BESSELJ 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.

BESSELJ evaluates the ordinary Bessel function of the first kind, Jₙ(x). It is an oscillatory special function. It is different from modified Bessel I and from second-kind Bessel Y; its output can be positive, negative or zero. Release entry.

=BESSELJ(X; Order)

Both X and Order are required. The add-in passes an integer order to its helper, which rejects negative integer orders. Use deliberate nonnegative integer orders. Negative X is accepted through the integer-order parity rule. Fractional input to Order is a conversion case, not a request for a fractional-order evaluation.

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.

At x=1 and order 1, =BESSELJ(1;1) gives approximately 0.440050585745. At x=2 it is approximately 0.576724807757. Label both the input argument and order when comparing rows; changing either can change the sign or magnitude.

An illustrative worksheet puts x in A2 and the integer order in B2, with =BESSELJ(A2;B2) in C2. Retain more precision than a chart label displays, especially if subsequent calculations subtract nearby Bessel values.

For nonnegative integer n, Jₙ(−x)=(−1)^n Jₙ(x). Thus the order-1 value at −2 is the negative of the value at +2. The helper explicitly remembers this sign while computing with the absolute input. At zero, J₀(0)=1 and Jₙ(0)=0 for positive n.

The recurrence J₀(2)+J₂(2)=J₁(2) follows from Jₙ₋₁(x)+Jₙ₊₁(x)=2nJₙ(x)/x. It supplies an independent cross-order check at n=1 and x=2. The general recurrence requires x≠0; do not introduce a division by zero into a check of the valid zero-input cases.

The implementation uses a bounded iterative algorithm and can select an asymptotic approximation when estimated work is very large and the order permits it. That branch is source evidence, not a claim that the ordinary examples test its extreme-input accuracy.

A negative result is not automatically an error: ordinary Bessel J oscillates. A number close to a root may have a small absolute error but a relatively large percentage error. Inspect the scale before using relative tolerances.

The fractional-order diagnostics should not be copied as a general fractional-order mathematical identity. An explicit negative integer order is rejected even though mathematical identities can relate negative and positive orders outside this worksheet contract.

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 BESSELJ 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.