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[de], 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