Even More Complex (Numbers)

The questions below are due on Monday April 07, 2025; 10:00:00 PM.

You are not logged in.
Please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.mit.edu) to authenticate, and then you will be redirected back to this page.

1) Justifying Euler's Formula

Consider the following series expansions:

e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \ldots

\cos x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots

\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{(2n+1)}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} +\ldots

Use these expansions to verify Euler's formula, \cos(\theta) + j\sin(\theta) = e^{j\theta}. Show your work.

2) Trig Functions in Terms of Complex Exponentials

Use Euler's formula to verify the following:

\cos\left(\theta\right) = {{e^{j\theta} + e^{-j\theta}} \over 2}

\sin\left(\theta\right) = {{e^{j\theta} - e^{-j\theta}} \over 2j}

Show your work.

3) Polar Representation of Complex Numbers

Consider a number a_0 + b_0j. We can think of this number as a special kind of vector (called a "phasor") in the complex plane, as shown below. This vector has a magnitude (let's call it r) and an angle (\phi) in the complex plane.

We can express this number in the form re^{j\phi} for some values of r and \phi that are both real-valued. Determine r and \phi in terms of a_0 and b_0, and also determine a_0 and b_0 in terms of r and \phi.

4) Operations on Complex Numbers

Note that rectangular form is a convenient form when considering addition or subtraction complex numbers and polar form is a convenient form when considering multiplication or exponentiation.

Consider adding two numbers a_1 + b_1j and a_2 + b_2j. What is the resulting number, and what are its real and imaginary parts?

Consider multiplying two numbers r_1e^{j\phi_1} and r_2e^{j\phi_2}. What is the resulting number, and what are its magnitude and phase?

Consider taking a number r_0e^{j\phi_0} and raising it to a real-valued power x. What is the resulting number, and what are its magnitude and phase?

Then, in your own words, summarize:

How do the real and imaginary parts of the sum of two complex numbers relate to the real and imaginary parts of those numbers?
How do the magnitude and phase of the product of two complex numbers relate to the product of those numbers?
When raising a complex number to a real-valued power, how do the magnitude and phase of the result relate to the magnitude and phase of the original number?

Upload your answers as a single PDF file, including both your answer and your work. Please do not include any identifying information in your submission so that we can grade the submissions anonymously.

No file selected