E-Powers Homework Help: Understanding Why e^[i(phi)] * e^[-i(phi)] = 1

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Start date Nov 29, 2005
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Homework

In summary, e^[i(phi)] is a complex number with a real component of e^phi and an imaginary component of e^phi*i. When multiplied by its conjugate, e^[-i(phi)], it equals 1 due to Euler's formula. This equation is related to the unit circle and has various applications in mathematics and science, including simplifying calculations involving trigonometric functions and understanding periodic phenomena and wave behavior.

Nov 29, 2005

asdf1

why does e^[i(phi)] * e^[-i(phi)]=1 instead of e^[(phi)^2]?

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Nov 29, 2005

Homework Helper

Well for powers, so for e-powers in particular, we have that [itex]e^a \cdot e^b = e^{a + b}[/itex]. Applying that here gives e^0 which is of course 1.

Nov 30, 2005

asdf1

thank you very much!

Nov 30, 2005

Homework Helper

No problem

1. What is e^[i(phi)]?

e^[i(phi)] is the exponential function raised to the power of the imaginary number i multiplied by the angle phi. In other words, it is a complex number with a real component of e^phi and an imaginary component of e^phi*i.

2. What does e^[i(phi)] * e^[-i(phi)] equal to?

When you multiply two complex numbers, you multiply their real components and add their imaginary components. In this case, e^[i(phi)] * e^[-i(phi)] equals e^0 = 1, since the imaginary components cancel each other out.

3. Why does e^[i(phi)] * e^[-i(phi)] equal to 1?

This is because of Euler's formula, which states that e^[i(phi)] = cos(phi) + i*sin(phi). When you multiply this by e^[-i(phi)], you get [cos(phi) + i*sin(phi)] * [cos(-phi) + i*sin(-phi)]. Using trigonometric identities, this simplifies to cos^2(phi) + sin^2(phi) = 1.

4. How is e^[i(phi)] * e^[-i(phi)] related to the unit circle?

The unit circle is a circle on the Cartesian plane with a radius of 1, centered at the origin. When you plot the real and imaginary components of e^[i(phi)], it forms a point on the unit circle. Similarly, e^[-i(phi)] forms a point on the unit circle, but in the opposite direction. When you multiply these two points, they cancel each other out and form the point (1,0) on the unit circle, which corresponds to the real number 1.

5. How is this equation useful in mathematics and science?

This equation is widely used in mathematics and science, particularly in fields such as physics, engineering, and signal processing. It is used to simplify complex calculations involving trigonometric functions and complex numbers. It also has applications in understanding periodic phenomena and wave behavior. Additionally, it is a fundamental concept in understanding the properties of exponential and logarithmic functions.