6.3 Elementary Transcendental Functions
Log function
\[ log(x) = \int_1^x \frac{1}{t} dt, \; x > 0 \]
Transcendental number \(e\)
a number x which satisfies
\[ \int_1^x \frac{1}{t} dt = 1 \]
i.e \(log(e) = 1\)
or
\[ e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n \]
Exponential function
\[ e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]
or the inverse of log(x)
- Reference: Walter Rudin, Principles of Mathematical Analysis. 3e. p178-9. McGraw Hill. 1976
Relationship between exponential and trigonometric function
\[ e^{ix} = 1 + \frac{x^1}{1!}i - \frac{x^2}{2!} - \frac{x^3}{3!}i + \cdots = cos(x) + i sin(x) \]
\[ cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \] \[ sin(x) = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]
\[ cos(x) = \frac{e^{ix} + e^{-ix}}{2} \] \[ sin(x) = \frac{e^{ix} - e^{-ix}}{2} \]
\[ e^{i \pi} = cos(\pi) + i sin(\pi) = -1 \] \[ e^{i \pi} + 1 = 0 \]
Hyperbolic funtion
\[ cosh(x) = \frac{e^x + e^{-x}}{2} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} \] \[ sinh(x) = \frac{e^x - e^{-x}}{2} = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \]
Question
How many different numbers can be presented in a computer?
Question
- \(y = x^2\), x=rational number. Is this differentiable?
- \[ y = \begin{cases} x^2 & \quad \text{if } x \text{ is rational number} \\ 0 & \quad \text{if } x \text{ is irrational number} \end{cases} \] \(\int_{-1}^{1} y dx =\) ?