Bessel functions

x²y'' + xy' + (x²-ν²)y = 0

which naturally appears in problems displaying cylindrical symmetries.

The Bessel function of the first kind is defined for parameter ν being a real number:

J_ν(x) = x^ν Sum_m=0:inf (-1)^mx^2m / 2^2m+νm!Γ(ν+m+1)

Parameter ν is often an integer, so that ν = n; we then talk about the Bessel function of the first kind of order n:

J_n(x) = xⁿ Sum_m=0:inf (-1)^mx^2m / 2^2m+nm!(n+m)!

Bessel functions of the second kind, or Neumann's function, are also solutions to the Bessel equation:

Y_ν(x) = (1/sinνπ) (J_ν(x)cosνπ - J_-ν(x))

When ν = n, the definition for the Bessel function of the second kind of order n becomes:

Y_n(x) = lim_ν->n Y_ν(x)

When the solution seeked must be complex for real values of x, Bessel functions of the third kind or Hankel functions are used:

H_ν⁽¹⁾ = J_ν(x) + iY_ν(x)
H_ν⁽²⁾ = J_ν(x) - iY_ν(x)

Modified Bessel function I_ν(x) = i^-νJ_ν(ix) is a solution to the modified Bessel equation

x²y'' + xy' - (x² + ν²)y = 0

Of course, there is also the Modified Bessel function of the second kind (also often called of the third kind for a reason unknown to me):

K_ν(x) = (π/2sinνπ)(I_-ν(x) - I_ν(x))

There are entire books devoted to the properties of Bessel functions; however, here's a short and useful list:

• J_-n(x) = (-1)ⁿJ_n(x);
• (d/dx)(x^νJ_ν(x)) = x^νJ_ν-1(x);
• (d/dx)(x^-νJ_ν(x)) = -x^-νJ_ν+1(x);
• J_ν-1(x) + J_ν+1(x) = (2ν/x)J_ν(x);
• J_ν-1(x) - J_ν+1(x) = 2J'_ν(x);
• J_1/2(x) = sqrt(2/πx)sinx;
• J_-1/2(x) = sqrt(2/πx)cosx

Primary source: Advanced Engineering Mathematics, Erwin Kreyszig

The Bessel functions may be defined by means of the generating function:
g(x,t)=e^x/2*(t-1/t) = J_ntⁿ
Equating coefficients of t^n on both sides we obtain the series expansion of the Bessel functions:
J_n = sum((-1)^k * (x/2)^(n+2k)/(k!(n+k)!))

Differentiating the generating function w.r.t t we get
(1+t^(-2))g(x,t)=sum(n*J_ntⁿ)
equating coefficients of t^n on both sides we obtain the recursion relation
(2n/x)*J_n = J_n-1 + J_n+1 Differentiating the generating function w.r.t x gives
2*J'_n = J_n-1 - J_n+1

These recursion relations may now be used to prove that J_n satisfies Bessel's differential equation:
x²J_n + xJ_n+(x² - n²)J_n = 0 The generating function may be used to prove many other properties. For exampe parity
J_n(-x)=(-1)^{nn(x) J_-n(x) = (-1)nn(x)}

^{If Bessel's differential equation is divided by x it becomes self-adjoint and thus we expect its eigenfunctions to be orthogonal(with a weighting function of x). Orthogonality for the Bessel functions is rather special. Here is not J_n and J_m which are orthogonal but J_p(a_n) and J_p(a_m) which are orthogonal where a_n and a_m are two zeroes of J_p. Thus we may expand any function in a series of zeroes of one Bessel function.}

^{Arfken is a good reference for Special functions in general. Of course the Bible of special functions is the book by Abramowitz.}