🛠️ Modified Spherical Bessel Function First Kind 計算機

This tool tabulates and graphs the modified spherical Bessel function of the first kind, i_v(x), for a fixed order v over a sequence of x values. Starting from an initial x, it adds a fixed step a chosen number of times, producing rows x_k = initialX + k · stepX for k = 0, 1, \dots, loopCount-1, and evaluates i_v(x_k) for each.

The modified spherical Bessel function is defined through the modified (cylindrical) Bessel function of the first kind I by

For low non-negative integer orders there are convenient hyperbolic closed forms: i_0(x) = ((\sinh(x)) ÷ (x)), i_1(x) = ((x \cosh x - \sinh x) ÷ (x^2)), i_2(x) = (((x^2+3) \sinh x - 3x \cosh x) ÷ (x^3)). Higher integer orders follow the recurrence i_{n+1}(x) = i_{n-1}(x) - ((2n+1) ÷ (x)) i_n(x). For general real v the calculator evaluates I_{v+((1) ÷ (2))}(x) from its power series using the Gamma function.

Enter the order v (for example 0, 1 or a half-integer like 0.5), the initial x value, the increment, and how many rows you want. The result shows a two-column table of x and i_v(x); the first value is highlighted at the top. Use a small step such as 0.1 for a smooth curve.

With v = 0, initialX = 0, stepX = 0.1, loopCount = 51, the function i_0(x) = ((\sinh(x)) ÷ (x)) is used. The first row at x = 0 gives the limiting value 1. At x = 1,

At x = 5 (the last row), so the curve rises smoothly from 1 to about 14.84.