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 ′ + q ( x ) y = 0 {\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
y j = ∑ n = 0 ∞ a j , n ( x − x 0 ) n + σ j , a 0 ≠ 0 j = 1 , 2. {\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 σ j {\displaystyle \sigma _{j}} can be determined from the equation. In the case when σ 1 − σ 2 {\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
F ( d y d z , y , z ) = 0 {\displaystyle F\left({\frac {dy}{dz}},y,z\right)=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.
References
External links Media related to Lazarus Immanuel Fuchs at Wikimedia Commons Jeremy Gray (1984). "Fuchs and the theory of differential equations". Bulletin of the AMS. New Series. 10 (1): 1–26. doi:10.1090/S0273-0979-1984-15186-3. MR 0722855. G. B. Mathews (1902) Lazarus Fuchs Nature 66:156,7 (#1702). O'Connor, John J.; Robertson, Edmund F., "Lazarus Fuchs", MacTutor History of Mathematics Archive, University of St Andrews Lazarus Fuchs at the Mathematics Genealogy Project
