Numbers BESSELY Function: Positive Inputs, Orders and Recurrence Checks

• Introduced: The original introduction release is not established by the checked official references.
• Availability: Documented for Numbers in the Apple Functions Help 15.4 edition; no final supported-version endpoint is established.
• Scope: The help edition is not an app build number. Historical Mac, iPad, iPhone and iCloud parity remains separately unverified.
• Checked: October 14, 2026, against the official BESSELY reference. The numerical examples were independently calculated; no native Numbers execution was performed.

BESSELY(pos-x-value, n-value)

Both arguments are required. pos-x-value is a number greater than zero. n-value is a nonnegative order; its decimal portion is ignored. Apple explicitly gives the same result for orders 3 and 3.9. This selects an integer order, not a requested number of result decimals. See the function-specific reference.

“Second kind” identifies the function family, not order 2. BESSELY(x,0), BESSELY(x,1) and BESSELY(x,2) are all second-kind functions. BESSELJ is the separate first-kind family, with a different contract for x. The two names are not aliases.

Create SecondKind, with six columns and eleven rows, one header row, no header columns and no footer rows. Put x, Order, Value, Rounded, Expected rounded and Difference in A1:F1. Enter A, B and E below. Keep C, D and F empty for formulas.

Enter =BESSELY(A2,B2) in C2, =ROUND(C2,10) in D2 and =D2-E2 in F2. Fill those formula columns through row 11. Rows 2–3 reproduce Apple’s examples at ten decimal places; the rest use independent evaluations of the defined mathematical function. A negative result does not violate the input rule: positive x can yield negative Yₙ(x).

Keep C unrounded and use ROUND only in the comparison column. F is expected to be zero at the chosen display precision; a discrepancy should be inspected rather than hidden by increasing rounding. The published Apple examples do not promise accuracy for every order or argument.

Duplicate SecondKind as ChangedOrderY. Change only B3 to 4.1 and update E3 to −0.0910410710. C3 now represents Y₄(25), approximately −0.091041070990093, while row 2 remains Y₃(25). Changing the order can change the sign as well as the magnitude.

On a separate DomainChecks table with two columns and four rows, put x and Formula in A1:B1. Enter −25 in A2 and 0 in A3. In B2 enter =BESSELY(A2,3) and fill to B3. Both cases are expected to produce an error under Apple’s stated domain; the entry explicitly illustrates the negative case and excludes zero. No particular spreadsheet error label is asserted here.

Do not put these deliberate invalid cells into the earlier numerical comparisons. Replacing an invalid input with ABS would change the mathematical question, and ABS(0) still leaves a forbidden zero. If the sign of a measured or computed argument is meaningful, inspect its origin instead of silently removing that sign.

The NIST recurrence, equation 10.6.1, gives Y₀(x) + Y₂(x) = 2Y₁(x)/x. Rearranging it reconstructs order 2 from orders 0 and 1. This is a mathematical cross-check; Apple does not disclose its internal evaluation algorithm.

Create OrderRecurrence, with seven columns and five rows. Put x, Y0, Y1, Y2, Reconstructed Y2, Difference and Absolute difference in A1:G1. Enter 1,2,5,25 in A2:A5, then enter B2:G2 and fill through row 5:

B2: =BESSELY(A2,0)
C2: =BESSELY(A2,1)
D2: =BESSELY(A2,2)
E2: =2*C2/A2-B2
F2: =D2-E2
G2: =ABS(F2)

F and G are zero in exact arithmetic. Keep the full stored B:C values in the reconstruction, rather than retyping the shortened decimals in this table. Near cancellation or at extreme arguments, a small residual is not a complete error estimate, because the component evaluations may have their own errors.

Create ArgumentScale, with six columns and three rows. Put 1 in A2 and enter:

B2: =BESSELY(A2,0)
C2: =BESSELY(2*A2,0)
D2: =2*B2
E2: =C2-D2
F2: =ABS(E2)

B2 is approximately 0.0882569642, C2 0.5103756726, and D2 0.1765139284. E2 and F2 are approximately 0.3338617442. Scaling the argument changes where the function is evaluated; scaling the returned value is a different operation. This simple control helps catch a misplaced factor in a larger formula.

The function page does not fully specify numeric-text coercion, blank references, negative fractional orders, large-order stability or exact error tokens. Its integer-order rule should not be extended to arbitrary noninteger-order scientific formulas. Use the separate function family and domain required by your mathematical model.

These examples use English function names, comma separators and uniquely named Numbers tables. Follow your formula editor’s displayed separators if the document locale differs. A2 refers to a local table cell; use SecondKind::A2 when referring to that input from another table. No result here establishes imported Excel-file or individual mobile-client parity.