(1-i)^4i

Jasonbradley

4 min read

·

Jun 16, 2024

8

Share

Exploring the Complex Power: (1-i)^4i

This article delves into the intriguing mathematical concept of (1-i)^4i, where 'i' represents the imaginary unit (√-1). This expression involves raising a complex number to a complex power, making it a fascinating problem to tackle.

Understanding Complex Exponentiation

Before delving into the specifics of (1-i)^4i, let's establish a solid understanding of complex exponentiation.

Complex exponentiation utilizes Euler's Formula, which links exponential and trigonometric functions:

e^(iθ) = cos(θ) + i sin(θ)

Where:

• e is the base of the natural logarithm
• i is the imaginary unit (√-1)
• θ is the angle in radians

This formula allows us to express complex numbers in exponential form, simplifying calculations and providing a deeper understanding of their properties.

Solving (1-i)^4i

Let's break down the process of evaluating (1-i)^4i:

1. Express (1-i) in Polar Form:

   To use Euler's formula effectively, we need to express (1-i) in polar form, which involves its magnitude (r) and angle (θ):

   • r = |1-i| = √(1² + (-1)²) = √2
   • θ = arctan(-1/1) = -π/4 (Since (1-i) lies in the 4th quadrant)

   Therefore, (1-i) = √2 * e^(-iπ/4)

2. Apply Exponent Properties:

   We can now rewrite (1-i)^4i as:

   (√2 * e^(-iπ/4))^4i = (√2)^4i * (e^(-iπ/4))^4i

3. Simplify:

   • (√2)^4i = 2^2i = e^(2i*ln(2)) (Using the property that a^b = e^(b*ln(a)))
   • (e^(-iπ/4))^4i = e^(iπ) = -1

4. Final Result:

   Combining the simplified terms, we get:

   (1-i)^4i = e^(2iln(2)) * (-1) = -e^(2iln(2))

This result, -e^(2i*ln(2)), is a complex number in exponential form. It can further be expressed in rectangular form (a + bi) by utilizing Euler's formula again.

Significance and Applications

While the solution of (1-i)^4i might appear abstract, complex exponentiation has significant applications in various fields:

• Electrical Engineering: Analyzing alternating current circuits, impedance, and wave propagation.
• Quantum Mechanics: Describing wave functions and probability amplitudes.
• Signal Processing: Representing and manipulating signals in the frequency domain.

Understanding complex exponentiation lays a foundation for exploring these advanced concepts in mathematics, physics, and engineering.

Note: This article provides a general approach to solving (1-i)^4i. The exact interpretation and further simplification of the result may depend on the context and specific applications.