IMSECH

What it does

IMSECH(inumber)

IMSECH(inumber) returns the complex hyperbolic secant, mathematically the reciprocal of IMCOSH. It returns complex text even if the imaginary component is zero.

Try it

=IMSECH("1") gives real text about "0.648054273663885", the reciprocal of cosh(1). The all-real input keeps the answer real; the general function also accepts a nonzero imaginary coordinate.

Quirks

Complex arguments may be real numbers or text such as "2-3i". Use lowercase i or j; operands in one call must agree on the suffix. The current parser rejects malformed complex text with #NUM! and logical values with #VALUE!. A complex result is text, so ordinary worksheet arithmetic does not automatically treat it as a pair of numeric components. A zero denominator gives #NUM! in the current reciprocal helper. Near a pole, small changes in the input can cause large changes in the result.

How to implement

Parse the two coefficients and suffix before doing arithmetic; use the common complex formatter afterward. Retain suffix consistency, suppress a zero component and omit the coefficient one before the imaginary unit. Formatting and parsing have their own fifteen-significant-digit rules, so testing only the algebra misses observable behaviour.

First compute the unformatted complex IMCOSH pair (c,d). If abs(c)>=abs(d), use r=d/c, v=1/(c+d*r), and return (v,-r*v). Otherwise use r=c/d, v=1/(d+c*r), and return (r*v,-v). Reject (0,0) before division. The current reciprocal forms one reciprocal and multiplies a component by it; calling generic IMDIV with numerator one forms a different rounded graph. Keep this distinction when reproducing recorded results.

Rust code

46 imsech "IMSECH" (1, 1) [V "inumber"] |s, a, sv| { unary_complex(s, a, |z| z.sech()) }

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

The record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.

Recorded environments and comparison details.

Alternatives

IMCOSH supplies the denominator; IMDIV divides arbitrary complex inputs; COMPLEX creates a value from coefficients.

Named variants

No named variants are recorded yet.