Lazarus Fuchs

Lazarus Fuchs
File:Lazarus Fuchs (HeidICON 33457) (cropped).jpg Fuchs in 1884
Born	(1833-05-05)5 May 1833 Moschin, Grand Duchy of Posen, Kingdom of Prussia
Died	26 April 1902(1902-04-26) (aged 68) Berlin, Kingdom of Prussia, German Empire
Alma mater	University of Berlin
Known for	Fuchs relation Fuchs's theorem Fuchsian groups Fuchsian model Fuchsian theory Picard–Fuchs equation
Scientific career
Institutions	University of Greifswald University of Heidelberg University of Berlin University of Göttingen
Doctoral advisor	Karl Weierstraß
Doctoral students	Gerhard Hessenberg Edmund Landau Hermann Schapira Ludwig Schlesinger Issai Schur Theodor Vahlen Ernst Zermelo

Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German mathematician who made important contributions to the field of linear differential equations. He was born in Moschin in the Grand Duchy of Posen (modern-day Mosina, Poland) and died in Berlin, Germany. He was buried in Schöneberg in the St. Matthew's Cemetery. His grave in section H is preserved and listed as a grave of honour of the State of Berlin.

Contribution

He is the eponym of Fuchsian groups and functions, and the Picard–Fuchs equation. A singular point a of a linear differential equation

$y^{″}+p(x)y^{\prime }+q(x)y=0$

is called Fuchsian if p and q are meromorphic around the point a, and have poles of orders at most 1 and 2, respectively. According to a theorem of Fuchs, this condition is necessary and sufficient for the regularity of the singular point, that is, to ensure the existence of two linearly independent solutions of the form

$y_j=\sum _{n=0}^{\mathrm{\infty }}{a}_{j,n}(x-x_0{)}^{n+{\sigma }_j},a_0\ne 0j=1,2.$

where the exponents ${\sigma }_j$ can be determined from the equation. In the case when ${\sigma }_1-{\sigma }_2$ is an integer this formula has to be modified.

Another well-known result of Fuchs is the Fuchs's conditions, the necessary and sufficient conditions for the non-linear differential equation of the form

$F(\frac{dy}{dz},y,z)=0$

to be free of movable singularities.

An interesting remark about him as a teacher during the period of his work at the Heidelberg University pertains to his manner of lecturing: his knowledge of the mathematics he was assigned to teach was so deep that he would not prepare before giving a lecture — he would simply improvise on the spot, while exposing the students to the train of thought taken by mathematicians of the finest degree.

Lazarus Fuchs was the father of Richard Fuchs, a German mathematician.

Selected works

• Über Funktionen zweier Variabeln, welche durch Umkehrung der Integrale zweier gegebener Funktionen entstehen, Göttingen 1881.
• Zur Theorie der linearen Differentialgleichungen, Berlin 1901.
• Gesammelte Werke, Hrsg. von Richard Fuchs und Ludwig Schlesinger. 3 Bde. Berlin 1904–1909.

External links

• Error creating thumbnail: File missing Media related to at Wikimedia Commons
• G. B. Mathews (1902) Lazarus Fuchs Nature 66:156,7 (#1702).
• Lazarus Fuchs at the Mathematics Genealogy Project