IMARCSINH Function in Gnumeric: Syntax, Examples and Branch Cuts

• Introduced: 1.1.6, according to the official release history. This was a development-series release.
• Availability: Supported in Gnumeric 1.12.62, based on the function-specific registration.
• Scope: Desktop; the fn-complex function module must be included and enabled.
• Checked: October 14, 2026. Source and independently calculated examples checked; Gnumeric 1.12.62 was not run.

IMARCSINH returns the principal complex inverse of hyperbolic sine. It operates on the whole complex value, not by applying the real inverse separately to each coefficient. The forward relation is sinh(w)=z; selecting one inverse branch is necessary because the forward function is not one-to-one. Function reference.

=IMARCSINH(z)

z is one required complex number: a quoted complex string, a cell holding one, or the result of COMPLEX(real,imaginary). There is no second argument or default value. For example:

=IMARCSINH("1+2i")
=IMARCSINH(COMPLEX(1,2))
=IMARCSINH(A2)

Quote a literal such as "1+2i"; do not enter a bare 1+2i expression. COMPLEX accepts an optional i/j suffix. The examples use comma separators inside multi-argument functions; local formula conventions can differ.

Enter each column-A formula in the stated row, then enter its column-B formula. Results are mathematical reference values rounded to nine decimal places. Any residual column measures the reference calculation’s forward reconstruction.

Use =IMREAL(B2) and =IMAGINARY(B2) to compare coefficients. A different i/j suffix or display precision does not by itself change the complex value.

The imaginary-axis rays below −i and above +i are branch cuts. The principal formula Log(z+sqrt(z²+1)) requires matching logarithm and square-root branches. Values near ±i are sensitive. For a real input, the mathematical inverse hyperbolic sine remains real. Documented branch behavior.

Compare COMPLEX(0.001,1.001) and COMPLEX(-0.001,1.001) on opposite sides of the same cut. Compare real and imaginary coefficients, not just formatted strings. Exact-cut signed-zero conventions and extreme-input behavior need checking in your installed version.

The forward check IMSINH(IMARCSINH(z))≈z is useful for finite nonsingular inputs. Reversing that composition need not recover an arbitrary original argument: the inverse returns the principal branch.

For a modeled response H=sinh(u), IMARCSINH(H) gives a principal u. Shifts by 2πi and symmetry-related alternatives can produce the same H. Choose the candidate satisfying the model’s continuity and boundary conditions.

Use A4’s 1+2i sample from the table. In C4 enter =IMSINH(B4) to reconstruct the input, and in D4 enter =IMABS(IMSUB(C4,A4)) for the residual. The independent reference reconstruction is within 1e−12 of the input; your numerical tolerance should reflect the model and input scale.

=IMSINH(IMARCSINH(A4))
=IMABS(IMSUB(IMSINH(IMARCSINH(A4)),A4))

A second mathematical check

The relation asinh(z)=−i·asin(iz) offers another check when principal branches are consistent. For A4’s sample:

=IMPRODUCT("0-i",IMARCSIN(IMPRODUCT("0+i",A4)))

Use the direct IMSINH reconstruction as the primary check. Algebraic rearrangements across a cut can select a different inverse branch.

The 1.12.62 complex parser maps malformed complex text to #NUM!, whereas the web manual’s generic invalid-input note says #VALUE!. The pinned parser rejects other unsupported types with #VALUE!; coercion before that helper can affect blanks or Booleans. Use an explicit complex string or COMPLEX result when diagnosing an input problem.

Near the branch points described above, a small input change can produce a large inverse change. The example rows avoid exact-cut ambiguity; retain that distinction when checking a branch-sensitive model.

• IMSINH
• IMREAL
• IMAGINARY
• IMABS
• COMPLEX

• IMARCSINH manual entry
• Gnumeric 1.12.62 IMARCSINH help and evaluator
• One-argument registration in the examined release
• Release complex inverse algorithms