📐 Bessel Function First Kind Table 計算機

This tool tabulates the Bessel function of the first kind, written J_v(x), for a fixed order v while sweeping the argument x. You choose a starting x value, an increment, and how many rows to generate, and the calculator returns a clean two-column table of x versus J_v(x). Bessel functions of the first kind appear throughout physics and engineering: vibrations of a circular drum, heat conduction in cylinders, electromagnetic waves in waveguides, and signal processing (FM modulation sidebands).

Enter the Order v (any real number — 0, 1, 2, fractional like 0.5, or negative). Set the Initial value of x, the Increment (the spacing between successive x values; may be negative for a descending sweep or zero to repeat one point), and the Number of repetitions (how many rows, from 1 up to 10000). Row i uses x = startX + i × stepX.

The function is defined by the power series

where \Gamma is the gamma function. The calculator evaluates this series term by term using a stable recurrence: each term is obtained from the previous one by multiplying by -(x^2/4) / ((k+1)(k+v+1)), which avoids factorial overflow. The gamma function is computed with the Lanczos approximation so that non-integer and negative orders work. For negative integer order it uses the identity J_{-n}(x) = (-1)^n J_n(x).

With v = 0, startX = 0, stepX = 0.2, loopCount = 6 the table gives J_0(0) = 1, J_0(0.2) ≈ 0.990025, J_0(0.4) ≈ 0.960398, J_0(0.6) ≈ 0.912005, J_0(0.8) ≈ 0.846287, and J_0(1.0) ≈ 0.765198 — matching the standard tabulated value J_0(1) = 0.7651976866.

The table below lists the Bessel function of the first kind J_v(x) for orders v=0,1,2 at several standard arguments. Values are rounded to six decimal places and follow from the series J_{v}(x)=\sum_{k=0}^{\infty}(((-1)^{k}) ÷ (k! \Gamma(v+k+1)))≤ft(((x) ÷ (2))\right)^{2k+v}.

x	J_0(x)	J_1(x)	J_2(x)
0	1.000000	0.000000	0.000000
0.5	0.938470	0.242268	0.030604
1	0.765198	0.440051	0.114903
2	0.223891	0.576725	0.352834
3	−0.260052	0.339059	0.486091
5	−0.177597	−0.327579	0.046565
10	−0.245936	0.043473	0.254630

As a worked check, evaluate J_0(1): the leading terms give 1-\tfrac{(0.5)^2}{1}+\tfrac{(0.5)^4}{4}-\tfrac{(0.5)^6}{36}+\dots = 1-0.25+0.015625-0.000434+\dots≈ 0.765198.

Notable Zeros (Roots)

The positive zeros are the values of x where J_v(x)=0; they set drum modes, waveguide cutoffs and similar boundary conditions.

Root index s	s-th zero of J_0	s-th zero of J_1
1	2.404826	3.831706
2	5.520078	7.015587
3	8.653728	10.173468
4	11.791534	13.323692

Note that x=0 is a zero of J_v for every order v>0, but it is not counted among the positive roots above.

Order v

The parameter (here the form field order) that selects which member of the Bessel family is computed. It may be any real number — integer orders arise in cylindrical problems, half-integer orders v=n+\tfrac12 give the spherical Bessel functions.

Argument x

The independent variable at which J_v is evaluated. In this table it starts at startX and advances by stepX for loopCount rows.

Gamma function \Gamma

The continuous extension of the factorial, with \Gamma(n+1)=n! for non-negative integers. It appears in the denominator \Gamma(v+k+1) of the series so that non-integer orders are well defined.

Bessel function of the first kind J_v(x)

The solution of Bessel's differential equation x^2 y''+x y'+(x^2-v^2)y=0 that remains finite at the origin (for v≥ 0). It is given by the power series in the formula above.

Zeros / roots

The values of x at which J_v(x)=0. Each order has infinitely many positive zeros, increasingly evenly spaced and asymptotically separated by π.

Half-integer (spherical) order

When v=n+\tfrac12, J_v relates to the spherical Bessel functions j_n(x)=\sqrt{\tfrac{π}{2x}} J_{n+1/2}(x), which describe radial parts of wave equations in spherical coordinates.

Recurrence-term ratio

Successive terms of the series satisfy ((a_{k+1}) ÷ (a_k))=((-(x/2)^2) ÷ ((k+1)(v+k+1))), which is used internally to generate each term from the previous one and to assess convergence.

A few facts help read the columns your sweep produces:

• Starting values. J_0(0)=1, while J_v(0)=0 for every order v>0. So a table beginning at x=0 starts at 1 only for the zeroth order.
• Oscillation with decay. For large x, J_v(x)≈\sqrt{\tfrac{2}{π x}}\cos\!≤ft(x-\tfrac{vπ}{2}-\tfrac{π}{4}\right). The function oscillates like a phase-shifted cosine while its amplitude decays as 1/√(x). Successive maxima therefore shrink slowly as x grows.
• Sign changes mark zeros. Wherever a column changes sign between two rows, a root of J_v lies in that interval (e.g. J_0 switches sign between x=2 and x=3, bracketing its first zero ≈ 2.4048). For large arguments consecutive zeros are spaced by about π.
• Physical nodes. Those zeros correspond to physical boundary conditions: the radial modes of a vibrating circular drumhead, cutoff frequencies of cylindrical waveguides, and field patterns in optical fibers are all indexed by zeros of J_v.
• Magnitude. For fixed x, higher orders v start near zero and rise more slowly; for small x the leading behavior is J_v(x)\sim ((1) ÷ (\Gamma(v+1)))≤ft(((x) ÷ (2))\right)^{v}, so larger v stays smaller until x becomes comparable to v.

These observations follow from the established series and asymptotic forms above and apply to any order you enter.