BESSELK Function (Gnumeric)

• Introduced: Gnumeric 0.32 release notes explicitly record this function being added.
• Supported in: Gnumeric 1.12.62, with no retirement established.
• Scope: Desktop Gnumeric with the Engineering plugin; English comma-separated formulas are shown.
• Checked: October 14, 2026. Reference and boundary tables plus the explicit advanced-check formula were recalculated through 1.12.57 ssconvert. Current 1.12.62 source was inspected; that later release was not executed.

BESSELK evaluates the modified Bessel function of the second kind, Kα(x). It is different from both ordinary Bessel Y and modified Bessel I. The ordinary real-valued examples here use positive x; this function is a special mathematical term, not a probability. Versioned function help.

=BESSELK(x,order)

Both x and order are required numeric arguments. Fractional order is retained, rather than truncated. Use positive x for the ordinary real-valued function. Zero or a negative x produced #NUM! in the tested cases. Function implementation.

Ordinary results are rounded to twelve decimal places; tiny nonzero values use scientific notation. The eight reference rows were independently checked.

Put x=1 and x=2 in A2:A3 and order 1 in B1. Filling =BESSELK(A2,$B$1) down gives the corresponding literal reference values, approximately 0.601907230197 and 0.139865881817. This decreasing pair is a useful sanity check when a worksheet requires K rather than the growing modified I function.

The cell-reference layout follows the tested literal cases; it is not a separate GUI-session claim. Keep the order and the special-function family visible in labels so a copied formula cannot silently substitute BESSELI or BESSELY.

At positive x, the half-order identity is K½(x)=SQRT(PI()/(2×x))×EXP(−x). At x=2 this gives approximately 0.119937771968. It is an independent check for the half-order row.

The recurrence K2(2)−K0(2)=K1(2) supplies another check. The executed residual =BESSELK(2,2)-BESSELK(2,0)-BESSELK(2,1) was approximately 2.775557561563E-17, rather than exact zero. Judge such a residual against the sizes of the terms; requiring exact equality would wrongly reject a numerically sound calculation.

The help says a negative order returns #NUM!, but the tested =BESSELK(2,-1) returned 0.13986588181652246, equal to the order-1 result. The worksheet entry passes the real order directly to the helper, and the tested case agrees with K−1(x)=K1(x). This is not evidence for every negative fractional order.

Order 1.9 is also retained: =BESSELK(1,1.9) is approximately 1.436076074798. Values close to x=0 can grow very large, while large positive x can produce tiny results. The small reference set does not establish universal underflow, overflow or extreme-order accuracy.

F5 below is an actually empty cell. The listed errors are native error results, not mathematical zeroes.

• Worksheet entry
• Numeric or argument-handling source

The retained workbook results include a deliberately stale SUM cache and an uncached SUM formula, both recalculated correctly. Native evidence covers the named 1.12.57 build and listed cases; cell-reference layouts and other explanatory compositions are instructions or independent mathematical checks unless explicitly identified as executed. Other locales, direct GUI entry and later-release output were not tested.