IMSEC Function (Gnumeric)

IMSEC evaluates complex secant for one complex-number argument. Compute the reciprocal of complex cosine, useful for reciprocal identities, complex-wave ratios and exploratory trigonometric models. 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 sec(z) = 1/cos(z). 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.

=IMSEC(z)

Use the spelling IMSEC. 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.

=IMSEC("1+2i")
=IMSEC(COMPLEX(1,2))
=IMSEC("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 IMSEC 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.

Reciprocal cosine, not inverse cosine

Gnumeric’s Complex manual defines IMSEC(z) = 1 / IMCOS(z). The word secant here means the reciprocal of cosine. It does not mean arccosine, which answers a different mathematical question. The documented IMARCCOS function is the inverse cosine in the functional sense (with complex branch cuts). Confusing the two leads to entirely different results.

The cosine in this formula is the complex cosine, not a separately applied real cosine on each coefficient. For z = x + yi, cos(z) = cos(x) cosh(y) − i sin(x) sinh(y). Hence sec(z) must be the reciprocal of that entire complex quantity. Computing 1/COS(IMREAL(z)) ignores the imaginary component and is not generally equivalent.

Where secant becomes unstable

Cosine is zero at z = π/2 + kπ for integer k. As a result, complex secant has poles at those points. A small cosine magnitude makes the reciprocal very large. For a numerical model, check =IMABS(IMCOS(A2)) before taking the reciprocal if values may approach a pole. The display may be affected by rounding and by the precision of π entered in the sheet.

Worked reciprocal check: Put =COMPLEX(1,2) in A2, =IMSEC(A2) in B2 and =IMDIV(1,IMCOS(A2)) in C2. The results should match mathematically; in D2 =IMPRODUCT(B2,IMCOS(A2)) should be close to 1+0i. If the input is complex, compare components numerically, not formatted text.

Example with a pure imaginary input: =IMSEC(COMPLEX(0,1)) represents 1/cosh(1), a positive real number less than one. This follows from cos(i) = cosh(1). It illustrates why a nonzero imaginary part changes the size of the reciprocal.

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

=IMPRODUCT(IMSEC(A2),IMCOS(A2))

Expected: 1 + 0i away from poles, within floating-point tolerance. 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(1,IMCOS(A2))

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

Secant is mathematically undefined at zeros of cosine. Near those zeros, cancellation and finite-precision arithmetic can create large output sensitivity. An error message at a pole, if any, is a runtime behavior to verify rather than guess. The upstream code calculates an inverse of complex cosine, supporting the mathematical identity, but these exact examples were not executed in the 1.12.62 application.

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: IMSEC 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.

IMCOS, IMDIV, IMCOT, IMCSC, IMABS, IMARCCOS, IMPRODUCT, 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.