BESSELY 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.

BESSELY evaluates the ordinary Bessel function of the second kind, Yₙ(x), under this add-in’s positive-real-input contract. It is also called a Neumann function. Negative output values are ordinary valid results and do not mean the input was outside the domain. Release entry.

=BESSELY(X; Order)

Two arguments are required. X must be strictly positive and the converted integer Order must be nonnegative. The wrapper explicitly rejects nonpositive X and negative integer orders. Fractional Order inputs pass through integer conversion rather than requesting a real-order Y 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.

For x=1 and order 1, =BESSELY(1;1) has mathematical reference approximately −0.781212821300. At x=2 and order 1 it is approximately −0.107032431541. Label the function family clearly: ordinary Y and modified K are both second-kind functions but have different definitions.

If X is in A2 and Order in B2, =BESSELY(A2;B2) follows the same literal example. Check positivity of X before filling a formula down; a missing or zero input is not a valid approximation to a positive one.

The ordinary recurrence Yₙ₋₁(x)+Yₙ₊₁(x)=2nYₙ(x)/x gives Y₀(2)+Y₂(2)=Y₁(2). It is an independent mathematical check for integer order 1 and x=2. The recurrence should be checked with a numerical tolerance, not exact equality of rounded cells.

The implementation computes base orders with iterative formulas, then constructs higher orders through recurrence. It also has asymptotic branches for very large positive X. Those branches are visible in source; the modest table does not exercise or certify them.

The mathematical function is singular at zero. Moving close to zero can greatly increase the magnitude for some orders. A chart or report should retain the input scale and not interpret a large negative value as a failed probability calculation.

The positive-X restriction is deliberate. Although mathematical continuation and parity conventions exist in broader treatments, this wrapper rejects negative X. It does not provide a general complex continuation.

A negative integer order is also rejected. The fractional diagnostics record this program’s argument conversion only. When transferring formulas, compare the exact order and domain conventions rather than just the function’s spelling.

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 BESSELY 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

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.