IMARCCOTH Function (Gnumeric)

• Introduced: 1.1.6, according to the official release history. This was a development-series release.
• Availability: Supported in Gnumeric 1.12.62, based on the function-specific registration.
• Scope: Desktop; the fn-complex function module must be included and enabled.
• Checked: October 14, 2026. Official source and independent mathematical examples checked; Gnumeric 1.12.62 was not run.

For nonzero z, the release computes atanh(1/z), using the principal branch of IMARCTANH. Both real and imaginary coefficients matter. The forward relationship is IMCOTH(IMARCCOTH(z))≈z for finite nonsingular inputs. Function reference; release algorithm.

An inverse returns one representative because the forward function is not one-to-one. Applying the functions in the reverse order need not recover an arbitrary original argument. “Inverse” does not mean 1/IMCOTH(z).

=IMARCCOTH(z)

There is one required argument and no optional flag. Use a quoted complex literal such as "1+2i", a cell containing complex text, or COMPLEX(real,imaginary). A real numeric input represents a complex number with zero imaginary part. COMPLEX has an optional "i"/"j" suffix; its default is "i".

=IMARCCOTH(COMPLEX(1,2))
=IMARCCOTH(A2)

Do not type an unquoted 1+2i arithmetic expression. The examples use decimal points and commas inside multi-argument functions; use your installation’s formula conventions.

Enter each formula in its stated row. The expected values below are independent mathematical references rounded to nine decimal places. Use =IMREAL(B2) and =IMAGINARY(B2) to compare coefficients rather than formatted strings.

The branch points are at z=±1. The parent arctanh cuts map to real-axis segments between −1 and +1, excluding the reciprocal singularity at zero. A real input with magnitude above 1 has a real principal inverse; a real input inside that interval can require a complex branch. Near ±1, small input changes can cause large output changes.

Use nonzero z and avoid z=±1, where the inverse is singular. Unlike IMARCCOT, this release routine has no explicit zero-input exception before taking the reciprocal. Pinned implementation.

The mapped cut descriptions follow the reciprocal algorithm and the parent inverse’s documented branches. They do not establish every exact-cut or signed-zero display convention in every installed build. Compare values just to either side when a model crosses a cut.

Suppose a boundary-response model is R=coth(γL), where γL is dimensionless. Put 1+2i in A2, calculate =IMARCCOTH(A2) in B2, and retain both coefficients. If length L is 2 units, =IMDIV(B2,"2") gives a candidate γ per unit length. Because coth is periodic by iπ, additional candidates differ by iπ/L.

In a separate scratch area, use these formulas with that A2/B2 example:

=IMDIV("1",IMTANH(B2))
=IMCOTH(B2)
=IMABS(IMSUB(IMCOTH(B2),A2))

The reciprocal hyperbolic tangent reconstructs the normalized boundary response. The second formula reconstructs z, and the final formula measures the residual. The independent calculation for 1+2i reconstructs both coefficients within 1e−12. Select a tolerance appropriate to the scale and input uncertainty; a correct identity does not validate a physical model.

The release parser maps malformed complex strings to #NUM!. The web manual’s generic invalid-input note instead says #VALUE!; use the pinned source when diagnosing this release. Unsupported types can reach a value error, and argument coercion before the helper can affect empty cells or Booleans. Start with explicit COMPLEX inputs when troubleshooting.

Check that the response is dimensionless where the model requires it, preserve both coefficients, and avoid replacing a branch-sensitive complex input with its magnitude. These Gnumeric-specific names and branch conventions should be checked separately when transferring a workbook to another program.

• IMARCCOTH syntax and purpose
• Gnumeric 1.12.62 function help and evaluator
• Release-pinned reciprocal/inverse algorithm