IMSECH in Excel: Verify Complex Hyperbolic Secant across Different Axes

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

=IMSECH(inumber)

The required argument uses x+yi or x+yj text. Construct it with COMPLEX when the coefficients are numeric. Hyperbolic secant is the reciprocal of complete hyperbolic cosine; it is not an inverse hyperbolic cosine.

IMSEC is a separate identity based on ordinary complex cosine. Even if some special inputs produce related values, the two functions have different denominators and cannot be exchanged in a general formula.

The examples use explicit complex construction for real-axis inputs too. They do not assume undocumented general plain-number or blank coercion.

Enter the two numeric input columns in A2:B5. Put =COMPLEX(A2,B2) in C2 and =IMSECH(C2) in D2; fill down. Use =IMREAL(D2) and =IMAGINARY(D2) in E and F.

The expected coefficients are rounded to nine decimals and explain E:F, rather than adding input columns to A:B. Retain calculated D values for later formulas.

Hyperbolic secant of a finite real argument is positive and at most one. The last input is imaginary and has a real output greater than one. Applying the real-only bound to arbitrary complex inputs would reject this valid mathematical result.

Compare D2 with:

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

The independent expected quantity matches the first row. IMCOSH calculates the complete denominator.

For the first input, hyperbolic cosine has squared magnitude approximately 2.151248998. Its reciprocal divides the conjugate coefficients by that shared value. The imaginary sign change is essential; taking separate scalar reciprocals does not calculate hyperbolic secant.

For every primary row, multiplying IMSECH(C2) by IMCOSH(C2) has independent target one with zero imaginary coefficient. All selected denominators are nonzero.

For a+bi, a component model uses denominator COSH(2a)+COS(2b). Its real numerator is 2COSH(a)COS(b), while its imaginary numerator is negative 2SINH(a)SIN(b).

For the first row, calculate:

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

Expected coefficients are 0.629485781 and −0.261904347, rounded. The placement of a and b differs from the ordinary complex-secant expression.

This is an independently derived algebraic model. The dedicated Microsoft page supplies the function definition and an example, rather than this expanded component equation.

Hyperbolic cosine is half the sum of exp(z) and exp(−z). Its reciprocal for the first row can therefore be checked with:

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

The independent quantity again matches the primary result. The numerator two is required because the denominator sum is twice hyperbolic cosine.

IMEXP supplies complete complex exponentials. This identity is useful for checking arithmetic and missing factors; it does not promise that the rearranged formula has the same stability as the native function at extreme scales.

In G2 construct a complete hyperbolic-tangent ratio:

=IMDIV(IMSINH(C2),IMCOSH(C2))

Then compare:

=IMSUM(IMPRODUCT(D2,D2),IMPRODUCT(G2,G2))

The independent target is one with zero imaginary coefficient. For the first input, the hyperbolic-secant square has coefficients approximately 0.327658462 and −0.329730126; the ratio square has approximately 0.672341538 and 0.329730126.

The imaginary terms cancel. This construction uses IMSINH explicitly and does not assume an additional IMTANH identity is available in Excel. Squares of magnitudes would discard the cross terms required by this relation.

Constructing real input two with =IMSECH(COMPLEX(2,0)) gives mathematical quantity near 0.265802229. Imaginary input 2i instead gives a real quantity near −2.402997962, because sech(bi)=1/COS(b).

Complete input negation leaves hyperbolic secant unchanged. Conjugation instead reverses its imaginary output sign. Both tests can detect a wrong sign in a manually built reciprocal.

Microsoft specifies #NUM! for unsupported complex-text format and #VALUE! for a logical argument. Its =IMSECH("4+3i") example has coefficients approximately −0.0362534969158689 and −0.00516434460775318, corroborated independently.

Mathematical poles lie at imaginary pi/2 plus integer multiples of pi times i. Native errors at those points, finite PI() approximations, extreme numerical limits and a complete coercion matrix remain unverified. Use explicit component tolerances for the moderate worked cases rather than exact display-string comparisons.

• Microsoft IMSECH 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.