IMPOWER Function

The IMPOWER function in Excel raises a complex number to a given power. This function is particularly useful in mathematical, engineering, and scientific applications that involve complex number manipulation, such as solving equations or analyzing electrical circuits.

Key Features of IMPOWER:

• Computes the result of raising a complex number to a specified power.
• Accepts complex numbers in the form a+bi or a+bj, where a is the real part and b is the imaginary part.
• Returns the result as a complex number in the same format: c+di.

Syntax:

IMPOWER(inumber, number)

• inumber: The complex number to be raised to the power. This input can be:

  • A text string such as "3+2i".
  • A reference to a cell containing a valid complex number.
  • Created with the COMPLEX(real_num, imaginary_num) function.

• number: The exponent to which the complex number is raised. This can be a real number or a reference to a cell containing the number.

Formula and Calculation:

For a complex number z = a+bi and a power n, the result is calculated as:

(a+bi)^n = |z|^n * (cos(nθ) + i * sin(nθ))

Where:

• |z|: The magnitude (absolute value) of the complex number, |z| = √(a² + b²).
• θ: The phase angle of the complex number, θ = atan(b/a) (arctangent of b/a).
• The result is expressed in polar form and converted back to rectangular form as c+di.

Examples:

1. Cube of a Complex Number:
   =IMPOWER("1+2i", 3)
   For the complex number 1+2i raised to the power of 3, the result is:
   Result: -11+2i

2. Square of a Complex Number:
   =IMPOWER("3+4i", 2)
   For 3+4i, the result is:
   Result: -7+24i

3. Raising a Real Number to a Power:
   =IMPOWER(5, 2)
   For the real number 5 squared, the result is:
   Result: 25 (since there is no imaginary part).

4. Complex Number Raised to Zero:
   =IMPOWER("1+1i", 0)
   Any number raised to the power of 0 results in:
   Result: 1 (independent of the number's real and imaginary parts).

5. Using a Reference for Input:
   If cell A1 contains "2-3i" and cell A2 contains 4, then:
   =IMPOWER(A1, A2)
   Result: -119-120i

Notes:

• If inumber is not a valid complex number or the format is incorrect, Excel returns a #VALUE! error.
• Negative or fractional powers of complex numbers are supported by IMPOWER. The function handles these cases using logarithms and polar coordinates.

IMPOWER - Precision Differences Between Excel and Codcel

Function

IMPOWER raises a complex number to a given power.

Observed Difference

When computing IMPOWER for purely imaginary bases, the Codcel Rust implementation produces slightly different floating-point noise than Excel. The mathematically-zero component shows a tiny non-zero residual, and the exact value of that residual differs between platforms.

Examples

i^4 (should mathematically equal 1)

Platform	Result
Excel	1-2.45029690981724E-16i
Codcel	1-2.44929359829471E-16i

(2i)^3 (should mathematically equal -8i)

Platform	Result
Excel	-1.47017814589034E-15-8i
Codcel	-1.46957615897483E-15-8i

In both cases the "noise" component (the one that should be exactly zero) differs slightly.

Why This Happens

IMPOWER uses the polar form to compute complex powers:

z^n = r^n * (cos(n*theta) + i*sin(n*theta))

For i^4, the computation becomes:

r = 1, theta = pi/2
z^4 = cos(2*pi) + i*sin(2*pi)

Mathematically, sin(2*pi) = 0 and cos(2*pi) = 1. But in floating-point arithmetic, sin(2*pi) is not exactly zero - it produces a tiny residual on the order of 10^-16. The exact value of this residual depends on the platform's implementation of sin:

• Excel (Windows MSVC): sin(2*pi) = -2.45029690981724E-16
• Rust (macOS/Linux libm): sin(2*pi) = -2.44929359829471E-16

Both values are correct to within 1 ULP of the true result (zero), but they differ because sin and cos are implemented using different polynomial approximations on each platform.

The same principle applies to (2i)^3, where cos(3*pi/2) produces a tiny non-zero residual that differs between platforms.

Why Not Just Zero It Out?

Excel itself preserves this floating-point noise rather than rounding it to zero. Codcel follows the same behavior - showing the actual computed result rather than applying a cleanup threshold. This ensures transparency: what you see is exactly what the floating-point computation produced.

Impact

• The "noise" values are on the order of 10^-15 to 10^-16, which is effectively zero.
• The mathematically-significant components (1 for i^4, -8i for (2i)^3) match exactly.
• No real-world calculation would be affected, as these noise values are 15+ orders of magnitude smaller than the meaningful result.

What You Can Do

• If comparing results between Excel and Codcel, treat components smaller than 1e-14 relative to the largest component as effectively zero.
• Alternatively, compare floating-point values with a tolerance rather than comparing formatted strings.
• If your application requires detecting when a result is "purely real" or "purely imaginary", use a threshold like abs(component) < 1e-10 * abs(other_component) rather than checking for exact zero.

Applications:

• Engineering: Used in electrical and control systems for signal analysis and feedback design.
• Mathematics: Solves polynomial equations and complex transformations.
• Physics: Analyzes waveforms and resonance involving complex impedance.
• Data Science: Supports transformations of multi-dimensional datasets.

Related Functions:

• IMPRODUCT: Multiplies two or more complex numbers.
  Example: =IMPRODUCT("2+3i", "4-5i") → 23+2i
• IMDIV: Divides one complex number by another.
  Example: =IMDIV("3+4i", "1+2i") → 2-1i
• IMSUM: Adds two or more complex numbers.
  Example: =IMSUM("2+3i", "1-2i") → 3+1i
• IMABS: Returns the magnitude (absolute value) of a complex number.
  Example: =IMABS("3+4i") → 5

Summary:

The IMPOWER function enables users to perform complex number power calculations efficiently in Excel. It is a critical tool for mathematical computations, signal processing, and other applications requiring complex arithmetic. Its ability to handle both real and imaginary components ensures robust functionality across various disciplines.