Difference between revisions of "E"
From specialfunctionswiki
| Line 1: | Line 1: | ||
| − | The number $e$ | + | The number $e$ is the number such that |
| − | $$ | + | $$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$ |
| − | + | This means, by definition, that $\log(e)=1$, where $\log$ denotes the [[logarithm]]. | |
| − | |||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
Revision as of 06:48, 4 June 2016
The number $e$ is the number such that $$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$ This means, by definition, that $\log(e)=1$, where $\log$ denotes the logarithm.
Properties
Theorem: The folllowing formula holds: $$e=\displaystyle\lim_{k \rightarrow \infty} \left( 1 + \dfrac{1}{k} \right)^k,$$ where $e$ denotes Euler's constant.
Proof: █
Theorem: The real number $e$ is irrational.
Proof: proof goes here █
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.16
