IMCSCH in Excel: Distinguish Complex Hyperbolic Cosecant from Cosecant

• 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 IMCSCH reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMCSCH(inumber)

The required argument uses x+yi or x+yj text. Use COMPLEX when the real and imaginary coordinates already occupy numeric cells.

IMCSCH is distinct from IMCSC, which takes the reciprocal of ordinary complex sine. Neither is an inverse function that returns an original argument. Similar names and similar small-input values do not make them interchangeable.

The examples below construct complex inputs explicitly. They do not infer acceptance of every plain numeric, logical, blank or mixed-suffix representation from another complex function.

Enter the two numeric input columns in A2:B5. Put =COMPLEX(A2,B2) in C2, =IMCSCH(C2) in D2, =IMREAL(D2) in E2 and =IMAGINARY(D2) in F2. Fill down.

Expected coefficients are rounded to nine decimals. The last two table columns explain E:F; they are not extra input values for A:B.

Increasing only the imaginary coordinate from 0.5 to two changes the sign of the real output. A calculation based solely on the input’s magnitude cannot reproduce this directional effect.

Compare D2 with:

=IMDIV(COMPLEX(1,0),IMSINH(C2))

The independently expected quantity matches D2. IMSINH supplies the complete denominator; IMDIV retains both coefficients during division.

For the first row, the squared magnitude of hyperbolic sine is approximately 1.610946693. The reciprocal divides its conjugate coefficients by this shared number. Dividing each coefficient independently by itself or taking two scalar reciprocals changes the operation.

Fill this alternative down and compare both extracted components within a stated tolerance. A target such as 1E-9 is suitable for checking this moderate independent fixture, not a blanket native precision claim.

For input a+bi, an independent component model uses denominator COSH(2a)−COS(2b). The real numerator is 2SINH(a)COS(b); the imaginary numerator is −2COSH(a)SIN(b).

For the first row, calculate:

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

These give independently expected coefficients 0.640204967 and −0.459228271, rounded. The hyperbolic functions act on a, whereas the ordinary trigonometric functions act on b in radians.

This expanded equation is independently derived from the reciprocal definition. Microsoft’s dedicated page does not print this component equation, so it is not described as a quoted vendor formula.

Hyperbolic sine equals half the difference of the exponentials of z and −z. For the first row, check:

=IMDIV(COMPLEX(2,0),IMSUB(IMEXP("1+0.5i"),IMEXP("-1-0.5i")))

The mathematical target is the same 0.640204967−0.459228271i. The numerator two accounts for the half in hyperbolic sine.

IMEXP computes the exponentials and IMSUB subtracts complete quantities. This alternative can help detect a missing factor of two; it is not guaranteed to have the same numerical conditioning as the native function for tiny or extreme arguments.

For every primary row, multiplying IMCSCH(C2) by IMSINH(C2) has independent target one with zero imaginary coefficient. The selected inputs have nonzero denominators.

For input 0.5i, IMCSCH has target −2.085829643i, while IMCSC has target −1.919034751i. The first is negative i divided by SIN(0.5); the second is negative i divided by SINH(0.5).

At 0.1i their respective imaginary coefficients are approximately −10.016686132 and −9.983352757. The closeness is a small-input mathematical effect, not evidence of alias identity.

A real-axis control constructed with =IMCSCH(COMPLEX(2,0)) instead has mathematical quantity approximately 0.275720565, with imaginary coefficient zero. Its denominator is real SINH(2), not SIN(2).

Complete input negation negates hyperbolic cosecant. Conjugating the input conjugates the output. Checking both transformations helps catch incorrect reciprocal signs.

Microsoft documents #NUM! for an argument outside the supported complex format and #VALUE! for a logical argument. Its =IMCSCH("4+3i") control has coefficients approximately −0.036275889628626 and −0.0051744731840194, corroborated independently.

Hyperbolic sine is mathematically zero at integer multiples of pi times i. Those points are reciprocal poles, but the dedicated reference does not specify their native errors or the behavior of approximate PI()-based inputs. Overflow, underflow and a complete coercion matrix remain unverified.

• Microsoft IMCSCH 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.