Understanding the Expressions of ( frac{1}{z^2} ) and ( 1-z^2 ) in Exponential Form

In this article, we will delve into the mathematics of expressing ( frac{1}{z^2} ) and ( 1-z^2 ) in terms of exponential forms, using Euler's formula and trigonometric identities. We will explore the step-by-step process and provide explanations for each step to help you build a strong understanding of these concepts.

Given: ( z e^{itheta} )

First, let's start with the given condition z e^{itheta}. Based on Euler's formula, we have:

[ e^{itheta} cos(theta) isin(theta) ]

Calculating ( z^2 )

To find z^2, we use the property of exponents:

[ z^2 (e^{itheta})^2 e^{2itheta} ]

Using Euler's formula, this can be expressed as:

[ z^2 e^{2itheta} cos(2theta) isin(2theta) ]

Expression of ( frac{1}{z^2} ) in Exponential Form

Next, we need to express frac{1}{z^2} ). Since z^2 e^{2itheta}), we have:

[ frac{1}{z^2} e^{-2itheta} ]

In exponential form, this can be written as:

[ frac{1}{z^2} e^{-2itheta} cos(-2theta) isin(-2theta) cos(2theta) - isin(2theta) ]

Note that (cos(theta)) is an even function and (sin(theta)) is an odd function, so (cos(-2theta) cos(2theta)) and (sin(-2theta) -sin(2theta)).

Expression of ( 1 - z^2 ) in Exponential Form

Now, let's express (1 - z^2) in exponential form. Using the expression for (z^2), we have:

[ 1 - z^2 1 - e^{2itheta} ]

Using the property of exponentials, we can rewrite this as:

[ 1 - z^2 1 - (cos(2theta) isin(2theta)) ]

This can also be written using Euler's formula:

[ 1 - z^2 1 - e^{2itheta} 1 - (cos(2theta) isin(2theta)) ]

More Detailed Expressions

Let's delve further into the expressions for ( frac{1}{z^2} ) and (1 - z^2) by expanding them using trigonometric identities:

For ( frac{1}{z^2} ), we already have:

[ frac{1}{z^2} e^{-2itheta} cos(2theta) - isin(2theta) ]

To express it in terms of cosine, we use the double angle identity:

[ cos(2theta) cos^2(theta) - sin^2(theta) ] [ sin(2theta) 2sin(theta)cos(theta) ]

Therefore:

[ frac{1}{z^2} cos^2(theta) - 2isin(theta)cos(theta) - sin^2(theta) ]

And for (1 - z^2), we have:

[ 1 - z^2 1 - (cos(2theta) isin(2theta)) ]

Expanding it, we get:

[ 1 - z^2 1 - (cos^2(theta) - sin^2(theta) i(2sin(theta)cos(theta))) ]

Using the identity (cos^2(theta) sin^2(theta) 1):

[ 1 - z^2 1 - (1 sin^2(theta) - 2sin^2(theta) - 2isin(theta)cos(theta)) ]

Simplifying, we get:

[ 1 - z^2 -i(2sin(theta)cos(theta)) -2isin(theta)cos(theta) ]

Conclusion

In conclusion, we have explored the different expressions of ( frac{1}{z^2} ) and ( 1 - z^2 ) in exponential form. Using Euler's formula and trigonometric identities, we have derived the expressions that provide a deeper understanding of these complex mathematical concepts. These expressions are not only useful for theoretical purposes but also have practical applications in various fields, such as electrical engineering and quantum mechanics.