IMSINH Function (Gnumeric)

IMSINH evaluates complex hyperbolic sine for one complex-number argument. Find the hyperbolic sine of a complex value; useful in wave propagation, transfer functions and exponential solutions. It is not an ordinary real-only function, and its result may have both real and imaginary components. M1, S1.

For z = x + yi, the complex unit i satisfies i² = −1. The primary mathematical identity is sinh(z) = (exp(z) − exp(−z))/2. Complex arithmetic is important even if the source data begin as ordinary real values.

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.

=IMSINH(z)

Use the spelling IMSINH. The documented function accepts one argument, not an arbitrary list or separate real,imag parameters. Create a value with =COMPLEX(1,2) first if your inputs are coefficients. COMPLEX(x,y,i) also documents a third optional suffix of "i" or "j", defaulting to "i". An inspected upstream helper accepts numeric values and complex strings, but no release-specific claims about coercion of booleans or formatting are made here. M1, S1.

The formulas below use commas and decimal points in the English-language notation used in the Gnumeric manual and examples. Your configured locale may use different argument separators or decimal conventions. That locale behavior has not been independently exercised, so consult the formula editor on the installed program before copying across locales.

=IMSINH("1+2i")
=IMSINH(COMPLEX(1,2))
=IMSINH("3+4i")

The two first expressions represent the same mathematical input. Expected numeric coefficients are listed below, but the exact number of digits, i/j suffix and error display are not claimed as tested Gnumeric outputs.

In an empty sheet, enter each expression in column A and enter the corresponding IMSINH formula into column B. The first column deliberately uses COMPLEX(real,imag) to remove ambiguity from hand-typed complex-number syntax.

All finite expected coefficients above were calculated independently using the mathematical complex operations in Python 3’s cmath module on October 13, 2026. No native Gnumeric execution was conducted. Small differences in rounded displays are expected across numerical libraries; compare numerical real/imaginary components rather than literal complex text.

For a direct text-input check, enter "1+2i" as a quoted argument in the function formula. For a j-suffix check, construct the input with =COMPLEX(1,2,"j"). The inspected upstream code carries a parsed suffix into the complex output representation, but the exact 1.12.62 application display has not been tested. M1, S1.

Why the formula is not an ordinary real sine

For z = x + yi, sinh(z) = sinh(x) cos(y) + i cosh(x) sin(y). The imaginary component uses an ordinary sine, while the real component uses an ordinary cosine. All trigonometric angles in these identities are in radians. A spreadsheet formula such as =IMSINH("1+2i") is not the same as =SINH(1) plus two, and it is not the same as =IMSIN("1+2i"). This distinction matters when analysing alternating-current signals or transfer functions.

For purely imaginary z = yi, the identity simplifies to sinh(yi) = i sin(y). For purely real z = x, it simplifies to the ordinary real sinh(x). For a negative input, sinh(−z) = −sinh(z), an odd-function property useful as a sign check. Each is an independent mathematical identity; the in-application rounding and text representation still require native validation.

Interpreting the complex answer

A result such as -0.48905626+1.40311925i is a pair of coefficients, not two cells. Use IMREAL and IMAGINARY to extract components. Use IMABS if you need the output magnitude. If the suffix is j, it represents the same mathematical imaginary unit; the inspected source preserves the incoming suffix where applicable, but the current release’s display behavior was not executed.

Worked engineering use: A hypothetical propagation constant is γL = 0.5 + 0.25i (dimensionless). Put =COMPLEX(0.5,0.25) in A2, =IMSINH(A2) in B2, and =IMABS(B2) in C2. For this positive-real input, B2 has the independently calculated value approximately 0.50489571+0.27897913i; the table instead includes the distinct input -0.5+0.25i. C2 represents the output magnitude. This is a mathematical illustration, not a claim about a particular cable, transmission line or Gnumeric test.

Cross-check by exponential identity: Compare =IMSINH(A2) with =IMDIV(IMSUB(IMEXP(A2),IMEXP(IMPRODUCT(A2,-1))),2). The expressions are mathematically equivalent. Complex floating-point rounding may make their printed coefficients differ slightly; compare components at a reasonable tolerance, not as exact text.

For a second, structurally different validation in the worksheet, use the following documented-function identity:

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

Expected: 1 + 0i (identity cosh²(z) − sinh²(z) = 1). This is an algebraic check rather than a second implementation of the same source code. Where the comparison uses division, it is valid only away from mathematical poles. If a worksheet displays a string, first extract real and imaginary components with IMREAL/IMAGINARY instead of attempting unsupported direct arithmetic on arbitrary complex text.

An alternative expression for the same result is:

=IMDIV(IMSUB(IMEXP(A2),IMEXP(IMPRODUCT(A2,-1))),2)

This is intentionally presented as a mathematical cross-check for a future native reviewer, not a guaranteed drop-in replacement under every locale or Gnumeric build.

Numerical precision and mathematical domain

Hyperbolic functions involve exponentials. Inputs with very large real parts can overflow, and subtracting large nearly equal terms in the exponential identity can lose significant digits. Such numerical concerns are mathematical/floating-point limitations, not observed application-specific failures. For small |z|, sinh(z) ≈ z; at z = 0 the exact mathematical result is zero.

The official 1.12.62 code returns #NUM! if its internally computed complex result is invalid, while its published manual’s generic invalid-input note says #VALUE!. Do not interpret the catalogue’s Exhaustive test label as evidence for any specific worksheet example or on every platform.

• Software studied: Gnumeric desktop spreadsheet. The Gnumeric team announced version 1.12.62 in September 2026; source packages and platform-specific builds can differ. This manuscript is not certified against a locally installed copy on Linux, Windows or macOS. R1.

• Upstream catalogue: IMSINH appears as Complete / Exhaustive in the currently accessible official catalogue. These labels summarize the upstream catalogue; they neither identify a specific package build nor replace native testing. C1.

• Excel and other spreadsheets: The inspected source’s function-help block includes an Excel-compatibility statement, whereas the published web manual does not explicitly certify each of these functions as Excel compatible. Therefore, the source claim is labelled release-source documentation, not proven cross-product byte-for-byte equivalence. Excel/LibreOffice behavior was not used to deduce Gnumeric outputs. S1, M1.

• Implementation evidence: The official manual confirms the function and syntax, the catalogue assigns a testing classification, and the inspected 1.12.62 source evaluates the expected complex operation. None alone means this exact workbook example ran in the 1.12.62 release.

IMCOSH, IMTANH, IMEXP, IMSUB, IMDIV, IMABS, IMREAL, IMAGINARY, COMPLEX.

No native Gnumeric execution, application screenshots, locale tests or workbook round-trip checks were performed. Source review and independent calculations do not replace application-level testing.