IMCOTH

GOOGLE'S OWN PUBLISHED RESULT, quoted verbatim from the Formula/Result table in the article body: =IMCOTH(COMPLEX(4,1)) -> 0.999720649533931-0.000609900253822228i. Derived independently as coth(4+i) = 0.999720649533930749036738985046491656585 - 0.0006099002538222275728563166231837522767836i, which rounds to the published 15 significant digits exactly. INDEPENDENT DERIVATION. Every figure quoted from these three pages was recomputed here at 40 decimal digits with mpmath before it was written down -- coth/tanh/log evaluated directly on the complex argument -- and all nine of Google's published results reproduce EXACTLY at the 15 significant digits the pages print, correctly rounded. Nothing on these pages needed correcting. NOTE ON THE RESULT FORMAT, WHICH THE PAGE NEVER STATES. None of the three pages contains a sentence describing the shape of the returned complex value -- no "a+bi" rule, no i-versus-j statement, no precision or rounding rule. The format is visible ONLY in the literal result strings the pages print. That is why this file asserts the string form in exactly one case per function and asserts the numeric components through IMREAL/IMAGINARY everywhere else: a difference in formatting and a difference in the mathematics are different findings and should not be able to masquerade as each other. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256. BATCH PROVENANCE (batch H, the first Sheets/LibreOffice-only batch). Every function in this batch has x == false in docs/data/compat.json: it is a Google Sheets function that Microsoft does not document at all, so this corpus makes NO claim about it in that engine and nothing here is measured against that vendor's documentation. The authority is Google's own support.google.com function page, cited by full URL and by the date it was read -- Google publishes no version number for these pages, so a bare URL dates nothing. WHAT GOOGLE ACTUALLY PRINTS, WHICH IS LESS THAN IT LOOKS: most of these pages carry a 'Sample Usage' block of FORMULAS WITH NO RESULTS. Where a value below is Google's own published output the note says so; where it is derived from the page's stated semantics the note says that instead, and says from which sentence. No value in this batch was taken from a search snippet, a blog or a mirror. LIBREOFFICE: probed on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3) in five storage spellings -- plain, _xlfn., COM.MICROSOFT., ORG.OPENOFFICE. and _xlfn.ORG.OPENOFFICE. -- and #NAME? under every one of them, so the expected values below describe Google Sheets and the LibreOffice column records absence.

GOOGLE'S OWN PUBLISHED RESULT: =IMCOTH(3.5) -> 1.00182542850644. Derived independently as coth(3.5) = 1.001825428506443467329849802769891235631. ROUND-wrapped at 14 places, which is where the published figure ends. The `number` argument description is what licenses a bare real here: "a real number interpreted as a complex number with imaginary parts equal to 0". Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

GOOGLE'S OWN PUBLISHED RESULT: =IMCOTH("3+2i") -> 0.996757796569358+0.00373971037633696i. This case asserts the REAL part of that value, taken through IMREAL so that a formatting difference cannot be mistaken for a numerical one. Derived independently as Re coth(3+2i) = 0.996757796569358310460968797117470718332. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

The imaginary half of the same published result, 0.00373971037633696, derived independently as Im coth(3+2i) = 0.003739710376336956660117408691902576240006. Rounded at 17 decimal places, which is 15 significant digits for a number of this size. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

A STRUCTURAL assertion with no derived constant. The page defines the function only by name -- "returns the hyperbolic cotangent of the given complex number" -- and its sibling page defines IMTANH as the hyperbolic tangent; coth = 1/tanh is what those two names mean, so the two spellings must agree in the same engine whatever either computes. Compared through IMABS of the difference, because complex results are strings and cannot be subtracted with `-`. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256. Google's IMTANH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366655.

GOOGLE'S OWN PUBLISHED RESULT, quoted verbatim from the Formula/Result table in the article body: =IMCOTH(COMPLEX(4,1)) -> 0.999720649533931-0.000609900253822228i. Derived independently as coth(4+i) = 0.999720649533930749036738985046491656585 - 0.0006099002538222275728563166231837522767836i, which rounds to the published 15 significant digits exactly. INDEPENDENT DERIVATION. Every figure quoted from these three pages was recomputed here at 40 decimal digits with mpmath before it was written down -- coth/tanh/log evaluated directly on the complex argument -- and all nine of Google's published results reproduce EXACTLY at the 15 significant digits the pages print, correctly rounded. Nothing on these pages needed correcting. NOTE ON THE RESULT FORMAT, WHICH THE PAGE NEVER STATES. None of the three pages contains a sentence describing the shape of the returned complex value -- no "a+bi" rule, no i-versus-j statement, no precision or rounding rule. The format is visible ONLY in the literal result strings the pages print. That is why this file asserts the string form in exactly one case per function and asserts the numeric components through IMREAL/IMAGINARY everywhere else: a difference in formatting and a difference in the mathematics are different findings and should not be able to masquerade as each other. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256. BATCH PROVENANCE (batch H, the first Sheets/LibreOffice-only batch). Every function in this batch has x == false in docs/data/compat.json: it is a Google Sheets function that Microsoft does not document at all, so this corpus makes NO claim about it in that engine and nothing here is measured against that vendor's documentation. The authority is Google's own support.google.com function page, cited by full URL and by the date it was read -- Google publishes no version number for these pages, so a bare URL dates nothing. WHAT GOOGLE ACTUALLY PRINTS, WHICH IS LESS THAN IT LOOKS: most of these pages carry a 'Sample Usage' block of FORMULAS WITH NO RESULTS. Where a value below is Google's own published output the note says so; where it is derived from the page's stated semantics the note says that instead, and says from which sentence. No value in this batch was taken from a search snippet, a blog or a mirror. LIBREOFFICE: probed on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3) in five storage spellings -- plain, _xlfn., COM.MICROSOFT., ORG.OPENOFFICE. and _xlfn.ORG.OPENOFFICE. -- and #NAME? under every one of them, so the expected values below describe Google Sheets and the LibreOffice column records absence.

GOOGLE'S OWN PUBLISHED RESULT: =IMCOTH(3.5) -> 1.00182542850644. Derived independently as coth(3.5) = 1.001825428506443467329849802769891235631. ROUND-wrapped at 14 places, which is where the published figure ends. The `number` argument description is what licenses a bare real here: "a real number interpreted as a complex number with imaginary parts equal to 0". Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

GOOGLE'S OWN PUBLISHED RESULT: =IMCOTH("3+2i") -> 0.996757796569358+0.00373971037633696i. This case asserts the REAL part of that value, taken through IMREAL so that a formatting difference cannot be mistaken for a numerical one. Derived independently as Re coth(3+2i) = 0.996757796569358310460968797117470718332. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

The imaginary half of the same published result, 0.00373971037633696, derived independently as Im coth(3+2i) = 0.003739710376336956660117408691902576240006. Rounded at 17 decimal places, which is 15 significant digits for a number of this size. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

A STRUCTURAL assertion with no derived constant. The page defines the function only by name -- "returns the hyperbolic cotangent of the given complex number" -- and its sibling page defines IMTANH as the hyperbolic tangent; coth = 1/tanh is what those two names mean, so the two spellings must agree in the same engine whatever either computes. Compared through IMABS of the difference, because complex results are strings and cannot be subtracted with `-`. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256. Google's IMTANH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366655.

GOOGLE'S OWN PUBLISHED RESULT, quoted verbatim from the Formula/Result table in the article body: =IMCOTH(COMPLEX(4,1)) -> 0.999720649533931-0.000609900253822228i. Derived independently as coth(4+i) = 0.999720649533930749036738985046491656585 - 0.0006099002538222275728563166231837522767836i, which rounds to the published 15 significant digits exactly. INDEPENDENT DERIVATION. Every figure quoted from these three pages was recomputed here at 40 decimal digits with mpmath before it was written down -- coth/tanh/log evaluated directly on the complex argument -- and all nine of Google's published results reproduce EXACTLY at the 15 significant digits the pages print, correctly rounded. Nothing on these pages needed correcting. NOTE ON THE RESULT FORMAT, WHICH THE PAGE NEVER STATES. None of the three pages contains a sentence describing the shape of the returned complex value -- no "a+bi" rule, no i-versus-j statement, no precision or rounding rule. The format is visible ONLY in the literal result strings the pages print. That is why this file asserts the string form in exactly one case per function and asserts the numeric components through IMREAL/IMAGINARY everywhere else: a difference in formatting and a difference in the mathematics are different findings and should not be able to masquerade as each other. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256. BATCH PROVENANCE (batch H, the first Sheets/LibreOffice-only batch). Every function in this batch has x == false in docs/data/compat.json: it is a Google Sheets function that Microsoft does not document at all, so this corpus makes NO claim about it in that engine and nothing here is measured against that vendor's documentation. The authority is Google's own support.google.com function page, cited by full URL and by the date it was read -- Google publishes no version number for these pages, so a bare URL dates nothing. WHAT GOOGLE ACTUALLY PRINTS, WHICH IS LESS THAN IT LOOKS: most of these pages carry a 'Sample Usage' block of FORMULAS WITH NO RESULTS. Where a value below is Google's own published output the note says so; where it is derived from the page's stated semantics the note says that instead, and says from which sentence. No value in this batch was taken from a search snippet, a blog or a mirror. LIBREOFFICE: probed on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3) in five storage spellings -- plain, _xlfn., COM.MICROSOFT., ORG.OPENOFFICE. and _xlfn.ORG.OPENOFFICE. -- and #NAME? under every one of them, so the expected values below describe Google Sheets and the LibreOffice column records absence.

GOOGLE'S OWN PUBLISHED RESULT: =IMCOTH(3.5) -> 1.00182542850644. Derived independently as coth(3.5) = 1.001825428506443467329849802769891235631. ROUND-wrapped at 14 places, which is where the published figure ends. The `number` argument description is what licenses a bare real here: "a real number interpreted as a complex number with imaginary parts equal to 0". Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

GOOGLE'S OWN PUBLISHED RESULT: =IMCOTH("3+2i") -> 0.996757796569358+0.00373971037633696i. This case asserts the REAL part of that value, taken through IMREAL so that a formatting difference cannot be mistaken for a numerical one. Derived independently as Re coth(3+2i) = 0.996757796569358310460968797117470718332. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

The imaginary half of the same published result, 0.00373971037633696, derived independently as Im coth(3+2i) = 0.003739710376336956660117408691902576240006. Rounded at 17 decimal places, which is 15 significant digits for a number of this size. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256.

A STRUCTURAL assertion with no derived constant. The page defines the function only by name -- "returns the hyperbolic cotangent of the given complex number" -- and its sibling page defines IMTANH as the hyperbolic tangent; coth = 1/tanh is what those two names mean, so the two spellings must agree in the same engine whatever either computes. Compared through IMABS of the difference, because complex results are strings and cannot be subtracted with `-`. Google's IMCOTH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366256. Google's IMTANH page, read live on 2026-08-31 at https://support.google.com/docs/answer/9366655.