Lazarus Fuchs

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.

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.

• 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.