An Accidental Hint at Euler's Equation

I had this thought at lunch today. It's not a sophisticated treatment of the subject, but I found it amusing.

We know that when you differentiate a sinusoid twice, you get back a scaled and negated version of the original. Here the scaling factor is unity.

\[ y = sin(t) \\ \ddot{y} = -sin(t) = -y\]

We also know that double-differentiating an exponential gets us back a scaled (but not negated) version of the original. Here again, the scaling factor is unity.

\[ y = e^t \\ \ddot{y} = e^t = y\]

These two behaviors are tantalizingly similar. So how might we get the exponential to behave like the sinusoid with respect to double-differentiation? Maybe we could throw in the square root of negative one.

\[ y = e^{j t} \\ \ddot{y} = j^2 e^{j t} = -e^{j t} = -y\]

Making the exponent complex makes the exponential behave like a sinusoid with respect to double-differentiation. Hence, we've stumbled onto something like Euler's equation (shown below for reference).

\[ e^{j t } = cos \, t + j \, sin \, t\]

Note by Steven Chase
2 weeks, 2 days ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Good Lord. Thanks sir for posting these.

Md Zuhair - 1 week, 3 days ago

Log in to reply

Glad you liked it

Steven Chase - 1 week, 2 days ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...