Kelvin Functions of the First Kind Calculator (ber, bei)

bei_v(x)	0.249566
ber'_v(x)	-0.062446
bei'_v(x)	0.497397

What are Kelvin functions of the first kind?

The Kelvin functions of the first kind, written ber_v(x) and bei_v(x), are the real and imaginary parts of the Bessel function J_v evaluated at a phase-rotated argument: $\mathrm{ber}_v(x) + i\cdot\mathrm{bei}_v(x) = J_v(x\cdot e^{i3\pi/4})$. They appear in problems with cylindrical symmetry and oscillating fields, classically in the analysis of the skin effect in electrical conductors and in heat-conduction and elasticity problems. This calculator returns ber_v(x), bei_v(x), and their first derivatives ber'_v(x) and bei'_v(x) for any real order v and real argument x.

Two oscillating curves labeled ber and bei plotted against x, growing in amplitude — The Kelvin functions ber_0(x) and bei_0(x) oscillate with growing amplitude as x increases.

How to use the calculator

Enter the order v (any real number; v = 0 is the most common) and the argument x (a real number). Press calculate. The hero box shows ber_v(x), and the table lists bei_v(x) and both derivatives. The series converges quickly for x up to roughly 20; for very large x cancellation reduces accuracy and an asymptotic expansion would be preferable.

The formula explained

The functions are computed from the convergent complex power series shown above, where Γ is the gamma function (evaluated here with a Lanczos approximation):

$$\mathrm{ber}_\nu(x) + i\,\mathrm{bei}_\nu(x) = \left(\frac{x}{2}\right)^{\nu} e^{\,i\,3\nu\pi/4} \sum_{k=0}^{\infty} \frac{\left(\frac{i\,x^{2}}{4}\right)^{k}}{k!\,\Gamma(\nu+k+1)}$$

Terms are accumulated with the recurrence $\text{term}_k = \text{term}_{k-1}\cdot(i x^2/4) / [k(\nu+k)]$, using two real accumulators for the real and imaginary parts. The derivatives use the exact relation $\mathrm{ber}'_\nu = (\mathrm{ber}_{\nu+1}+\mathrm{bei}_{\nu+1})/\sqrt{2} + (\nu/x)\mathrm{ber}_\nu$ and $\mathrm{bei}'_\nu = (\mathrm{bei}_{\nu+1}-\mathrm{ber}_{\nu+1})/\sqrt{2} + (\nu/x)\mathrm{bei}_\nu$, so the next-order functions are also evaluated with the same definition.

Complex plane showing a ray at angle 3 pi over 4 from the origin — Kelvin functions arise from the Bessel function J_v evaluated along the ray x·e^{i3π/4} in the complex plane.

Worked example (v = 0, x = 1)

For v = 0 the series reduce to $\mathrm{ber}_0(x) = \sum(-1)^{k}(x/2)^{4k}/[(2k)!]^2$ and $\mathrm{bei}_0(x) = \sum(-1)^{k}(x/2)^{4k+2}/[(2k+1)!]^2$. At x = 1 this gives $\mathrm{ber}_0(1) \approx 0.984382$ and $\mathrm{bei}_0(1) \approx 0.249566$, matching standard tables (Abramowitz & Stegun 9.9).

FAQ

What is the valid range of x? The series implementation is reliable for roughly $0 \le x \le 20$. Beyond that, floating-point cancellation degrades accuracy.

What happens at x = 0? For v = 0, $\mathrm{ber}_0(0) = 1$ and $\mathrm{bei}_0(0) = 0$, with both derivatives 0. For v > 0 the functions tend to 0; for v < 0 they may diverge.

Can I use non-integer order? Yes. Any real v is supported as long as v+1 is not a negative integer (a gamma-function pole).