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

$$\mathrm{lb}(x) \equiv \log_{\rm 2}(x)$$

$$\ln(x) \equiv \log_{\rm e}(x)$$

$$\lg(x) \equiv \log_{10}(x)$$

Special values

$$\log_{a}(1) = 0$$

$$\log_{a}(a) = 1$$

Change of base

$$\log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)} = \frac{1}{\log_{x}(a)}$$

Identities

$$\log_{a}(x y) = \log_{a}(x) + \log_{a}(y)$$

$$\log_{a}(x / y) = \log_{a}(x) - \log_{a}(y)$$

$$\log_{a}(x^{y}) = y \log_{a}(x)$$

Inverse

$$y = f(x) = \log_{a}(x) \quad \rightarrow \quad x = f^{-1}(y) = a^{y}$$

Derivative and integral

$$\frac{\mathrm{d}}{\mathrm{d}x} \ln(x) = \frac{1}{x}$$

$$\frac{\mathrm{d} \log_{a}(x)}{\mathrm{d}x} = \frac{1}{\ln(a) x}$$

$$\int \ln(x) \mathrm{d}x = x \ln(x) - x$$

Exponential

Special bases

$$\exp(x) \equiv \mathrm{e}^{x}$$

Special values

$$a^{0} = 1$$

$$a^{1} = a$$

Change of base

$$a^{x} = b^{x \log_{b}(a)}$$

Identities

$$a^{x + y} = a^{x} a^{y}$$

$$a^{x - y} = \frac{a^{x}}{a^{y}}$$

$$a^{x y} = (a^{x})^{y} = (a^{y})^{x}$$

Inverse

$$y = f(x) = a^{x} \quad \rightarrow \quad x = f^{-1}(y) = \log_{a}(y)$$

Derivative and integral

$$\frac{\mathrm{d}}{\mathrm{d}x} a^{x} = a^{x} \ln(a)$$

$$\int a^{x} \mathrm{d}x = \frac{a^{x}}{\ln(a)}$$

$$\frac{\mathrm{d}}{\mathrm{d}x} \exp(x) = \exp(x)$$

$$\int \exp(x) \mathrm{d}x = \exp(x)$$

Gaussian function

Definition

$$G(x) \equiv A \exp \! \left( -\frac{(x - x_{0})^{2}}{2 \sigma^{2}} \right)$$

Special values

$$G(x_{0}) = A$$

$$G \! \left( x_{0} \pm \sqrt{2 \ln(2)} \sigma \right) = \frac{A}{2}$$

Derivative and integral

$$\frac{\mathrm{d}}{\mathrm{d}x} G(x) = -\frac{x - x_{0}}{\sigma^{2}} G(x)$$

$$\int G(x) \, \mathrm{d}x = -\sqrt{\frac{\pi}{2}} A \sigma \, \mathrm{erf} \! \left( \frac{x_{0} - x}{\sqrt{2} \sigma} \right)$$

$$\int \limits_{-\infty}^{\infty} G(x) \, \mathrm{d}x = A \sigma \sqrt{2 \pi}$$

Properties

$$\frac{\mathrm{FWHM}}{\sigma} = 2 \, \sqrt{2 \ln(2)} \approx 2.3548$$

$$\frac{w_{20}}{\sigma} = 2 \, \sqrt{2 \ln(5)} \approx 3.5882$$

$$\int \limits_{-\infty}^{\infty} G(x) \, \mathrm{d}x = \sqrt{\frac{\pi}{4 \ln(2)}} \, A \times \mathrm{FWHM} \approx 1.0645 \, A \times \mathrm{FWHM}$$

Related functions

$$\delta(x) = \lim \limits_{a \to \infty} \sqrt{\frac{a}{\pi}} \exp(-a x^{2})$$

$$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int \limits_{0}^{x} \exp(-y^{2}) \, \mathrm{d}y$$

Trigonometric functions

Definition

$$\sin(\alpha) \equiv \frac{\mathrm{opp}}{\mathrm{hyp}}$$

$$\cos(\alpha) \equiv \frac{\mathrm{adj}}{\mathrm{hyp}}$$

$$\tan(\alpha) \equiv \frac{\sin{\alpha}}{\cos{\alpha}}$$

$$\csc(\alpha) \equiv \frac{1}{\sin{\alpha}}$$

$$\sec(\alpha) \equiv \frac{1}{\cos{\alpha}}$$

$$\cot(\alpha) \equiv \frac{1}{\tan{\alpha}}$$

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

$$\sin(\alpha) = -\sin(\alpha + \pi) = \cos \! \left(\frac{\pi}{2} - \alpha \right)$$

$$\cos(\alpha) = \cos(-\alpha) = -\cos(\alpha + \pi) = \sin \! \left(\frac{\pi}{2} - \alpha \right)$$

$$\tan(\alpha) = \tan(\alpha + \pi) = \cot \! \left(\frac{\pi}{2} - \alpha \right)$$

$$\cot(\alpha) = \cot(\alpha + \pi) = \tan \! \left(\frac{\pi}{2} - \alpha \right)$$

$$\sin^{2}(\alpha) + \cos^{2}(\alpha) = 1$$

$$\sin(\alpha) = \frac{1}{2 \mathrm{i}} \left( \mathrm{e}^{\mathrm{i} \alpha} - \mathrm{e}^{-\mathrm{i} \alpha} \right)$$

$$\cos(\alpha) = \frac{1}{2} \left( \mathrm{e}^{\mathrm{i} \alpha} + \mathrm{e}^{-\mathrm{i} \alpha} \right)$$

$$\cos(\alpha) + \mathrm{i} \sin(\alpha) = \exp(\mathrm{i} \alpha)$$

Further information

The Wikipedia list of trigonometric identities contains an extensive and useful list of relations between trigonometric functions.