IMCOSH in Excel: Compute Complex Hyperbolic Cosines and Check Even Symmetry

• Introduced: the Microsoft function catalogue marks this identity as an Excel 2013 addition. An exact first build or preview date is not established.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official IMCOSH reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMCOSH(inumber)

Inumber is required. Microsoft describes x+yi or x+yj text and recommends COMPLEX for numeric coefficient inputs.

A value outside the stated complex format has documented #NUM! behavior; logical input has documented #VALUE! behavior. The dedicated reference does not provide a complete accepted-shortcut, blank or competing-error table.

The component model above comes from the mathematical complex hyperbolic cosine definition. It is an independent checking method; the dedicated page itself provides the function contract and a 4+3i worked result rather than this full equation.

Enter the first column as text in A2:A5. Put =IMCOSH(A2) in B2 and fill down. Extract real and imaginary output coefficients into C and D.

The last row reduces to COS(1), not COSH(1). The imaginary-axis input carries an ordinary trigonometric argument in the component identity.

The second and third rows are conjugate results. Their real coefficients agree, while their imaginary signs differ. Preserve both coefficients if the output is later used in multiplication or division.

The displayed numbers are rounded independent expectations. They are not native Excel string captures or guarantees of a specific number of returned digits.

For z=1+i, calculate:

=COSH(1)*COS(1)
=SINH(1)*SIN(1)

Expected coefficients are approximately 0.833730025 and 0.988897706. Unlike the complex ordinary-cosine equation, this imaginary term has a positive sign.

For a cell-driven check, use IMREAL(A2) for the hyperbolic arguments and IMAGINARY(A2) for the ordinary trigonometric arguments. IMREAL and IMAGINARY return numeric coordinates for this purpose.

A formula that merely applies COSH to each coordinate separately would produce neither the documented complex operation nor the correct cross-component structure.

Hyperbolic cosine is half the sum of the exponentials of z and −z. For z=1+i, reproduce that definition with:

=IMDIV(IMSUM(IMEXP("1+i"),IMEXP("-1-i")),COMPLEX(2,0))

The two exponentials have coefficients approximately (1.468693940,2.287355287) and (0.198766110,−0.309559876). Half their sum gives approximately (0.833730025,0.988897706).

IMEXP and IMSUM make the intermediate quantities inspectable. Constructing the real divisor explicitly keeps the formula’s complex-input intent clear.

This structural identity does not guarantee equal numerical stability for direct exponential evaluation at extreme real coefficients. Keep the independent example within its moderate scale.

Hyperbolic cosine is even: negating both coefficients of 1+i gives −1−i, whose independently expected output is the same as that of 1+i.

Conjugation is different. Changing only the imaginary sign gives 1−i, whose output has negative imaginary coefficient. The two transformations should not be treated as equivalent sign corrections.

For real-only input 1+0i, the expected real result is COSH(1), approximately 1.543080635. For imaginary-only input 0+1i, it is COS(1), approximately 0.540302306. Both have mathematical imaginary coefficient zero.

For z=1+i, compute c=IMCOSH(z) and s=IMSINH(z). The mathematical difference c squared minus s squared has real coefficient one and imaginary coefficient zero.

Use complete complex products:

=IMSUB(IMPRODUCT(IMCOSH("1+i"),IMCOSH("1+i")),IMPRODUCT(IMSINH("1+i"),IMSINH("1+i")))

The squared quantities have matching imaginary coefficients, so subtraction cancels them. This identity is not obtained by subtracting squares of magnitudes; that would discard the phase information before the subtraction.

For these inputs, the difference of squared magnitudes instead equals COS(2), approximately −0.416146837. It is a different valid mathematical expression with a different answer.

Microsoft’s explicit 4+3i result has coefficients near −27.0349456030742 and 3.85115333481178, corroborated by the independent component equation.

The named logical and invalid-format errors do not establish native overflow, underflow, error precedence or all alternate parsing forms. Tiny cancellation residuals may remain in approximate identity checks; compare numeric components using a stated tolerance rather than exact displayed-string equality.

All worked outputs were independently modeled. No native Excel execution was performed.

• Microsoft IMCOSH reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.