Families of solutions to related differential equations
Bessel functions describe the radial part of vibrations of a circular membrane .
Bessel functions , first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel , are canonical solutions y (x ) of Bessel's differential equation
x
2
d
2
y
d
x
2
+
x
d
y
d
x
+
(
x
2
−
α
2
)
y
=
0
{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}
for an arbitrary
complex number
α
{\displaystyle \alpha }
, which represents the
order of the Bessel function. Although
α
{\displaystyle \alpha }
and
−
α
{\displaystyle -\alpha }
produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly
smooth functions of
α
{\displaystyle \alpha }
.
The most important cases are when
α
{\displaystyle \alpha }
is an integer or half-integer . Bessel functions for integer
α
{\displaystyle \alpha }
are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates . Spherical Bessel functions with half-integer
α
{\displaystyle \alpha }
are obtained when solving the Helmholtz equation in spherical coordinates .