IMSECH in Google Sheets: Check Hyperbolic Secant without Real-Only Limits

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

=IMSECH(number)

Google accepts real numeric inputs, COMPLEX results and numeric x+yi text. A numeric real input has imaginary coefficient zero. The dedicated reference does not establish a complete native type-classification rule for scalar-looking results.

Hyperbolic secant is the reciprocal of complete hyperbolic cosine. It is distinct from ordinary IMSEC and is not inverse hyperbolic cosine.

Invalid complex input produces a documented error without a specified code. Blank, logical and mixed-representation behavior is not fully described by this page.

Enter only the first two numeric columns in A2:B5. Put =COMPLEX(A2,B2) in C2 and =IMSECH(C2) in D2, filling down. Use IMREAL and IMAGINARY on D to populate E:F.

Expected coefficients are rounded independent calculations. The final result exceeds one even though hyperbolic secant of a finite real argument is positive and at most one. Its input is imaginary, so the real-only restriction does not apply.

The second row’s imaginary coefficient also has magnitude greater than one. Clipping either result to a real-only range would change the quantity.

Compare D2 with:

=IMDIV(COMPLEX(1,0),IMCOSH(C2))

The independent quantities match D for the selected rows. IMCOSH gives a first-row denominator approximately 1.254434717+0.203444415i.

Its squared magnitude is approximately 1.614996089. Dividing the conjugate coordinates by that shared value produces hyperbolic secant, including the negative imaginary coefficient.

Two separate scalar reciprocals would be a different operation. Test both components after IMDIV, rather than relying on a matching real part or magnitude alone.

For input a+bi, a component model uses denominator COSH(2a)+COS(2b). Its real numerator is 2COSH(a)COS(b); its imaginary numerator is negative 2SINH(a)SIN(b).

For the first row, use:

=2*COSH(0.75)*COS(0.25)/(COSH(1.5)+COS(0.5))
=-2*SINH(0.75)*SIN(0.25)/(COSH(1.5)+COS(0.5))

The independently expected results are 0.776741644 and −0.125972079. This expanded expression is mathematically derived, not an equation printed in the dedicated Google reference.

For the moderate worked inputs, compare both component differences within an explicit tolerance such as 1E-9. Retain calculated values rather than propagating the rounded table coefficients.

Hyperbolic cosine equals half the sum of the exponentials of z and −z. Its reciprocal for C2 can therefore be checked with:

=IMDIV(COMPLEX(2,0),IMSUM(IMEXP("0.75+0.25i"),IMEXP("-0.75-0.25i")))

The independent target matches D2. The numerator two accounts for the half in the hyperbolic-cosine definition. IMEXP supplies complete quantities before IMSUM combines them.

For each primary row, multiplying IMSECH(C2) by IMCOSH(C2) gives mathematical one with imaginary coefficient zero. The chosen inputs avoid zero denominators.

Compare complete squares of hyperbolic secant and IMTANH:

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

The independent target is one. For the first input, the secant square is approximately 0.587458617−0.195695520i, and the tangent square is approximately 0.412541383+0.195695520i.

The imaginary coefficients cancel. A square of a magnitude removes the cross term and is not a substitute for either complete complex product.

Numeric =IMSECH(0.5) has mathematical target approximately 0.886818884. Imaginary 0.25i instead gives approximately 1.032085024, because sech(bi)=1/COS(b). At 2i, the independent real output is approximately −2.402997962.

Complete negation leaves the function unchanged, while conjugation conjugates its output. These distinct controls expose sign errors in a manual reciprocal.

Google’s documented numeric 3.5 result is approximately 0.0603397441201677. Its 3+2i control has coefficients approximately −0.0416749644111443−0.0906111371962376i, corroborated independently.

Mathematical poles occur on imaginary pi/2 plus integer multiples of pi times i. Native error codes, finite-PI boundary outcomes and extreme numerical precision are not established by the dedicated page. Those limits remain explicit rather than inheriting another spreadsheet product’s behavior.

• Google IMSECH 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.