IMCSCH

What it does

IMCSCH(inumber)

IMCSCH(inumber) returns the complex hyperbolic cosecant, mathematically the reciprocal of IMSINH. It returns complex text, including when the answer has only a real component.

Try it

=IMCSCH("1") gives real text about "0.850918128239322", the reciprocal of sinh(1). At zero the denominator is zero and the current reciprocal helper returns #NUM!.

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 IMSINH 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

40 imcsch "IMCSCH" (1, 1) [V "inumber"] |s, a, sv| { unary_complex(s, a, |z| z.csch()) }

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

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

Named variants

No named variants are recorded yet.