(E_\alpha\) in Lesson 08.

Émile Borel’s method of summation, based on the Laplace transform, converges in a smaller region, the Borel polygon, which is always contained in the star.

The function \(E_\alpha\)

For the general reader. The exponential function solves the simplest growth and decay equations. In 1903 Mittag-Leffler introduced a family of functions \(E_\alpha\) that interpolate between the exponential and other elementary functions, as a tool for his summation method. A century later they became the basic special functions of fractional calculus, describing anomalous diffusion and the slow relaxation of viscoelastic materials, polymers and glasses.

Special cases: \(E_1(z)=e^z\), \(E_2(z^2)=\cosh z\), \(E_2(-z^2)=\cos z\), and \(E_{1/2}(z)=e^{z^2}\operatorname{erfc}(-z)\). By Stirling’s formula \(E_\alpha\) is entire of order \(1/\alpha\).

These asymptotics are what Lesson 07 needed: as \(\alpha\to0^+\), the sector where \(E_\alpha\) is exponentially large closes up on the positive real axis, and elsewhere \(E_\alpha(z)\to-\sum_{k\ge1}z^{-k}=\frac1{1-z}\).

Fractional relaxation

For \(0<\alpha\le1\), the Caputo fractional derivative of a function \(y\) on \(t>0\) is \(D^\alpha y(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t\frac{y'(s)}{(t-s)^{\alpha}}\,ds\) for \(\alpha<1\), and \(D^1y=y'\). On powers, \(D^\alpha t^\beta=\frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}\) for \(\beta>0\), and \(D^\alpha1=0\).

Several variables: the Cousin problem

For the general reader. In one complex variable, Mittag-Leffler’s theorem always works. In two or more variables, singularities are no longer isolated points but whole surfaces, and the question becomes whether local data can be glued into one global function. Pierre Cousin posed this in 1895. It does not always have a solution, and the search for the right conditions led to sheaf cohomology, one of the central tools of twentieth-century geometry.

Mittag-Leffler data

Let \(X\) be a complex manifold with an open cover \(\{U_i\}\). Cousin I data are meromorphic functions \(f_i\) on \(U_i\) such that \(f_i-f_j\) is holomorphic on \(U_i\cap U_j\). A solution is a global meromorphic \(f\) with \(f-f_i\) holomorphic on each \(U_i\). Prescribing principal parts at isolated points is the special case of Lesson 06.

Idea of (i). The differences \(g_{ij}=f_i-f_j\) are holomorphic on overlaps and satisfy \(g_{ij}+g_{jk}=g_{ik}\): they form a Čech 1-cocycle with values in \(\mathcal O\). If \(H^1(X,\mathcal O)=0\) it is a coboundary, \(g_{ij}=h_i-h_j\) with \(h_i\) holomorphic on \(U_i\). Then \(f=f_i-h_i\) agrees on overlaps and solves the problem.

Idea of (ii). Using a partition of unity, the cocycle splits smoothly as \(g_{ij}=\varphi_i-\varphi_j\) with \(\varphi_i\) smooth. Then \(\bar\partial\varphi_i=\bar\partial\varphi_j\) on overlaps, so these patch to a global smooth \((0,1)\)-form \(v\,d\bar z\). If \(u\) solves \(\partial u/\partial\bar z=v\), the functions \(h_i=\varphi_i-u\) are holomorphic and \(h_i-h_j=g_{ij}\). In one variable \(\partial u/\partial\bar z=v\) is always solvable on any open set; for compactly supported \(v\) the solution is

and an exhaustion argument with Runge’s theorem, the same one as in Lesson 06, handles general \(v\). This proof, from Lars Hörmander’s book, reduces Mittag-Leffler’s theorem to the solvability of the \(\bar\partial\)-equation.

The Mittag-Leffler condition

For the general reader. The proof of Mittag-Leffler’s theorem builds a function in stages, one compact region at a time, and makes the stages fit by small corrections. Nicolas Bourbaki and later algebraists isolated the abstract pattern: when do compatible pieces of information at each stage come from a single global object? The answer, a condition on images in an inverse system, carries Mittag-Leffler’s name throughout algebra, topology and algebraic geometry.

Towers and their limits

A tower of abelian groups is a sequence \(A_1\leftarrow A_2\leftarrow A_3\leftarrow\cdots\) of homomorphisms \(f_n:A_{n+1}\to A_n\). Define \(d:\prod_nA_n\to\prod_nA_n\) by \(d(a)_n=a_n-f_n(a_{n+1})\). Then

A short exact sequence of towers \(0\to A\to B\to C\to0\) gives, by the snake lemma applied to \(d\), an exact sequence

Proof sketch. If all \(f_n\) are surjective, solve \(a_n-f_n(a_{n+1})=b_n\) by choosing \(a_1=0\) and then each \(a_{n+1}\) as a preimage of \(a_n-b_n\) (Exercise 10.2). In general, the stable images \(A_n'\) form a tower with surjective maps, and the quotient tower \(A_n/A_n'\) has \(\varprojlim^1=0\) because its maps are eventually zero along any composite, which lets one write down a solution of \(d(a)=b\) as a finite sum at each level.

Topological version

Applied to the spaces of functions meromorphic near the compact sets \(K_n\) of Lesson 06 with the prescribed principal parts, with the topology of uniform convergence, density of the maps is exactly Runge’s theorem, and the theorem returns Mittag-Leffler’s. The condition appears in the Milnor exact sequence for homotopy groups and in Alexander Grothendieck’s theory of formal functions.

Stockholm and Acta Mathematica, 1881–1911

The journal

The idea of a Scandinavian mathematics journal came from Sophus Lie in 1881. Mittag-Leffler made it international. Acta Mathematica appeared in 1882, funded by subscriptions, an appeal to donors and his wife’s fortune; King Oscar II was its first subscriber. Its first volume opened with Henri Poincaré’s Théorie des groupes fuchsiens, and Poincaré’s papers on automorphic functions filled much of the early volumes. In 1883 Mittag-Leffler printed French translations of Georg Cantor’s papers on set theory, which German journals had been reluctant to accept. Leopold Kronecker, Cantor’s fiercest critic, refused to publish in Acta. Mittag-Leffler was protective of Cantor but also cautious: in 1885 he persuaded him to withdraw a paper on order types, judging it far ahead of its time. Cantor took the advice badly and their friendship cooled.

Mittag-Leffler edited Acta for 45 years. He judged papers shrewdly, published French and German authors side by side in the years after 1870, when their national journals barely spoke to each other, and wrote tirelessly to bring in the best work. G. H. Hardy, who knew him late in life, called him a mathematician of the front rank and a fine judge of the quality of others’ work.

Kovalevskaya

As soon as he held the Stockholm chair, Mittag-Leffler worked to bring Sofia Kovalevskaya there. The private Stockholms högskola could appoint a woman where the state universities could not. She arrived late in 1883, began lecturing early in 1884, and in June 1884 was appointed professor for five years. She joined the editorial board of Acta. His sister Anne Charlotte Leffler became her close friend, and together they wrote the play The Struggle for Happiness (1887). After Kovalevskaya won the Prix Bordin in 1888, he secured her appointment as professor for life in 1889. When she died in February 1891, he wrote about her life and work and defended her scientific reputation.

The King Oscar prize

In 1885 Mittag-Leffler announced in Acta a prize competition for the sixtieth birthday of King Oscar II of Sweden and Norway. The jury was Mittag-Leffler, Weierstrass and Hermite. The first of four questions asked for series solutions of the \(n\)-body problem converging uniformly for all time. Twelve entries arrived by June 1888, and on the king’s birthday, 21 January 1889, the prize went to Poincaré for his memoir on the three-body problem. While the memoir was being printed, the editor Edvard Phragmén queried some convergence arguments. Poincaré found a far deeper error, and the printed copies had to be recalled and destroyed. Mittag-Leffler handled the affair with discretion; the corrected memoir filled Acta 13 in 1890 and became the starting point of chaos theory.

The Stockholm school

In Stockholm he gathered a remarkable group: Kovalevskaya; Edvard Phragmén, later known for the Phragmén–Lindelöf principle; Ivar Bendixson, of the Poincaré–Bendixson theorem; Helge von Koch, whose snowflake curve appeared in 1904 and who took his doctorate with Mittag-Leffler in 1892; and Ivar Fredholm, who studied with him in Stockholm and published his theory of integral equations in Acta in 1903. Mittag-Leffler wanted the högskola to remain a research school. Bendixson, who wanted a more conventional university, prevailed in the arguments over its future. Mittag-Leffler retired from his chair in 1911.

Djursholm, death, legacy, 1889–1927

From 1889 Mittag-Leffler bought land at Djursholm, a new garden suburb north of Stockholm, and by April 1891 the family villa was complete. It was enlarged around 1900 and 1906. The house held one of the great mathematical libraries, with books going back to the sixteenth century, and from 1904 to 1914 he hosted a celebration there each December during the Nobel festivities. Hardy, who visited him in the country, saw in him both the cosmopolitan scholar and the rural landowner. He was also a businessman: he helped found an insurance company in 1882 and later invested in industrial ventures.

Mittag-Leffler was active in the early Nobel Prizes. In 1903 he alerted Pierre Curie that the nomination for the physics prize named only Pierre and Henri Becquerel; Pierre insisted, and Marie Curie shared the prize. He later campaigned, without success, for a physics prize for Poincaré. Stories that Alfred Nobel left out mathematics because of a rivalry with Mittag-Leffler have no basis in the evidence.

Honours came from across Europe. He was elected to the Royal Swedish Academy of Sciences in 1883, an honorary member of the London Mathematical Society in 1892, a foreign member of the Royal Society in 1896 and a member of the Paris Academy of Sciences in 1900. He spoke at the International Congresses of Mathematicians of 1900 and 1908, and received honorary degrees from Oxford, Cambridge, Aberdeen, St Andrews, Bologna and Christiania (Oslo).

The Institute

On his seventieth birthday, 16 March 1916, Gösta and Signe Mittag-Leffler made public their will. It left the villa, the library and their fortune to a foundation to promote pure mathematics in the Scandinavian countries, with a research institute, professorships and fellowships. The foundation was placed under the Royal Swedish Academy of Sciences in 1919. The war and inflation eroded the capital; Signe died in 1921, and the crash of 1922 brought Mittag-Leffler close to bankruptcy. He died at Djursholm on 7 July 1927, aged 81.

Torsten Carleman directed the institute until his death in 1949, after which it was for nearly twenty years little more than a library. In 1969 Lennart Carleson revived it as Mittag-Leffler had intended, a centre for visiting researchers and research programmes, which it remains. Acta Mathematica is still published by the institute and is among the most selective journals in mathematics.

Legacy

• Complex analysis: the Mittag-Leffler theorem, with Weierstrass’s product theorem, is the foundation of the theory of meromorphic functions on arbitrary domains.
• Summation and continuation: the Mittag-Leffler star and summation method, and the functions \(E_\alpha\) and \(E_{\alpha,\beta}\), now central to fractional calculus.
• Algebra and geometry: the Cousin problems and the Mittag-Leffler condition carry his idea of gluing local data into sheaf cohomology and homological algebra.
• Institutions: Acta Mathematica and the Institut Mittag-Leffler; a glacier on Spitsbergen also bears his name.

Connections

People linked to Gösta Mittag-Leffler across this collection. A name without a link has no page here yet. See Gösta Mittag-Leffler in the network of all 337 lives →

Teachers and advisors

• Karl Weierstrass
• Göran Dillner 1872

Students

• Ivar Fredholm
• Helge von Koch 1892
• Hjalmar Mellin 1882

Influenced by

• Charles Hermite 1873-1901
• Karl Weierstrass

Colleagues and collaborators

• Charles Hermite
• G. H. Hardy
• Gaston Darboux
• Georg Cantor
• Henri Poincaré
• Marie Curie
• Pierre Curie
• Sofia Kovalevskaya 1876-1891
• Sophus Lie
• Thomas Joannes Stieltjes 1885-1887
• Edvard Phragmén
• Ivar Bendixson

Rivals

• Leopold Kronecker

Also mentioned in this life

• Isaac Barrow
• Henri Becquerel
• Émile Borel
• Augustin-Louis Cauchy
• Michel Chasles
• Leonhard Euler
• Alexander Grothendieck
• Pierre-Simon Laplace
• Joseph Liouville
• John Milnor

Fields

• complex analysis 13 lives

On CoursesHub

• Complex Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Magnus Gösta Mittag-Leffler, MacTutor History of Mathematics.
2. Gösta Mittag-Leffler, Wikipedia.
3. Institut Mittag-Leffler, History and The prize competition of King Oscar II.
4. Obituary, The Times, July 1927 (MacTutor).
5. A. Stubhaug, Gösta Mittag-Leffler: A Man of Conviction, Springer, 2010; reviews.
6. G. H. Hardy, Gösta Mittag-Leffler, J. London Math. Soc. 3 (1928), 156–160.
7. Gösta Magnus Mittag-Leffler, Mathematics Genealogy Project.
8. J. Barrow-Green, Poincaré and the Three Body Problem, American Mathematical Society, 1997.

Mathematics

1. G. Mittag-Leffler, Sur la représentation analytique des fonctions monogènes uniformes d’une variable indépendante, Acta Math. 4 (1884), 1–79.
2. G. Mittag-Leffler, Sur la représentation analytique d’une branche uniforme d’une fonction monogène, five notes, Acta Math. 23–29 (1900–1905).
3. G. Mittag-Leffler, Sur la nouvelle fonction \(E_\alpha(x)\), C. R. Acad. Sci. Paris 137 (1903), 554–558.
4. L. E. Turner, The Mittag-Leffler theorem: the origin, evolution, and reception of a mathematical result, 1876–1884, Historia Mathematica 40 (2013), 36–83.
5. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1987, chapters 13 and 15.
6. G. H. Hardy, Divergent Series, Oxford, 1949 (Mittag-Leffler summation and the star).
7. R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, 2014; 2nd ed. 2020.
8. L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 3rd ed. 1990.
9. C. A. Weibel, An Introduction to Homological Algebra, Cambridge, 1994, §3.5 (\(\varprojlim^1\) and the Mittag-Leffler condition).