PlanMaker BESSELK: Orders, Truncation and Reproducible Controls

The PlanMaker 2026 manual, 2024 manual and 2021 manual were retrieved on October 15, 2026. They establish documented support in those manual families, not an original introduction or a last-working-version endpoint. Both historical endpoints remain unverified. Every individual desktop build and FreeOffice/mobile edition is outside this check.

No native PlanMaker execution was performed. Numerical expectations are independent high-precision mathematical calculations, not captured application results. English formulas use commas; verify names and separators in the installed language and locale. Displayed decimals are rounded reference values, not a promise of native last-bit accuracy.

=BESSELK(x, n)

The first argument x is the evaluation point. The second, n, is the order, not the number of sample points or requested decimal places. Both are supplied in every example.

All three checked PlanMaker references explicitly require n ≥ 0 and truncate fractional orders. Consequently n=1.9 selects order 1, not order 2 and not the true noninteger-order mathematical function. The order-zero case is explicitly within the documented inequality; it is not guessed from an ambiguous error table.

The examples use positive x and nonnegative orders. The short reference does not provide a full x-domain, overflow or native error matrix. Negative x, singular endpoints, very large inputs and validation precedence for a negative fractional order remain outside these controls. Truncating n does not imply truncating x.

Create BesselControls with x in A1, Order_zero in B1 and Order_one in C1. Enter these six x values in A2:A7. Fill B2 =BESSELK(A2,0) and C2 =BESSELK(A2,1) through row 7.

In a copy change A4 from 2 to 2.5. Its order-zero result becomes 0.0623475532, and its order-one result becomes 0.0738908163. The evaluation point retains its fractional part. The other rows are independent inputs and stay unchanged.

On a separate OrderCheck sheet put x=2 in F2. Put orders 0.9,1,1.9,2,2.9 in G2:G6, and fill H2 =BESSELK($F$2,G2) downward.

The independent results are 0.1138938727,0.1398658818,0.1398658818,0.2537597546,0.2537597546. The first uses order zero; 1 and 1.9 agree; 2 and 2.9 agree. The dollar signs keep the same evaluation point while the order changes.

Keep integer orders in routine formulas for clarity. The truncation example documents a specific PlanMaker rule, rather than recommending that a scientific model silently discard a meaningful fractional order. A model requiring a genuine order of 1.9 needs a different verified computational method.

Put Order_two in D1 on the original BesselControls sheet and fill D2 =BESSELK(A2,2) down. The mathematical recurrence is K_2(x)−K_0(x)=2*K_1(x)/x. The NIST recurrence reference supports this relationship with its modified-cylinder notation, whose phase factor gives the stated K signs.

In E2 enter =D2-B2-2/A2*C2 and fill through E7. Each independent residual has target 0. At x=2 the order-two value is 0.2537597546. Use full cell precision rather than the printed rounded table when checking the identity.

A small computed residual can reflect finite precision; a substantial discrepancy can reveal an incorrect function family, order or copied reference. The recurrence check is not a native precision certification or permission to propagate a numerically unstable recurrence to extreme orders.

Use x=1 at row 3 as a reference and fill F2 =C2/$C$3 through F7. The normalized order-one value is 1 at x=1, 0.2323711609 at x=2 and 0.0067196625 at x=5.

The positive sampled values decay as x increases. They do not grow like modified I values. K_1(2) is about 0.1398658818, while I_1(2) is about 1.5906368546. Substituting one modified Bessel family for the other changes the mathematical solution, even though both names contain “modified.”

This normalization is a numerical profile comparison. It does not establish physical boundary conditions or the validity of a particular engineering model. The mathematical K function is singular at zero; this guide does not assign a native error at that endpoint.

The checked manual does not supply exact error tokens for invalid orders, nonnumeric cells, blanks, arrays, or extreme evaluation points. No errors are borrowed from Excel or another spreadsheet. The documented truncation rule applies to n, while x is the independent evaluation variable.

If values differ, check the family letter, the order and whether a fractional order was intended. Reproduce both order columns and the changed-x control in the actual build. Record edition, platform, language and workbook format; source documentation plus independent calculations are not a native application test or a guarantee about import/export compatibility.