# Bessel function

families of solutions to related differential equations

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0}$

for an arbitrary complex number α (the order of the Bessel function). Although α and −α produce the same differential equation for real α, 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.

The most important cases are for α an integer or half-integer. Bessel functions for integer α 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 are obtained when the Helmholtz equation is solved in spherical coordinates.

## Quotes

• When I see equations, I see the letters in colors – I don’t know why. As I’m talking, I see vague pictures of Bessel functions from Jahnke and Ernde’s book, with light-tan j’s, slightly violet-bluish n’s, and dark brown x’s flying around. And I wonder what the hell it must look like to the students.
• Richard Feynman, What Do You Care What Other People Think? (1988), It’s as Simple as One, Two, Three…
• I remember... at a Board meeting at Cambridge, the subject of Bessel's functions came into the discussion... to include them in an examination syllabus. Their utility in connection with Applied Mathematics having been referred to, a very great Pure Mathematician who was present ejaculated—"Yes, Bessel's functions are very beautiful functions, in spite of their having practical applications." It would have been interesting to have heard what this great man would have said if he had known that Professor Perry would one day propose the desecration of these beautiful functions by recommending them as suitable playthings for young boys.
• [To use spherical harmonies or Bessel functions is] to be able to start in mathematics in Cambridge just about the place where some of the best mathematical men now end their studies forever.