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Definition: Exponential Function
$\forall x \in \R $, we define the exponential function $\exp(x) $ to be the real number
$$\exp(x) = \sum_{ n=0 }^{ \infty } \frac{ x^n }{ n! } $$
Theorem
: Basic Properties of Exponential
- $\forall x \in \R, \sum_{ n=0 }^{ \infty } \frac{ x^n }{ n! } $ is absolutely
convergent.
In particular, $\forall x \in \R, \exp (x), \exists \exp(x) \in \R $.
$\sum_{ n=0 }^{ \infty } \frac{ x^n }{ n! } $ has an infinite radius of convergence, and $\exp $ is a real analytic function on $(- \infty, \infty)$. - $\exp$ is differentiable on $\R $, and for every $x \in \R $, $\exp'(x) = \exp (x) $.
- $\exp$ is continuous on $\R $, and for every interval $[a,b] $, we have $\int_{ [a,b] } \exp (x) \d x = \exp (b) - \exp (a)$
- $\forall x, y \in \R, \exp (x + y) = \exp (x) \exp (y). $
- $\exp (0) = 1. \forall x \in \R, \exp (x) > 0, \exp (-x) = \frac{ 1 }{ \exp (x)}$
- $\exp$ is strictly monotone increasing.
Definition: Euler's Number
The number $e$ is defined to be
$$e \triangleq \exp (1) = \sum_{ n=0 }^{ \infty } = \frac{ 1 }{ 0! } + \frac{ 1 }{ 1! } + \frac{ 1 }{ 2! } + \frac{ 1 }{ 3! } + … $$
Proposition
$$\forall x \in \R, \exp (x) = e^x$$
Definition: Logorithm
We define the natural logarithm function $\ln : (0, \infty) \to \R$ to be the inverse of the exponential function. Thus $\exp \ln x = x $ and $\ln \exp x = x $.
Theorem
: Logarithm Properties
- $\forall x \in (0, \infty), \ln ' (x) = \frac{ 1 }{ x } $.
In particular, by the fundamental theorem of calculus, we have $\int_{ [a, b] } \frac{ 1 }{ x } \d x = \ln (b) - \ln (a)$ for any interval $[a, b] \subseteq (0, \infty)$ - $\forall x, y \in (0, \infty), \ln (xy) = \ln (x) + \ln (y)$.
- $\forall x \in (i, \infty), \ln (1) = 0, \ln (\frac{ 1 }{ x }) = - \ln (x).$
- $\forall x \in (0, \infty) , y \in \R, \ln (x^y) = y \ln (x).$
- $\forall x \in (-1, 1), \ln (1-x) = - \sum_{ n=1 }^{ \infty } \frac{ x^n }{ n }.$
In particular, $\ln$ is analytic at $1$, with the power series expansion $$\ln (x) = \sum_{ n=1 }^{ \infty } \frac{(-1)^{n+1}}{ n } (x-1)^n $$
for $x \in (0,2) $, with radius of convergence $1$.
Example: Application of Abel's theorem
TODO: