IMCOSH 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. Documentation/source check; no new native test in Gnumeric 1.12.62.

Compute the hyperbolic cosine of a complex argument. Unlike ordinary complex cosine, the real coefficient is transformed hyperbolically and the imaginary coefficient enters ordinary sine and cosine factors.

Gnumeric’s official Complex manual documents this exact function name and signature; the official function catalogue separately lists it as Complete with Exhaustive upstream testing. “Complete” describes the catalogue’s implementation classification; “Exhaustive” describes the listed upstream tests, not a claim about the example worksheet below or every available platform/version. Official documentation, checked 2026-10-13.

Version-specific error distinction: The online 1.12 manual describes malformed complex inputs as #VALUE!. The official 1.12.62 source instead returns #NUM! when parsing a malformed complex string; other unsupported value types can still return #VALUE!. Any #VALUE! expectations below are the manual’s older documented rule, not verified 1.12.62 output. No native error test was performed.

=IMCOSH(z)

There are no additional arguments. z stands for a single complex value, not two separate real-number arguments. To construct a value from distinct coefficients use the separately documented COMPLEX function; in Gnumeric the related function’s arguments are COMPLEX(x,y,i) where the optional i/j suffix defaults to i. The simple one-argument examples here work independently of list-separator conventions, but COMPLEX’s comma notation reflects the English manual; check the installed locale before pasting multi-argument expressions. Official source, checked 2026-10-13.

Radian convention. Interpret angular inputs in the complex trigonometric formulas using radians. The real and imaginary components of the input are ordinary numeric coefficients; appending i marks the imaginary coefficient, not an angle unit. If starting with a real angle in degrees, convert it to radians first before constructing z; do not append i to a degree number and expect automatic conversion.

Definition: cosh(x+iy) = cosh(x) cos(y) + i sinh(x) sin(y). The y angle in cos(y) and sin(y) is in radians; the hyperbolic part acts on the real x. This is different from cos(x+iy) = cos(x)cosh(y) − i sin(x)sinh(y).

The mathematical formula explains the independent expected outputs below. It is not a separate assertion that an untested Gnumeric build has identical last-digit floating-point behavior in every environment.

Create an empty Gnumeric sheet with column headings: A = complex input, B = IMCOSH result, C = real component, D = imaginary coefficient. Enter the text values in A2:A6 (if automatic cell conversion interferes, explicitly format A cells as text), then enter the formulas in B2:D2 and fill them down. Do not type the illustrative expectation into the formula-result cells.

These values are rounded to nine decimal places. A complex-result cell may display additional digits, omit a zero coefficient, or represent imaginary units with another supported suffix; the table tests numeric coefficients, not exact string serialization. The official manual documents IMREAL and IMAGINARY for extracting those parts. If the input entered as 1+2i is not treated as a valid complex text representation on a particular system, explicitly enter it as a string or generate it with =COMPLEX(1,2); do not silently assume every locale parses input the same way. Documentation checked 2026-10-13.

A direct formula to try

=IMCOSH("1+2i")

Use =IMREAL(IMCOSH("1+2i")) and =IMAGINARY(IMCOSH("1+2i")) to compare the two independently expected coefficients from the table above. Both the compound formulas and the output-format details are review examples pending native confirmation, rather than proof that they were executed in Gnumeric.

For 1+2i, the mathematical expectation is approximately −0.642148125 + 1.068607421i. A complex hyperbolic cosine may therefore have a negative real coefficient and a nonzero imaginary coefficient; it should not be interpreted as a real-only hyperbolic growth measure.

Two complex identities provide useful independent cross-checks. First, cosh(−z) = cosh(z) (evenness). Compare =IMCOSH(A2) and =IMCOSH(IMNEG(A2)) for A2 = 1+2i. Second, cosh(z) = cos(iz). The latter can be expressed with complex multiplication:

=IMCOSH(A2)
=IMCOS(IMPRODUCT(A2,"0+1i"))

Both should have approximately the same real and imaginary coefficients, within numerical tolerance. For a pure imaginary input "0+1i", the mathematical identity cosh(i) = cos(1) predicts a real coefficient of about 0.540302306 and zero imaginary coefficient. These are independent calculations, not native Gnumeric test results.

Numerical tolerance tip: even for mathematically exact identities such as a sum being 1+0i, roundoff can leave a very small residual. Comparing text representations for equality is not a sound numerical test. Set a problem-appropriate absolute/relative tolerance for each numeric coefficient.

Gnumeric’s Complex manual documents IMCOSH, but does not attach an explicit Microsoft Excel compatibility statement to that entry. Do not infer identical implementation, availability in all other products, file-import behavior, or error codes. The 1.12.62 announcement reports broad improvements in XL interoperability without naming this function.

The official release material identifies Gnumeric 1.12.62 in September 2026, and the GNOME source archive lists the corresponding 1.12.62 tarball. This review uses the online official manual and catalogue as consulted 2026-10-13, alongside inspection of checksum-verified official 1.12.62 source; no source build was executed. Consequently, the evidence supports documented function availability in the contemporary Gnumeric project, not a claim that every Linux distribution, platform port, or historical build produces identical results. Official release notes · GNOME release archive.

IMSINH · IMTANH · IMCOS · IMREAL · IMAGINARY · IMPRODUCT · COMPLEX

1. Official Gnumeric Complex function manual — exact IMCOSH(z) signature, argument meaning, invalid-input #VALUE! note, specific compatibility wording where provided, and nearby related complex functions.
2. Gnumeric’s public function catalogue — contemporary listing and labels “Complete” and “Exhaustive”; these are classification/testing labels, not a runtime experiment by this research task.
3. Official Gnumeric 1.12.62 release announcement — project version and general Excel-compatibility improvements; not proof of this function’s exact release implementation.
4. GNOME source-release directory — available 1.12.62 source archive; official 1.12.62 source inspected after checksum verification; not compiled or executed.

Evidence limits: Official documentation and release-source inspection; independent mathematics. No native Gnumeric execution, application screenshots, locale tests or cross-application workbook round-trips.