MandysNotes

Euler’s Formula

13 May 2011

The real part of $e^{i\alpha} \$

is:

$\text{Re}(e^{i \alpha}) = \frac{ e^{i\alpha} + e^{-i\alpha}}{2}$

$=\sum_{k=0}^{\infty} (-1)^{k}\frac{\alpha^{2k}}{(2k)!} .$

The imaginary part of $e^{i\alpha} \$

is:

$\text{Im}(e^{i\alpha}) = \frac{ e^{i\alpha} - e^{-i\alpha}}{2i}$

$= \sum_{k=1}^{\infty}(-1)^{k -1}\frac{\alpha^{2k -1}}{(2k -1)!} .$

We define:

$\text{Re}(e^{i\alpha}) = \frac{ e^{i\alpha} + e^{-i\alpha}}{2} := \cos{\alpha} ;$

$\text{Im}(e^{i\alpha}) = \frac{ e^{i\alpha} - e^{-i\alpha}}{2i} := \sin{\alpha}.$

Therefore:

$e^{i\alpha} = \text{Re} (e^{i\alpha}) + i\text{Im}(e^{i\alpha}) =\cos{\alpha} + i \sin{\alpha}.$

This is Euler's Formula.

The Exponential of a Purely Imaginary Number

13 May 2011

Consider the exponential of a purely imaginary number:

$z = i\alpha$

with $\alpha \in \mathbb{R}\ .$

$e^{i\alpha} = \lim_{n \to \infty } (1 + \frac{i\alpha}{n})^{n}$

$= \sum_{k=0}^{\infty }\frac{i\alpha^{k}}{k!} .$

Multiplication of Exponentials

13 May 2011

Consider first the exponential of a constant, a:

$e^{a} = \lim_{n \to \infty} (1 + \frac{a}{n})^{n} = \sum_{k=0}^{\infty} \frac{a^{k}}{k!}.$

If b is another constant, what is $e^{a}e^{b} ?$

Let's go back to the definition of exponential:

$e^{a}e^{b} \ = \lim_{n \to \infty} f(n) .$

Where:

$f(n) = (1 + \frac{a}{n})^{n} (1 + \frac{b}{n})^{n} = ( 1 + \frac{ a + b}{n} + \frac{ab}{n^{2}})^{n}$

$= \sum_{k=0}^{n} \binom{n}{k} c^{k},$

The Exponential Function

11 May 2011

(Note: this is a quick and dirty introduction to exponentials. I will not bother with any epsilons or deltas, but for those of you who are concerned about such things, be assured that all convergence is uniform, all sequences are Cauchy, etc. The series that defines the exponential is about as well behaved as an infinite series can be.)

The function $e^{x}$is defined to be the limit as $$n$$ approaches infinity of $( 1 + \frac{x}{n})^{n}:$

exp(x) = $e^{x} = \lim_{n \to \infty} (1 + \frac{x}{n})^{n} .$