mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Fuchs' theorem, named after

Lazarus Fuchs Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-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 d ...

, states that a

second-order differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and ...

of the form

y'' + p(x)y' + q(x)y = g(x)

has a solution expressible by a generalised

Frobenius series In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form z^2 u'' + p(z)z u'+ q(z) u = 0 with u' \equiv \frac and u' ...

when

p(x)

q(x)

and

g(x)

are analytic at

x = a

a

is a

regular singular point In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at ...

. That is, any solution to this second-order differential equation can be written as

y = \sum_^\infty a_n (x - a)^, \quad a_0 \neq 0

for some positive real ''s'', or

y = y_0 \ln(x - a) + \sum_^\infty b_n(x - a)^, \quad b_0 \neq 0

for some positive real ''r'', where ''y''₀ is a solution of the first kind. Its radius of convergence is at least as large as the minimum of the radii of convergence of

p(x)

q(x)

and

g(x)

References

* . * {{Citation , last=Butkov , first=Eugene , title=Mathematical Physics , location=Reading, MA , publisher=Addison-Wesley , year=1995 , isbn=0-201-00727-4 . Differential equations Theorems in analysis

See also

References