BESSELY in Calligra Sheets: Bessel Function of the Second Kind

• Introduced: original introduction date unknown. Do not interpret a tested or inspected release as the first release.
• Source availability verified: Calligra 25.04.2 and 26.08.2 both register BESSELY with exactly two arguments. The inspected engineering and calculation-engine files are byte-identical between these two tags.
• Checked October 18, 2026: KDE handbook, 26.08.2 registration, and 25.04.2 implementation.
• Native verification: Calligra Sheets 25.04.2, Linux, October 18, 2026. Literal-formula cases were recalculated in the application and saved to ODS. The reference calculations remain separate from these observations; see the native table below. No native 26.08.2 execution is claimed.

=BESSELY(x; n)

Both arguments are mandatory. In the function wrapper, the worksheet order is x first, n second; the internal engine reverses them when calling its numerical routine. Do not reverse the worksheet arguments because a mathematics library writes its own call as function(order, x). The source checks n with an exact integer test; 1.5 is invalid, not order 1. Wrapper and registration · domain guard.

The English examples use a semicolon separator and a decimal point. If your locale parses formulas differently, use the function editor to check local syntax. Keep units in adjacent label cells rather than typing text such as 0.5 m into a numeric argument.

An annular vibration model can combine J₀(k r) and Y₀(k r). An annulus has a positive inner radius, so the Y singularity at r = 0 can be excluded from the model. The values below are basis evaluations, not a solved structural design.

1. Put the label k (1/m) in A1 and 2 in B1.
2. Put r (m) in A3, x = k*r in B3, order n in C3 and Yn(x) in D3.
3. Enter 0.5 in A4 and 1 in A5. These are radii in metres.
4. Enter =$B$1*A4 in B4 and fill it down to B5. Expected x values are 1 and 2; the inverse-metre and metre units cancel.
5. Enter 0 in both C4 and C5, selecting order zero.
6. Enter =BESSELY(B4;C4) in D4 and fill down to D5. Display at least nine decimal places before comparing.

Y₀ is positive at both sample points here, but it is not always positive. Do not interpret these values as probabilities or a monotone response.

These calculations check the mathematics of the examples. The native table separately records application results for equivalent literal formulas. These checks do not establish accuracy for all inputs, and the step-by-step cell-reference layout itself was not separately executed.

For a positive inner radius a = 0.5 m and k = 2 1/m, define g(r) = J₀(2r)Y₀(1) − Y₀(2r)J₀(1). This combination is constructed to be zero at r = a.

1. Put Radial combination in E3.
2. In E4 enter =BESSELJ(B4;0)*BESSELY(1;0)-BESSELY(B4;0)*BESSELJ(1;0).
3. Fill the formula down to E5.
4. E4 is expected to be 0 (the two products coincide). E5 is approximately -0.370778363504.

A negative result is a signed basis amplitude, not an error. The expression enforces only the stated inner-boundary zero; it does not solve an outer boundary or determine an eigenfrequency. Keep the inner radius positive and avoid substituting zero into the Y terms.

The four explicit domain-error formulas in this table were also tested in native 25.04.2 and displayed #VALUE!. Missing/extra-argument behavior here remains source-derived; it was not part of this function’s native fixture. Text/blank/Boolean coercion and localized error labels have not been established by these numerical examples. Validate numeric inputs rather than relying on implicit conversion. Calligra argument-count check · Bessel domain and dispatch.

Zero is a singularity, not a regular test point

The shared guard allows x = 0, but the internal Y routine returns HUGE_VAL at that point. For this reason, =BESSELY(0;0) must not be presented as an ordinary finite result. Native Calligra Sheets 25.04.2 displayed INF for this exact zero-input case; that is not a finite valid Bessel value. For Y, the real-valued limit from positive x is negative infinity for nonnegative integer orders, so the implementation sentinel must not be mistaken for the mathematical signed limit. See the numerical routines.

Numerical precision is not identical to the reference calculation

An independent review compiled the isolated numerical routines from the KDE source, separately from the application. That isolated probe is not a native Calligra test; a subsequent native 25.04.2 run independently reproduced the values listed below. That source-algorithm check found BESSELY(4;2) ≈ 0.215903601900667, whereas the independent mathematical reference is ≈ 0.215903594603615. At the supported upper-order boundary, BESSELJ(1;28) was ≈ 1.211320740471416 × 10⁻³⁸ versus the reference ≈ 1.211364502417112 × 10⁻³⁸, a relative difference of about 0.00361%. Therefore, do not interpret the many displayed reference digits in this guide as a promise of equal application accuracy. Compare against the reference and the observed application result separately, especially for higher orders. A fixed absolute tolerance such as 10⁻¹² would incorrectly accept zero in place of these tiny nonzero values.

This guide describes Calligra Sheets, not an Excel-compatible contract. In particular, preserve the 0–28 integer-order restriction, the x-domain rule and the worksheet argument order when moving formulas. The source also registers COM.SUN.STAR.SHEET.ADDIN.ANALYSIS.GETBESSELY as an alternate name for this same function; it does not represent a different Bessel family or merit a duplicate guide. Registration is evidence of the alias, not proof that every imported workbook round-trips unchanged. Registration.

• KDE BESSELI reference: modified bessel function of the first kind.
• KDE BESSELJ reference: bessel function of the first kind.
• KDE BESSELK reference: modified bessel function of the second kind.

Reviewed October 18, 2026. Primary implementation evidence: KDE 26.08.2 engineering functions, KDE calculation engine, and KDE 25.04.2 engineering functions. The official function reference supplies the function name and signature. Worked model examples and numerical cross-checks are original calculations; they are not native Calligra observations. The native record below covers the displayed formulas in Calligra Sheets 25.04.2 only; it does not establish behavior for every build, locale or input range.

On October 18, 2026, the formulas below were loaded into the actual Linux application, recalculated, and saved to an ODS workbook. Results were read from that native-saved workbook, including error display text. The screenshot shows the real application; it was not synthesized. The independent mathematical values earlier in this guide remain reference values, and small differences are intentional evidence of the numerical approximation rather than transcription errors.

The display uses scientific notation: E-03 means multiply by 10⁻³. A small value in scientific notation is not zero. INF identifies the observed non-finite output, not an ordinary validated model result.

In particular, the displayed positive INF at x = 0 must not be interpreted as the mathematical one-sided Y₀ limit, which is negative infinity. Avoid the singular point in finite real calculations.