Quick Facts
Biography
Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a German mathematician who contributed important research in the field of linear differential equations. 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
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 at 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 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.
Lasarus 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.