BESSELK in ONLYOFFICE: Modified Bessel K: Decaying Radial Solutions

K is a second modified-Bessel solution associated with decay for positive x. Unlike I, it diverges toward x=0 for common nonnegative integer orders, so including x=0 would create a mathematically singular example.

Modified Bessel function of the second kind, with decay for positive x and a singularity at zero. The two arguments represent evaluation point and order, respectively; do not swap them. A Bessel order is conceptually different from the value of the function at x. Throughout this guide n is a nonnegative integer, a conservative input class that meets the vendor’s stated requirement that n be numeric and nonnegative. The vendor page does not explicitly document coercion of fractional, text or negative order inputs. Do not assume truncation or special integer treatment from Excel behavior; native verification is necessary.

For comparison, the currently documented engineering functions include BESSELI, BESSELJ, BESSELK, BESSELY. Those names are distinct operations, not confirmed aliases. Mathematical background is provided to explain calculations rather than asserting any undocumented internal algorithm.

=BESSELK(x,n)

Documented domain: strictly positive numeric x; nonnegative numeric order n. For BESSELK, the vendor explicitly rules out x≤0.

There is no optional argument in the documented signature, and the vendor does not list an omitted-order default. Decimal numeric x values are allowed by the mathematical definition; exact text/coercion outcomes and error strings require native testing.

1. In a new spreadsheet, label A1 x, B1 n (order), C1 BESSELK formula, D1 Expected mathematical result, E1 Absolute difference and F1 Tolerance pass. Write the numeric x and whole-number n values shown below into A2:B21. Never add unit suffixes directly to these numeric cells.
2. Enter =BESSELK(A2,B2) into C2, then fill down through C21. The references must move row by row; do not accidentally lock A2/B2 with $ signs. Paste expected numerical reference values into D2:D21 as static numbers. This guide did not run C2 in a native ONLYOFFICE editor.
3. Put =ABS(C2-D2) in E2, copy to E21, and enter =E2<0.0000001 in F2, filling down. This is a proposed native QA check, not a claim that the column passed in the application. For results near zero or very large, evaluate relative error and significant digits instead of blindly applying one absolute tolerance.
4. Find the function via Formula → Function → Engineering, or type it manually. The official insertion guide documents entering = first, using commas to separate function arguments in the English instructions, showing formulas and recalculating with F9. If your region/formula language differs, confirm locally rather than substituting separators or names on guesswork.

All 20 expected results were computed with SciPy’s Bessel implementation and independently cross-checked against mpmath at 70-digit precision, with agreement better than 1e-10 × max(1, |reference value|). This does not measure ONLYOFFICE accuracy and cannot establish that the editor uses either library. The decimal display in a particular build may differ.

The following are documented-shape formulas; each line’s trailing comment gives a mathematical expected value and must not be typed into ONLYOFFICE as part of the formula:

=BESSELK(1,0)   // expected 0.421024438241
=BESSELK(1,1)   // expected 0.601907230197
=BESSELK(2,0)   // expected 0.11389387275

Remove the // expected ... comments before entering the formulas. Do not infer that numerical values are certified native outputs.

The mathematical three-term relation for these samples is:

K₂(x) − K₀(x) = 2 K₁(x)/x (for x>0).

Enter numeric 2 into H2. In I2 enter this proposed formula:

=BESSELK(H2,2)-BESSELK(H2,0)-2*BESSELK(H2,1)/H2

The mathematical expected result is 0 (subject to small floating-point roundoff in a real implementation). The identity has been numerically cross-checked using independent mathematical libraries. H2 must not equal zero because the displayed recurrence divides by x. A zero residual in one test does not prove every input or version behaves correctly.

At x=0, the standard K functions are singular; vendor independently requires x>0. This is a statement about the mathematical functions, not a guaranteed native result or error code for a particular cell input. For functions requiring strictly positive x, do not use x=0 as a normal worksheet test.

A radially decaying, dimensionless basis can be represented by =BESSELK(H2*I2,0) when H2*I2 is strictly positive. Avoid evaluating this singular basis at the origin; physically meaningful models normally combine basis functions and boundary conditions.

For real model inputs, store length or physical scale in separate numeric cells and only feed the dimensionless product into the special function. A mathematical basis function is not a complete boundary-value solution; it does not automatically enforce boundary conditions, normalize measured data or validate a physical system. Compare independent analytic reference points before relying on outputs.

For large x, K becomes very small; for small positive x, it can become very large. Avoid zero and negative x as the vendor explicitly requires a positive evaluation point. Do not infer a native #NUM! diagnostic or precision guarantee.

Specific implementation constraints not verified: maximum numeric x/order, behavior for large orders or fractional n, finite floating-point precision, overflow/subnormal handling, exact displayed errors and localized function aliases. Keep test reports separate by web-hosted Docs build versus installed Desktop Editors build.

The official English, currently unversioned ONLYOFFICE Engineering function page and Insert Function catalogue establish documented name, category, purpose and arguments as of 2026-10-13 UTC. They do not establish the first release to implement BESSELK. Docs release history and Desktop Editors release history are distinct; the historical availability and exact calculation engine behavior for every release are UNKNOWN in this review.

Native test status: Docs web: NOT TESTED. Desktop Editors: NOT TESTED. Mobile: NOT TESTED. Historical releases: NOT TESTED. Excel, LibreOffice, or another product’s Bessel behavior is not evidence of ONLYOFFICE compatibility. Neither workbook import/export roundtrips nor screenshots have been produced. Locale/formula language and file-format interoperability must be verified on a target build.

BESSELI, BESSELJ, BESSELK, BESSELY; vendor function-entry instructions; advanced settings; Docs releases; Desktop releases. All URLs were checked as actual vendor references for this guide. These links point to the vendor references.