IMCOTH in Google Sheets: Check Complex Hyperbolic Cotangent Ratios

• Introduced: first introduction date is not established by the reviewed official history; no desktop release number is inferred.
• Availability: documented in the rolling Google Sheets service; these examples use the web worksheet workflow.
• Checked: October 14, 2026; official IMCOTH reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMCOTH(number)

Google permits a COMPLEX result, a real number interpreted with imaginary part zero, or numeric x+yi text. Thus a numeric call such as =IMCOTH(0.5) is within the documented input scope; it need not be converted to text merely to represent a real quantity.

Hyperbolic cotangent divides complete hyperbolic cosine by complete hyperbolic sine. It is distinct from IMCOT, which uses ordinary trigonometric functions, and from an inverse hyperbolic function that recovers an argument.

The dedicated reference says invalid complex input causes an error, but does not name its code or provide a full coercion matrix.

Enter the first two numeric columns in A2:B5. Use =COMPLEX(A2,B2) in C2 and =IMCOTH(C2) in D2; fill down. Extract D’s numeric coordinates into E and F with IMREAL and IMAGINARY.

The output columns are independent expectations rounded to nine decimals, not extra input data. The second row demonstrates that real-only bounds must not be imposed on arbitrary complex hyperbolic cotangent.

Compare D2 with:

=IMDIV(IMCOSH(C2),IMSINH(C2))

Fill down. Each expected quantity agrees with D. IMCOSH and IMSINH provide the complete numerator and denominator.

Dividing corresponding real coordinates alone would omit imaginary cross terms. A useful audit checks both real and imaginary differences against an explicit tolerance such as 1E-9 for this moderate fixture.

The source does not promise identical numerical conditioning for all equivalent formulas, especially near zeros of the denominator.

An independent model for a+bi uses denominator COSH(2a)−COS(2b), real numerator SINH(2a), and imaginary numerator negative SIN(2b).

For the first input, check:

=SINH(1.5)/(COSH(1.5)-COS(0.5))
=-SIN(0.5)/(COSH(1.5)-COS(0.5))

These produce the independent targets 1.443748574 and −0.325072379. This equation is mathematically derived from the named function, rather than quoted from a displayed Google equation.

Ordinary cotangent places the hyperbolic functions on the other coordinate. Confusing the two formulas changes both coefficients even when their names look similar.

For the first input, hyperbolic cotangent also equals:

=IMDIV(IMSUM(IMEXP("0.75+0.25i"),IMEXP("-0.75-0.25i")),IMSUB(IMEXP("0.75+0.25i"),IMEXP("-0.75-0.25i")))

The expected quantity matches D2. The one-half factors in hyperbolic cosine and sine cancel in this quotient. IMEXP exposes the complete exponential terms.

This construction is a check of the mathematics, not a recommendation that it is always the most stable numerical implementation.

Multiply IMCOTH(C2) by IMTANH(C2). For this nonsingular input, the independent target is one with imaginary coefficient zero.

The complete square relation is different: IMCOTH(C2) squared minus IMCSCH(C2) squared equals one. For the first row, those squares have coefficients approximately 1.978737893−0.938645567i and 0.978737893−0.938645567i.

Use IMPRODUCT for each full complex square and IMSUB for their difference. Magnitude squares discard the imaginary cross terms and do not test the same identity.

Numeric input 0.5 has mathematical result approximately 2.163953414. Imaginary input 0.25i instead has quantity near −3.916317365i, because coth(bi)=−i/TAN(b).

Complete negation negates the result; conjugation conjugates it. These transformations are useful sign checks for a manual quotient.

The official numeric =IMCOTH(3.5) control is approximately 1.00182542850644, consistent with the mathematical model. Hyperbolic sine has zeros at integer multiples of pi times i; the dedicated reference does not establish precise native pole errors or floating-PI outcomes. Those cases, mixed representations and extreme limits remain unverified.

• Google IMCOTH reference.
• Editorial testing standards.

Examples use fictional data, English function names and comma separators. Expected outputs are independent mathematical calculations or explicitly documented rules, not native Sheets captures. Check the spreadsheet locale when reproducing punctuation-sensitive formulas.