Mathematical functions
On this page I have collated some frequently used mathematical functions and their basic properties. To keep the page short and simple, I refrained from explicitly mentioning the range of allowed values for each equation, but this should be immediately obvious in most cases anyway. As usual, I cannot guarantee that the equations on this page are correct, so please use them with caution. Please let me know if you find any errors.
Logarithm
Special bases
\begin{equation} \mathrm{lb}(x) \equiv \log_{\rm 2}(x) \end{equation}
\begin{equation} \ln(x) \equiv \log_{\rm e}(x) \end{equation}
\begin{equation} \lg(x) \equiv \log_{10}(x) \end{equation}
Special values
\begin{equation} \log_{a}(1) = 0 \end{equation}
\begin{equation} \log_{a}(a) = 1 \end{equation}
Change of base
\begin{equation} \log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)} = \frac{1}{\log_{x}(a)} \end{equation}
Identities
\begin{equation} \log_{a}(x y) = \log_{a}(x) + \log_{a}(y) \end{equation}
\begin{equation} \log_{a}(x / y) = \log_{a}(x) - \log_{a}(y) \end{equation}
\begin{equation} \log_{a}(x^{y}) = y \log_{a}(x) \end{equation}
Inverse
\begin{equation} y = f(x) = \log_{a}(x) \quad \rightarrow \quad x = f^{-1}(y) = a^{y} \end{equation}
Derivative and integral
\begin{equation} \frac{\mathrm{d}}{\mathrm{d}x} \ln(x) = \frac{1}{x} \end{equation}
\begin{equation} \frac{\mathrm{d} \log_{a}(x)}{\mathrm{d}x} = \frac{1}{\ln(a) x} \end{equation}
\begin{equation} \int \ln(x) \mathrm{d}x = x \ln(x) - x \end{equation}
Exponential
Special bases
\begin{equation} \exp(x) \equiv \mathrm{e}^{x} \end{equation}
Special values
\begin{equation} a^{0} = 1 \end{equation}
\begin{equation} a^{1} = a \end{equation}
Change of base
\begin{equation} a^{x} = b^{x \log_{b}(a)} \end{equation}
Identities
\begin{equation} a^{x + y} = a^{x} a^{y} \end{equation}
\begin{equation} a^{x - y} = \frac{a^{x}}{a^{y}} \end{equation}
\begin{equation} a^{x y} = (a^{x})^{y} = (a^{y})^{x} \end{equation}
Inverse
\begin{equation} y = f(x) = a^{x} \quad \rightarrow \quad x = f^{-1}(y) = \log_{a}(y) \end{equation}
Derivative and integral
\begin{equation} \frac{\mathrm{d}}{\mathrm{d}x} a^{x} = a^{x} \ln(a) \end{equation}
\begin{equation} \int a^{x} \mathrm{d}x = \frac{a^{x}}{\ln(a)} \end{equation}
\begin{equation} \frac{\mathrm{d}}{\mathrm{d}x} \exp(x) = \exp(x) \end{equation}
\begin{equation} \int \exp(x) \mathrm{d}x = \exp(x) \end{equation}
Gaussian function
Definition
\begin{equation} G(x) \equiv A \exp \! \left( -\frac{(x - x_{0})^{2}}{2 \sigma^{2}} \right) \end{equation}
Special values
\begin{equation} G(x_{0}) = A \end{equation}
\begin{equation} G \! \left( x_{0} \pm \sqrt{2 \ln(2)} \sigma \right) = \frac{A}{2} \end{equation}
Derivative and integral
\begin{equation} \frac{\mathrm{d}}{\mathrm{d}x} G(x) = -\frac{x - x_{0}}{\sigma^{2}} G(x) \end{equation}
\begin{equation} \int G(x) \, \mathrm{d}x = -\sqrt{\frac{\pi}{2}} A \sigma \, \mathrm{erf} \! \left( \frac{x_{0} - x}{\sqrt{2} \sigma} \right) \end{equation}
\begin{equation} \int \limits_{-\infty}^{\infty} G(x) \, \mathrm{d}x = A \sigma \sqrt{2 \pi} \end{equation}
Properties
\begin{equation} \frac{\mathrm{FWHM}}{\sigma} = 2 \, \sqrt{2 \ln(2)} \approx 2.3548 \end{equation}
\begin{equation} \frac{w_{20}}{\sigma} = 2 \, \sqrt{2 \ln(5)} \approx 3.5882 \end{equation}
\begin{equation} \int \limits_{-\infty}^{\infty} G(x) \, \mathrm{d}x = \sqrt{\frac{\pi}{4 \ln(2)}} \, A \times \mathrm{FWHM} \approx 1.0645 \, A \times \mathrm{FWHM} \end{equation}
Related functions
\begin{equation} \delta(x) = \lim \limits_{a \to \infty} \sqrt{\frac{a}{\pi}} \exp(-a x^{2}) \end{equation}
\begin{equation} \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int \limits_{0}^{x} \exp(-y^{2}) \, \mathrm{d}y \end{equation}
Trigonometric functions
Definition
\begin{equation} \sin(\alpha) \equiv \frac{\mathrm{opp}}{\mathrm{hyp}} \end{equation}
\begin{equation} \cos(\alpha) \equiv \frac{\mathrm{adj}}{\mathrm{hyp}} \end{equation}
\begin{equation} \tan(\alpha) \equiv \frac{\sin{\alpha}}{\cos{\alpha}} \end{equation}
\begin{equation} \csc(\alpha) \equiv \frac{1}{\sin{\alpha}} \end{equation}
\begin{equation} \sec(\alpha) \equiv \frac{1}{\cos{\alpha}} \end{equation}
\begin{equation} \cot(\alpha) \equiv \frac{1}{\tan{\alpha}} \end{equation}
where $\mathrm{adj}$ is the adjacent side, $\mathrm{opp}$ is the opposite side, and $\mathrm{hyp}$ is the hypotenuse of a right triangle.
Special values
| α | sin(α) | cos(α) | tan(α) | csc(α) | sec(α) | cot(α) |
|---|---|---|---|---|---|---|
| $0$ | $0$ | $1$ | $0$ | $\infty$ | $1$ | $\infty$ |
| $\pi/4$ | $1/\sqrt{2}$ | $1/\sqrt{2}$ | $1$ | $\sqrt{2}$ | $\sqrt{2}$ | $1$ |
| $\pi/2$ | $1$ | $0$ | $\infty$ | $1$ | $\infty$ | $0$ |
| $3\pi/4$ | $1/\sqrt{2}$ | $-1/\sqrt{2}$ | $-1$ | $\sqrt{2}$ | $-\sqrt{2}$ | $-1$ |
| $\pi$ | $0$ | $-1$ | $0$ | $\infty$ | $-1$ | $\infty$ |
Identities
\begin{equation} \sin(\alpha) = -\sin(\alpha + \pi) = \cos \! \left(\frac{\pi}{2} - \alpha \right) \end{equation}
\begin{equation} \cos(\alpha) = \cos(-\alpha) = -\cos(\alpha + \pi) = \sin \! \left(\frac{\pi}{2} - \alpha \right) \end{equation}
\begin{equation} \tan(\alpha) = \tan(\alpha + \pi) = \cot \! \left(\frac{\pi}{2} - \alpha \right) \end{equation}
\begin{equation} \cot(\alpha) = \cot(\alpha + \pi) = \tan \! \left(\frac{\pi}{2} - \alpha \right) \end{equation}
Additional relations
\begin{equation} \sin^{2}(\alpha) + \cos^{2}(\alpha) = 1 \end{equation}
\begin{equation} \sin(\alpha) = \frac{1}{2 \mathrm{i}} \left( \mathrm{e}^{\mathrm{i} \alpha} - \mathrm{e}^{-\mathrm{i} \alpha} \right) \end{equation}
\begin{equation} \cos(\alpha) = \frac{1}{2} \left( \mathrm{e}^{\mathrm{i} \alpha} + \mathrm{e}^{-\mathrm{i} \alpha} \right) \end{equation}
\begin{equation} \cos(\alpha) + \mathrm{i} \sin(\alpha) = \exp(\mathrm{i} \alpha) \end{equation}
Further information
The Wikipedia list of trigonometric identities contains an extensive and useful list of relations between trigonometric functions.