IMCSCH Function (Calligra Sheets)

• Introduced: First release not established; the name is present in Calligra 2.4.0 source.
• Supported in: Calligra Sheets 26.08.2, with no retirement established.
• Scope: Desktop Sheets with the Engineering module. Examples use English semicolon-separated formulas.
• Checked: October 14, 2026. Formulas were executed through the official 25.04.2 engine library and checked against source and independent mathematics. This is not GUI-cell or workbook-import evidence; 26.08.2 was source-checked, not executed.

IMCSCH returns csch(z) = 1/sinh(z). It is a reciprocal hyperbolic sine rather than an inverse hyperbolic function. Its mathematical poles are at z = ikπ for integer k, including zero. Function help.

=IMCSCH(complexValue)

One complex value is required. Construct the components with COMPLEX(real;imaginary) to avoid ambiguity. Pure numeric inputs are real; recognized strings use the shared complex converter. Registration and implementation, with versioned help.

Results are shown as (real,imaginary) component pairs, rounded to twelve decimal places, not a literal cell format. Tiny or very large values use thirteen significant digits in scientific notation. A component rounded to zero is shown as 0; the stored value may still contain a tiny residual or signed zero. NaN means not-a-number; infinity and NaN are not finite results.

For z = i, csch(i)=−i csc(1), approximately 0−1.188395105778i. The negative imaginary sign follows from reciprocating i sin(1). It is not obtained by applying real CSCH separately to the two components.

A worksheet can use =IMCSCH(COMPLEX(A2;B2)) while retaining both numeric inputs. Pay particular attention when a model can approach the zero complex value.

At z = 0.5+0.25i, the executed product IMCSCH(z)×IMSINH(z) returned real component 0.999999999999999999946 and imaginary component about −2.710505431214E−20. This is consistent with the reciprocal identity at a non-pole input.

Csch is odd and satisfies csch(ib)=−i csc(b) for real b away from integer multiples of π. The relation explains why its imaginary-axis poles repeat even though its real-axis behavior looks like a rapidly decreasing reciprocal.

The exact advanced-check formula was:

=IMPRODUCT(IMCSCH(COMPLEX(0.5;0.25));IMSINH(COMPLEX(0.5;0.25)))

The exact-zero call returned (+Infinity,NaN) with the engine error flag unset. Such components are not a usable finite answer. The implementation directly divides one by the complex hyperbolic sine, without a worksheet-specific zero guard.

Malformed and empty text became zero and reached the same pole. IMCSCH(COMPLEX(0;PI())) instead returned a large finite negative imaginary component near −8.165619676598E15 because the represented π is slightly displaced from the exact pole. No GUI-cell formatting is inferred.

The complex conversion helper explains the text and Boolean cases above. The inspected Engineering module and complex converter are byte-identical between 25.04.2 and 26.08.2. That source comparison does not establish every locale, import path or extreme-value case.

Use COMPLEX to construct values from numeric components, IMABS for magnitude, and IMARGUMENT for principal phase.