Skip to content

IMSECH Function

The IMSECH function in Excel calculates the hyperbolic secant of a given complex number. This function is primarily used in mathematical, engineering, and scientific contexts involving complex numbers and hyperbolic trigonometric computations.

Key Features of IMSECH:

  • Computes the hyperbolic secant of a complex number in the form a+bi or a+bj.
  • Works with both real and complex numbers.
  • Returns the result as a complex number, even if the imaginary part is 0.

Syntax:

IMSECH(inumber)
  • inumber: The complex number whose hyperbolic secant needs to be calculated. This can be:
    • A string, such as "5+3i".
    • A cell reference containing a valid complex number.
    • A real number (treated as a complex number with an imaginary part of 0).

Formula Details:

The hyperbolic secant of a complex number is defined as:

sech(z) = 1 / cosh(z)

Where z is the complex number input, and cosh(z) is its hyperbolic cosine.

Examples:

  1. Calculating the Hyperbolic Secant of a Complex Number:
    =IMSECH("4+2i")
    The hyperbolic secant of 4+2i is:
    Result: -0.0362534969 - 0.0051643446i

  2. Calculating Hyperbolic Secant with a Real Input:
    =IMSECH(3)
    For real numbers, the hyperbolic secant is computed as 1/cosh(real part):
    Result: 0.0993279274

  3. Handling a Purely Imaginary Number:
    =IMSECH("0+3i")
    The hyperbolic secant of 3i is:
    Result: 0.9950547537 - 0.9950547537i

  4. Using a Cell Reference:
    If cell A1 contains "3-4i", then:
    =IMSECH(A1)
    The hyperbolic secant of 3-4i is:
    Result: -0.0417572559 - 0.0681247048i

  5. Complex Value From Formula:
    =IMSECH(COMPLEX(1, -1))
    If you create the complex number 1-i using the COMPLEX function, the hyperbolic secant of this value is:
    Result: 1.8508157177 - 0.313201924i

  6. Inputting Zero:
    =IMSECH(0)
    Since the hyperbolic secant of 0 is 1/cosh(0) and cosh(0) is 1:
    Result: 1

Notes:

  • If the input (inumber) is not formatted as a valid complex number, Excel will return a #VALUE! error.
  • The function uses radians for hyperbolic calculations. Use the RADIANS function to convert degrees to radians if needed.
  • Output will always be expressed as a complex number.

IMSECH - Precision Differences Between Excel and Codcel

Function

IMSECH returns the hyperbolic secant of a complex number, defined as 1 / cosh(z).

Observed Difference

When computing IMSECH for certain complex inputs, the Codcel Rust implementation may produce a result with a different 15th significant digit compared to Excel. In some cases, Excel's result appears to have one fewer significant digit because its 15th digit is a trailing zero that gets stripped.

Example

Input Excel Result Codcel Result
3+4i -0.065294027857947+0.0752249603027732i -0.0652940278579471+0.0752249603027732i

The real component shows: Excel ...57947 (14 visible digits, effectively ...579470 with trailing zero stripped) vs Codcel ...579471 (15 digits). The difference is in the 15th significant digit.

Why This Happens

IMSECH(a+bi) is computed as 1 / cosh(a+bi), which involves:

cosh(a+bi) = cosh(a)*cos(b) + i*sinh(a)*sin(b)
sech(a+bi) = 1 / cosh(a+bi)

This requires calls to cosh, sinh, cos, sin and then a complex division. Each of these operations can round differently at the bit level across platforms:

  • Excel uses the Windows MSVC runtime math library.
  • Codcel uses Rust's standard library / platform C math library.

In this case, the cascade of rounding differences results in the 15th significant digit differing. Excel computes ...579470 (which displays as ...57947 after trailing zero removal), while Codcel computes ...579471.

Why It Looks Like Fewer Digits

Both Excel and Codcel format numbers with up to 15 significant digits and strip trailing zeros. When the 15th digit happens to be 0, it gets removed, making the displayed result appear shorter. This is purely a display difference - both systems are working with 15-digit precision internally.

Impact

  • The difference is at the level of 1 part in 10^15 (one quadrillionth).
  • This is within the precision limit of IEEE 754 double-precision floating-point (~15.9 significant decimal digits).
  • No practical calculation would be affected by this difference.

What You Can Do

  • If comparing results between Excel and Codcel, use a tolerance of at least 1e-14 for each component rather than exact string equality.
  • When parsing complex number strings for comparison, convert components to floating-point values and compare numerically rather than comparing strings.

Applications:

  • Engineering: Useful in fields such as electronics and control systems involving hyperbolic functions.
  • Mathematics: Essential for evaluating equations involving hyperbolic trigonometric relationships.
  • Data Analysis: Supports extensions of periodic behavior modeling using complex hyperbolic functions.
  • IMCOSH: Returns the hyperbolic cosine of a complex number.
    Example: =IMCOSH("3+4i") → -6.5806630401 - 7.5815527427i
  • IMSINH: Returns the hyperbolic sine of a complex number.
    Example: =IMSINH("3+4i") → 6.5481200409 - 7.6192317203i
  • IMSEC: Computes the secant of a complex number.
    Example: =IMSEC("3+4i") → -0.0652940279 + 0.0752206537i
  • IMCOTH: Returns the hyperbolic cotangent of a complex number. Example: =IMCOTH("2+i") → 0.3799489623 - 0.2298488471i

Summary:

The IMSECH function in Excel is a powerful tool for performing hyperbolic trigonometric operations on complex numbers. Its use cases include advanced mathematical modeling, engineering solutions, and systems analysis involving hyperbolic computations.