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Lazarus Fuchs

Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German[1] mathematician who contributed important research in the field of linear differential equations.[2] He was born in Moschin (Mosina) (located in Grand Duchy of Posen) 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.

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

${\displaystyle y''+p(x)y'+q(x)y=0}$

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

${\displaystyle y_{j}=\sum _{n=0}^{\infty }a_{j,n}(x-x_{0})^{n+\sigma _{j}},\quad a_{0}\neq 0\,\quad j=1,2.}$

where the exponents ${\displaystyle \sigma _{j}}$ can be determined from the equation. In the case when ${\displaystyle \sigma _{1}-\sigma _{2}}$ 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

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

to be free of movable singularities.

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.

References

1. ^ O'Connor, John J.; Robertson, Edmund F., "Lazarus Immanuel Fuchs", MacTutor History of Mathematics archive, University of St Andrews.
2. ^ Wilczynski, E. J. (1902). "Lazarus Fuchs". Bull. Amer. Math. Soc. 9 (1): 46–49. doi:10.1090/s0002-9904-1902-00952-x. MR 1557937.
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