1. What is the Complex Power Formula?

The complex power formula (De Moivre's Theorem) allows us to raise complex numbers in polar form to any power. It's a fundamental operation in complex analysis with applications in engineering, physics, and mathematics.

2. How Does the Calculator Work?

The calculator uses the complex power formula:

Where:

• \( r \) — Magnitude (modulus) of the complex number
• \( θ \) — Angle (argument) in degrees
• \( n \) — The power to raise the complex number to
• \( e \) — Euler's number (~2.71828)
• \( i \) — Imaginary unit (√-1)

Explanation: Raising a complex number to a power involves raising its magnitude to that power and multiplying its angle by the power.

3. Importance of Complex Powers

Details: Complex powers are essential in electrical engineering (AC circuit analysis), quantum mechanics, signal processing, and solving differential equations. They simplify calculations involving periodic phenomena.

4. Using the Calculator

Tips: Enter the magnitude (r ≥ 0), angle in degrees (θ), and the power (n). The calculator will show results in both polar and rectangular (a + bi) forms.

5. Frequently Asked Questions (FAQ)

Q1: What happens when n is not an integer?
A: The formula works for any real number n, including fractions and irrational numbers, giving principal values.

Q2: How are negative angles handled?
A: Negative angles represent clockwise rotation. The calculator normalizes all angles to between -180° and 180°.

Q3: What if r = 0?
A: When r=0, the result is 0 for any n, except when n=0 which is undefined.

Q4: Can this calculate roots of complex numbers?
A: Yes, use fractional powers (e.g., 1/3 for cube root). Note there are multiple roots for fractional powers.

Q5: How accurate are the results?
A: Results are accurate to 4 decimal places, though exact symbolic forms might be preferred for theoretical work.