IMARCSECH 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 acosh(1/z), using the principal branch of IMARCCOSH. Both real and imaginary coefficients matter. The forward relationship is IMSECH(IMARCSECH(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/IMSECH(z).

=IMARCSECH(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".

=IMARCSECH(COMPLEX(1,2))
=IMARCSECH(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 parent arccosh cut, on real arguments below 1, maps to negative real z and positive real z above 1. Zero remains a reciprocal singularity; z=1 is a branch point. Exact-cut signed-zero conventions can determine the sign of an imaginary answer, so the table uses off-cut cases or nonambiguous real inputs.

Zero has no finite inverse-hyperbolic-secant value. A positive real response between 0 and 1 can have a real inverse; a real response above 1 generally requires a complex value. 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.

For a normalized response R=sech(γL), put 1+2i in A2 and =IMARCSECH(A2) in B2. This gives a candidate dimensionless propagation parameter γL. If L is 2 units, =IMDIV(B2,"2") gives γ per unit length. Cosh is even and complex-periodic, so ±γL+2πik can represent alternative branches.

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

=IMCOSH(B2)
=IMSECH(B2)
=IMABS(IMSUB(IMSECH(B2),A2))

The hyperbolic cosine should equal =IMDIV("1",A2). Compare both coefficients to confirm that the reciprocal response has been reconstructed. 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.

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