# Help with Complex Exponentials

• JJHK
In summary, the conversation discusses the multiplication of a complex number by ejθ and how it can be described as a positive rotation through the angle θ of the vector without altering its length. The conversation also discusses using exponential form to represent the complex number and using trigonometry to prove the rotation.
JJHK

## Homework Statement

Hello all, I'm currently reading through "Vibrations and Waves" by A.P French. I'm not very used to these type of books, so I could use a little bit of help right now:

Show that the multiplication of any complex number z by e is describable, in geometrical terms, as a positive rotation through the angle θ of the vector by which z is represented, without any alteration of its length.

## Homework Equations

z = a + jb

j = an instruction to perform a counterclockwise rotation of 90 degrees upon whatever it precedes.

cosθ + jsinθ = e

j2 = -1

## The Attempt at a Solution

so z(e)
= (a + jb)(cosθ + jsinθ)
= acosθ + (asinθ + bcosθ)j - bsinθ
= (acosθ - bsinθ) + (asinθ + bcosθ)j

Now I'm supposed to show that the above equation is a positive rotation through the angle θ of the vector by which z is represented, without any alteration of its length.

First, I'm going to prove that the length was not altered:

Original length of z, Lo:

Lo2 = a2 + b2
Lo = √(a2+b2)

Length of new z, L1:

L12 = (acosθ - bsinθ)2 + (asinθ + bcosθ)2
L12 = a2 + b2
L1 = √(a2+b2)

Therefore Lo = L1

Now, How do I prove the rotation? I'm not too strong with trig, maybe that's the problem? Can somebody help me out? Thank you!

Have you considered that your complex number can also be described in exponential form using the same relationship:

cosθ + jsinθ = e

if R = |a2 + b2| is the magnitude of the complex number, and ##\phi## = atan(b/a) is the argument of the complex number (equivalent vector's direction angle), then the vector r for the complex number can also be represented by:

r = R(cos##\phi## + jsin##\phi##) = R ej##\phi##

## 1. What are complex exponentials?

Complex exponentials are mathematical expressions of the form z = a+bi, where a and b are real numbers and i is the imaginary unit. They are used in various fields of science and mathematics, including signal processing, quantum mechanics, and electrical engineering.

## 2. How do you represent complex exponentials?

Complex exponentials can be represented in two forms: polar form and rectangular form. In polar form, z = re^(iθ), where r is the magnitude of the complex number and θ is the angle it makes with the positive real axis. In rectangular form, z = a+bi, where a and b are the real and imaginary parts of the complex number, respectively.

## 3. What are the properties of complex exponentials?

There are several properties of complex exponentials that are useful in calculations. These include the commutative, associative, and distributive properties, as well as the properties of conjugation and modulus. The properties of logarithms also apply to complex exponentials.

## 4. How do you perform operations with complex exponentials?

To perform operations with complex exponentials, you can use the properties of complex exponentials or convert them to their polar or rectangular forms. Addition and subtraction of complex exponentials is done by adding or subtracting their real and imaginary parts, while multiplication and division is done by multiplying or dividing their magnitudes and adding or subtracting their angles.

## 5. What are the applications of complex exponentials?

Complex exponentials have many applications in science and engineering, including in electrical circuits, signal analysis and processing, control systems, and quantum mechanics. They are also used in solving differential equations and in Fourier analysis.

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