BESSELY Function (Gnumeric)

• Introduced: First release not established. Fractional-order support is documented from 1.12.7.
• 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.

BESSELY evaluates the Bessel function of the second kind, Yα(x), also called a Neumann function. It is distinct from the first-kind J function. Negative results are meaningful, and the function is singular at zero rather than being a probability distribution. Versioned function help.

=BESSELY(x,order)

Both x and order are required. Use positive x for ordinary real-axis work. Fractional orders are supported; release notes explicitly record that extension in 1.12.7. The help’s integer-order label is therefore too restrictive for current Gnumeric. Function implementation.

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

For order 1, the literal cases at x=1 and x=2 are approximately −0.781212821300 and −0.107032431541. A worksheet can put those x values in A2:A3, order 1 in B1, and fill =BESSELY(A2,$B$1) downward.

Keep the sign. Taking ABS would change the mathematical function, not merely its display. A model using both J and Y should label each term clearly; the functions cannot be substituted simply because they share the same arguments.

The positive half-order identity Y½(x)=−SQRT(2/(PI()×x))×COS(x) gives approximately 0.234785710406 at x=2. It independently checks the fractional-order table entry.

The recurrence Y0(2)+Y2(2)=Y1(2) was also tested: =BESSELY(2,0)+BESSELY(2,2)-BESSELY(2,1) returned 0 in this run. Other arguments can leave small floating-point residuals. Near a zero of Y, absolute error and relative error answer different questions, so choose a tolerance that fits the use.

The tested =BESSELY(2,-1) returned approximately 0.107032431541, contradicting the help’s blanket negative-order error statement. The native =BESSELY(-2,1) also returned that positive value. Gnumeric’s helper implements a real parity extension for negative x with integer order; it does not return the full complex continuation of Y across the negative real axis.

Do not extrapolate that real extension to arbitrary negative fractional arguments. Zero x and an empty cell converted to zero both produced #NUM! here. For ordinary real mathematical tables, keep x>0 and document any deliberate use of the product-specific negative-x extension.

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.