HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 533

Bessel's equation The ordinary differential equation x2⋅(d2y/dx2) + x⋅ (dy/dx)+ (x2-ν2) ⋅y = 0, where the parameter ν is a nonnegative real number, is known as Bessel's differential equation. Solutions to Bessel's equation are given in terms of Bessel functions of the first kind of order ν: ∑ Jν (x) = xν ∞ ⋅ m=0 (−1)m ⋅ x 2m 22m+ν ⋅ m!⋅Γ(ν + m , + 1) where ν is not an integer, and the function Gamma Γ(α) is defined in Chapter 3. If ν = n, an integer, the Bessel functions of the first kind for n = integer are defined by ∑ J n (x) = xn ⋅ ∞ m=0 (−1)m ⋅ x 2m 22m+n ⋅ m!⋅(n + . m)! Regardless of whether we use ν (non-integer) or n (integer) in the calculator, we can define the Bessel functions of the first kind by using the following finite series: Thus, we have control over the function's order, n, and of the number of elements in the series, k. Once you have typed this function, you can use function DEFINE to define function J(x,n,k). This will create the variable @@@J@@@ in the soft-menu keys. For example, to evaluate J3(0.1) using 5 terms in the series, calculate J(0.1,3,5), i.e., in RPN mode: .1#3#5@@@J@@@ The result is 2.08203157E-5. Page 16-55