IMCOSH in Collabora Office: Check Complex Hyperbolic Cosine with Complete Square Identities

• Introduced: the original Collabora release and launch date are not established by the reviewed sources.
• Availability: documented in Collabora Office 26.04 desktop help.
• Platform scope: online and mobile deployment support is not inferred from the desktop documentation.
• Checked: October 14, 2026. Function-specific documentation and independent controls; no native Collabora execution.

=IMCOSH(Complex_number)

Collabora documents a+bi and a+bj strings, with numeric input also accepted when the imaginary part is zero. The result is always described as a string representing a complex number. A zero real or imaginary part is omitted from the display.

The numeric control =IMCOSH(2) returns approximately 3.76219569108363 as text. The result’s scalar-looking display should not be taken as proof of a numeric return type. A component extractor can supply a numeric measurement explicitly.

Format A2:A5 as Text and enter 0.5+0.75j, 2+j, 0.5-0.75j and 2j. Fill =IMCOSH(A2) down B2:B5. Extract real and imaginary coefficients into C and D with IMREAL and IMAGINARY.

These are nine-decimal independent expectations. Keep the original formulas for further calculations instead of copying their shortened displayed coefficients.

The final row is negative and purely real. Although real hyperbolic cosine is at least one, that bound does not apply to hyperbolic cosine of arbitrary complex arguments. The imaginary-axis input changes the mathematical reduction.

The official image gives real coefficient cosh(a)×cos(b) and imaginary coefficient sinh(a)×sin(b). The imaginary term has a positive sign.

For the first input, check:

=COSH(0.5)*COS(0.75)
=SINH(0.5)*SIN(0.75)

Expected coefficients are approximately 0.825071367 and 0.355198758. COSH and SINH act on a, while ordinary cosine and sine act on b in radians.

The complex ordinary-cosine equation uses a different arrangement and a negative imaginary term. Reusing that sign without checking which function is intended would calculate the wrong result.

Hyperbolic cosine is half the sum of the exponentials of z and −z. For the first row, calculate:

=IMDIV(IMSUM(IMEXP("0.5+0.75j");IMEXP("-0.5-0.75j"));COMPLEX(2;0;"j"))

Its independently expected coefficients match the primary table. IMSUM adds complete complex quantities before division by an explicitly constructed complex two.

This gives an inspectable alternative route to the same mathematical value. It does not guarantee that a manually expanded exponential implementation has the same precision or overflow behavior as IMCOSH at extreme input sizes.

Negating both coordinates gives −0.5−0.75j. Its hyperbolic cosine is unchanged, with approximate coefficients 0.825071367 and 0.355198758, because cosh is even.

Conjugating only the imaginary coordinate gives 0.5−0.75j and reverses the output’s imaginary sign. That is the third table row. A complete negation and a conjugation are therefore useful but different audit controls.

For real numeric input two, the result is near 3.762195691. For the imaginary-axis input 2j, it is cos(2), near −0.416146837. Both inputs have magnitude two, so magnitude alone cannot predict the output.

For A2, use complete complex products:

=IMSUB(IMPRODUCT(IMCOSH(A2);IMCOSH(A2));IMPRODUCT(IMSINH(A2);IMSINH(A2)))

The independently expected real coefficient is one, and the imaginary coefficient is zero. IMSINH supplies the second complex term.

Subtracting squares of magnitudes is a different operation. For the first input, that difference equals cos(1.5), approximately 0.070737202, rather than one. The complete complex square retains phase and cross terms that a magnitude calculation discards.

Use both real and imaginary component differences when checking the identity. An absolute tolerance such as 1e-9 is a practical declared target for this moderate fixture, not a guarantee for every input.

The official 4−3i control has coefficients approximately −27.0349456030742 and −3.85115333481178, consistent with the same equation. The source’s OpenDocument 1.2 statement identifies standard membership, not Collabora’s original introduction date.

The dedicated section does not give a complete malformed-text, blank, logical-value or native overflow matrix. Those cases are not filled in from another product. Documented string output and numeric-input acceptance are retained, while all new numerical outputs remain independently modeled rather than native-tested.

• Collabora Office 26.04: IMCOSH.
• Editorial testing standards.

Complete dedicated reference and its equation image reviewed. Examples are independently calculated controls or explicit documented statements, not native application captures. Formulas use English names and semicolon separators.