IMSECH in Collabora Office: Separate Complex Hyperbolic Secant from Real-Only Bounds

• Introduced: the original Collabora release and launch date are not established by the reviewed sources.
• Availability: documented in Collabora Office 26.04 desktop help.
• Platform scope: online and mobile deployment support is not inferred from the desktop documentation.
• Checked: October 14, 2026. Function-specific documentation and independent controls; no native Collabora execution.

=IMSECH(Complex_number)

Collabora accepts a+bi or a+bj text, with numeric input also permitted when the imaginary part is zero. The result is described as a string representing a complex quantity. A zero component is omitted from the display.

The numeric control =IMSECH(2) returns approximately 0.26580222883408 as text. IMREAL can supply a numeric real coefficient for a later calculation; the scalar-looking string itself retains the documented representation.

Hyperbolic secant is a reciprocal, not an inverse hyperbolic cosine that returns an input argument.

Format A2:A5 as Text and enter 0.5+0.75j, 2+j, 0.5-0.75j and 2j. Fill down =IMSECH(A2) in B, =IMREAL(B2) in C and =IMAGINARY(B2) in D.

The expected displays are independently calculated and rounded to nine decimals. Retain the actual B formulas for later checks.

For real arguments, hyperbolic secant lies between zero and one. The first and final complex inputs show why that real-only bound must not be imposed on arbitrary complex outputs.

Compare IMSECH(A2) with:

=IMDIV(COMPLEX(1;0;"j");IMCOSH(A2))

The expected coefficients match the table. For the first input, hyperbolic cosine is approximately 0.825071367+0.355198758j, with squared magnitude approximately 0.806908918.

Its reciprocal uses the conjugate divided by that shared squared magnitude. Both the sign change and the complete denominator are necessary; independent scalar reciprocals of the two coordinates would produce a different quantity.

IMCOSH provides the full denominator, and IMDIV keeps the complex division explicit.

The source image uses denominator cosh(2a)+cos(2b). Its real numerator is 2cosh(a)cos(b), and its imaginary numerator is negative 2sinh(a)sin(b).

For a=0.5 and b=0.75, check:

=2*COSH(0.5)*COS(0.75)/(COSH(1)+COS(1.5))
=-2*SINH(0.5)*SIN(0.75)/(COSH(1)+COS(1.5))

The independently expected coefficients are 1.022508673 and −0.440196842, rounded. The coordinates supplied to the hyperbolic and ordinary functions differ from the ordinary complex-secant formula.

Hyperbolic cosine is half the sum of exp(z) and exp(−z). Therefore its reciprocal can be modeled by:

=IMDIV(COMPLEX(2;0;"j");IMSUM(IMEXP("0.5+0.75j");IMEXP("-0.5-0.75j")))

Expected coefficients match the first row. The numerator two accounts for the one-half factor in hyperbolic cosine.

For every primary row, =IMPRODUCT(IMSECH(A2);IMCOSH(A2)) has mathematical target one, with zero imaginary coefficient. All denominators in this fixture are nonzero. The alternative expression is an independent check, not a guarantee of identical native numerical conditioning at extreme scales.

Construct t as =IMDIV(IMSINH(A2);IMCOSH(A2)) in F2. This is the complete complex hyperbolic-tangent ratio for the fixture; it does not require assuming a separate IMTANH function exists.

Then compare:

=IMSUM(IMPRODUCT(IMSECH(A2);IMSECH(A2));IMPRODUCT(F2;F2))

The independent target is one, with zero imaginary coefficient. IMSECH(A2) squared has coefficients approximately 0.851750726 and −0.900210178; F2 squared has approximately 0.148249274 and 0.900210178.

The imaginary terms cancel. IMSINH supplies a complete complex numerator, and the products retain the cross terms required by this identity.

Numeric real input two has a result near 0.265802229. Imaginary input 2j has expected real coefficient near −2.402997962, because sech(bj)=1/cos(b). Input j gives approximately 1.850815718, also outside the real-only positive-at-most-one restriction.

Hyperbolic secant is even under complete input negation. Conjugating only the imaginary coordinate instead conjugates the output, as shown in the third row.

Its mathematical poles occur at imaginary pi/2 plus integer multiples of pi×j. The dedicated page does not assign native errors or precision thresholds at such points. Its 4−3i control, approximately −0.0362534969158689+0.00516434460775318i, is corroborated independently.

OpenDocument 1.2 membership is stated separately from unknown original Collabora introduction. Invalid strings, mixed suffixes, blank coercion and extreme numerical limits remain unverified.

• Collabora Office 26.04: IMSECH.
• Editorial testing standards.

Complete dedicated reference and its equation image reviewed. Examples are independently calculated controls or explicit documented statements, not native application captures. Formulas use English names and semicolon separators.