BESSELJ 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 used below.
• Checked: October 14, 2026. The reference and boundary tables plus the explicit roundtrip or recurrence were recalculated through 1.12.57 ssconvert and compared with native formula records, source and independent arithmetic. 1.12.62 is source-checked only.

BESSELJ returns the ordinary Bessel function of the first kind, Jα(x). It is an oscillatory special function, not a cumulative probability and not the exponentially growing modified Bessel I function. Negative results can be mathematically meaningful. Versioned function help.

=BESSELJ(x,order)

Both x and order are required. Gnumeric accepts real, fractional orders, despite the help argument label describing an integer. Fractional-order support is explicitly recorded from version 1.12.7. A negative x with a noninteger order does not produce a real result through this worksheet function. Function implementation.

Noninteger results are rounded to twelve decimal places. All eight reference rows below were recalculated in the stated 1.12.57 build.

To inspect the order-1 term at x=0, 1 and 2, place those x values in A2:A4 and order 1 in B1. Fill =BESSELJ(A2,$B$1) down from B2. The corresponding tested literal values are 0, approximately 0.440050585745, and 0.576724807757.

Keep x and order in separate labeled cells so a copied formula cannot silently exchange them. These values describe a mathematical term; choosing a physical model or boundary condition is a separate step.

The half-order identity J½(x)=SQRT(2/(PI()×x))×SIN(x) gives approximately 0.513016136562 at x=2, matching the table. It also shows why this function is not the modified I function: the sine factor oscillates rather than growing like a hyperbolic sine.

The recurrence J0(2)+J2(2)=J1(2) supplies a second check. The executed formula =BESSELJ(2,0)+BESSELJ(2,2)-BESSELJ(2,1) returned 0. Do not assume every recurrence residual will be exactly zero after floating-point rounding.

The help’s blanket negative-order error statement does not match the tested order −1. =BESSELJ(2,-1) returned approximately −0.576724807757, consistent with J−1(x)=−J1(x). Negative x with integer order 1 also gives the negative of the positive-x result. The source helper treats negative x with fractional order differently, producing a nonnumeric result that becomes #NUM!.

=BESSELJ(1,1.9) is approximately 0.134386952219. The fractional order is not silently truncated. Tests here cover small arguments and orders; accuracy near roots and at extreme arguments requires a tolerance suitable for the application, not a fixed count of displayed decimals.

In the following tests F5 is an actually empty cell. The empty string "", when present, is an explicit string argument.

The 1.12.62 Bessel helper and worksheet entry explain the real-order and negative-x handling. Help and implementation differ on the restrictions identified above, so the tested cases are stated explicitly. The formula identities, saved results and two recalculation controls establish the named 1.12.57 run; no GUI entry, other locale or later-release execution is claimed.