BESSELK Function (OpenOffice Calc)

• Introduced: First release not established; documented by OpenOffice.org 3.3.
• Supported in: Apache OpenOffice 4.1.16; this identifies a verified release, not a final supported version.
• Scope: Desktop Calc with its Analysis add-in, English function names and semicolon-separated formulas.
• Checked: October 14, 2026. Eight reference formulas and the listed diagnostics were recalculated after XLSX import in 4.1.16 on Linux x86-64, Build 9816. Source and independent reference calculations were also checked.

BESSELK evaluates the modified Bessel function of the second kind, Kₙ(x), for the real positive-input contract used by this add-in. It typically decays as positive X grows, unlike modified Bessel I. It is not a cumulative probability. Release entry.

=BESSELK(X; Order)

Both arguments are required. X must be strictly positive, and the converted integer Order must be nonnegative. The wrapper rejects X≤0 and negative integer orders before returning a value. Use a deliberate integer order rather than assuming a fractional input is evaluated as a fractional-order function.

The native column records the saved 4.1.16 results. Ordinary numeric values are rounded to twelve decimal places, with scientific notation for tiny values. Stored values and display text were inspected separately; a twelve-decimal cell display can hide a tiny nonzero number. The native values are not claimed to match every printed mathematical digit; the source approximation explains the differences.

For x=1 and order 1, the high-precision mathematical reference is approximately 0.601907230197. For x=2 it is approximately 0.139865881817. The native values are shown separately in the table because the inspected implementation uses finite-coefficient approximations; do not relabel a mathematical reference as an exactly matching observed value.

A worksheet can put x in A2 and the integer order in B2, then use =BESSELK(A2;B2). Check that units and scaling in the surrounding model actually require the modified second-kind function rather than BESSELI or BESSELY.

The source evaluates K₀ and K₁ using separate polynomial approximations on different X intervals. Higher integer orders use a recurrence. The use of decimal coefficients with limited precision can matter before the last displayed digits of a twelve-decimal reference table.

At x=2, the mathematical recurrence K₂(2)−K₀(2)=K₁(2) gives an independent relation among orders. A small recurrence residual alone does not prove absolute accuracy: values derived from the same approximations can satisfy a recurrence while sharing a systematic error.

Another mathematical check is I₀(x)K₁(x)+I₁(x)K₀(x)=1/x for positive x. That composition is independent reference guidance, not an additional native execution assertion in this guide.

Zero is outside this function’s real finite domain and is rejected, not replaced by a very large finite value. Negative X is also rejected. Do not copy another implementation’s complex continuation or sign convention into this argument contract.

Small positive X can produce large values, while large positive X can lead to very small results. The table documents ordinary inputs and its actual differences from high-precision mathematics; it does not certify universal underflow or overflow accuracy.

In the saved diagnostics, Order=1.9 produced the order-1 value and Order=−0.9 produced the order-0 value. These observations are consistent with conversion toward zero in the tested path. Order=−1 was rejected. They should not be described as fractional-order mathematical evaluations.

These additional formulas were recalculated through the same import path. Error displays are retained as errors, even when an ODS numeric attribute stores zero.

The BESSELK guide for Gnumeric documents that product’s own syntax and tested behavior. A shared function name does not guarantee the same argument conversion, text parsing, branch or precision rules.

• Function-specific release entry
• Numeric algorithm or complex parser/formatter
• K₀ approximation and K₁ approximation

The saved ODS retained every tested formula identity. A deliberately stale SUM cache changed from 999999 to 3, and an uncached SUM returned 7 after hard recalculation. Native evidence covers the eight reference formulas and listed diagnostics only. Other explanatory compositions are independent mathematical checks; other releases, locales and extreme-value behavior were not established by this run.