Bessel functions are all solutions to the Bessel equation

x2y'' + xy' + (x22)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:

Jν(x) = xν Summ=0:inf (-1)mx2m / 22mm!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:

Jn(x) = xn Summ=0:inf (-1)mx2m / 22m+nm!(n+m)!

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

Yν(x) = (1/sinνπ) (Jν(x)cosνπ - J(x))

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

Yn(x) = limν->n Yν(x)

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

Hν(1) = Jν(x) + iYν(x)
Hν(2) = Jν(x) - iYν(x)

Modified Bessel function Iν(x) = iJν(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):

Kν(x) = (π/2sinνπ)(I(x) - Iν(x))

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

• J-n(x) = (-1)nJn(x);
• (d/dx)(xνJν(x)) = xνJν-1(x);
• (d/dx)(xJν(x)) = -xJν+1(x);
• Jν-1(x) + Jν+1(x) = (2ν/x)Jν(x);
• Jν-1(x) - Jν+1(x) = 2J'ν(x);
• J1/2(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)=ex/2*(t-1/t) = Jntn
Equating coefficients of t^n on both sides we obtain the series expansion of the Bessel functions:
Jn = 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*Jntn)
equating coefficients of t^n on both sides we obtain the recursion relation
(2n/x)*Jn = Jn-1 + Jn+1
Differentiating the generating function w.r.t x gives
2*J'n = Jn-1 - Jn+1

These recursion relations may now be used to prove that Jn satisfies Bessel's differential equation:
x2Jn + xJn+(x2 - n2)Jn = 0
The generating function may be used to prove many other properties. For exampe parity
Jn(-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 Jn and Jm which are orthogonal but Jp(an) and Jp(am) which are orthogonal where an and am are two zeroes of Jp. 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.

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