IMTANH in Google Sheets: Verify Complex Hyperbolic Tangent without Real-Only Bounds

• 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 IMTANH reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMTANH(number)

Google explicitly permits real numeric values, a COMPLEX result or x+yi text. A real numeric value represents an imaginary coefficient of zero. The source does not say that the visible representation alone establishes a particular native TYPE classification.

Hyperbolic tangent divides complete hyperbolic sine by complete hyperbolic cosine. It is distinct from ordinary IMTAN and from inverse hyperbolic tangent.

Invalid complex input causes a documented error, but the dedicated page does not assign a code or describe every blank, logical or text coercion case.

Enter the two numeric input columns in A2:B5. Put =COMPLEX(A2,B2) in C2 and =IMTANH(C2) in D2, filling down. In E and F, extract D’s coefficients with IMREAL and IMAGINARY.

Expected coefficients are rounded independent calculations. The second row’s real coefficient exceeds one. The familiar real TANH bound does not apply to every coordinate of a complex result.

Do not clip the second row to one or silently discard its imaginary coefficient. Either action changes the mathematical quantity.

Compare D2 with:

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

Fill down. The independently expected quantities match the primary table. IMSINH and IMCOSH supply the two complete values.

This division uses both coordinates and the denominator’s squared magnitude. Dividing only the real coefficients would calculate a different ratio.

For this moderate fixture, test both component differences against a declared threshold such as 1E-9. Keep calculated D values rather than re-entering the rounded display table.

For input a+bi, a component model has denominator COSH(2a)+COS(2b), real numerator SINH(2a) and imaginary numerator SIN(2b).

For the first row, use:

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

Expected coefficients are 0.659221242 and 0.148429319. The plus in the denominator distinguishes this model from hyperbolic cotangent.

The expanded equation is an independent mathematical derivation. Google’s dedicated page identifies the function and provides controls, rather than printing this component formula.

For the first row, calculate:

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

Its independent target matches D2. The common one-half factors cancel when hyperbolic sine is divided by hyperbolic cosine.

For the same nonsingular fixture, multiplying IMTANH(C2) by IMCOTH(C2) yields mathematical one with zero imaginary coefficient. Reciprocal recovery is a separate check from the exponential expansion.

For C2, compare the sum of complete squares:

=IMSUM(IMPRODUCT(IMTANH(C2),IMTANH(C2)),IMPRODUCT(IMSECH(C2),IMSECH(C2)))

The independent target is one with imaginary coefficient zero. The tangent square has coefficients approximately 0.412541383+0.195695520i. The hyperbolic-secant square has approximately 0.587458617−0.195695520i.

IMSECH supplies the reciprocal hyperbolic cosine. Full complex products preserve the imaginary terms that cancel; squares of magnitudes are different operations.

Numeric =IMTANH(0.5) has mathematical target approximately 0.462117157. Imaginary input 0.25i instead gives approximately 0.255341921i, following tanh(bi)=i TAN(b).

Complete input negation negates the function, while conjugation conjugates its output. The third primary row changes only the real input coordinate, explaining why its imaginary output retains the first row’s sign.

Google’s documented =IMTANH(3.5) control is approximately 0.998177897611199. Mathematical poles occur where complex hyperbolic cosine is zero, on imaginary pi/2 plus integer multiples of pi times i. Native error codes, approximate-PI behavior and extreme numerical limits are not established by this dedicated reference.

• Google IMTANH 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.