Functions Overview

This document summarizes functions supported by microMathmatics Plus

1 Constants

Set a value

$$a := 10$$ $$b := 60$$

Get value

$$a = 10.0$$ $$b = 60.0$$

Build-in constants

$$e = 2.71828$$

$${\pi} = 3.14159$$

$$pi = 3.14159$$

$$i = 0.0+1.0i$$

2 Operators

Addition ("+")

$$d := 10 + 70$$ $$d = 80.0$$

$$c := 1 + i$$ $$c = 1.0+1.0i$$

Subtraction ("-")

$$d := 10 - 70$$ $$d = -60.0$$

$$c := 1 - i$$ $$c = 1.0-1.0i$$

Multiplication ("*")

$$d := 10 \cdot 70$$ $$d = 700.0$$

$$c := i \cdot i$$ $$c = -1.0$$

Division ("/" or "รท")

$$d := a / b$$ or $$d := \frac{a}{b}$$

$$c := i / 2$$ $$c = 0.0+0.5i$$

Parenthesis

$$ \left( e\right) = 2.71828$$

Complex operators:

$$\left( a + b \right) \cdot \left( a - b \right) - \left( a \cdot a - b \cdot b\right) = 0.0$$

3 Common functions

Raise to power ("^")

$${100}^{0} = 1.0$$ $${2}^{2} = 4.0$$

The square root of the argument ("#" or "sqrt")

$$\sqrt{25} = 5.0$$ $$\sqrt{-1} = 0.0+1.0i$$

$$sqrt \left( 10\right) = 3.16228$$

The n-th root of the argument ("$")

$$\sqrt[\leftroot{-3}\uproot{3}3]{125} = 5.0$$

The product of all positive integers less than or equal to the argument ("!")

$$5! = 120.0$$

The absolute value of the real or complex argument ("|" or "abs")

$$ \left| 10 \right| = 10.0$$ $$ \left| -10 \right| = 10.0$$ $$abs \left( i\right) = 1.0$$

The complex conjugate of the complex argument ("~")

$$\overline{\left( 3 + 5i \right)} = 3.0-5.0i$$

$$\overline{\left( 1 - 2i \right)} = 1.0+2.0i$$

The real and imaginary parts of the complex argument

$$\Re\left( 3 + 5i \right) = 3.0$$ $$\Im\left( 3 + 5i \right) = 5.0$$

4 Trigonometric functions

Sine, cosine, tangent

$$sin \left( {\pi} / 2\right) = 1.0$$ $$cos \left( {\pi}\right) = -1.0$$ $$tan \left( 0\right) = 0.0$$

Cosecant, secant, cotangent

$$csc \left( {\pi} / 2\right) = 1.0$$ $$sec \left( -{\pi}\right) = -1.0$$ $$cot \left( {\pi} / 4\right) = 1.0$$

Inverse sine, cosine, tangent

$$asin \left( 1\right) = 1.5708$$

$$acos \left( -1\right) = 3.14159$$

$$atan \left( 0\right) = 0.0$$

Inverse cosecant, secant, cotangent

$$acsc \left( 1\right) = 1.5708$$

$$asec \left( -1\right) = 3.14159$$

$$acot \left( 1\right) = 0.785398$$

The arc tangent of y/x within the range [-pi..pi]

$$atan2 \left( 1,\, 5\right) = 0.197396$$

5 Hyperbolic functions and logarithms

The raising 'e' to the power of the argument

$$exp \left( 3\right) = 20.0855$$

$${e}^{3} = 20.0855$$

The natural logarithm of the argument

$$ln \left( {e}^{5}\right) = 5.0$$

Logarithm of number to the specified base

$$log \left( 256,\, 2\right) = 8.0$$

The base 10 logarithm of the argument

$$log10 \left( {10}^{5}\right) = 5.0$$

Hyperbolic sine, cosine, tangent

$$sinh \left( 0\right) = 0.0$$ $$cosh \left( 0\right) = 1.0$$ $$tanh \left( 0\right) = 0.0$$

Hyperbolic cosecant, secant, cotangent

$$csch \left( 1\right) = 0.850918$$

$$sech \left( 0\right) = 1.0$$

$$coth \left( 1\right) = 1.31304$$

6 Functions for real numbers

The most positive (closest to positive infinity) of the two arguments

$$max \left( 1,\, 2\right) = 2.0$$ $$max \left( 2,\, 1\right) = 2.0$$

The most negative (closest to negative infinity) of the two arguments

$$min \left( 1,\, 2\right) = 1.0$$ $$min \left( 2,\, 1\right) = 1.0$$

The remainder on dividing the first argument by the second

$$mod \left( 11,\, 2\right) = 1.0$$ $$mod \left( e,\, 2\right) = 0.718282$$

Calculates percentage (for example, 5% of 20)

$$perc \left( 20,\, 5\right) = 1.0$$

The pseudo-random value between zero and the argument

$$random \left( 1\right) = 0.935868$$ $$random \left( 10\right) = 7.23644$$

The most negative (closest to negative infinity) integer value greater than or equal to the argument

$$ceil \left( 1\right) = 1.0$$ $$ceil \left( 1.25\right) = 2.0$$

$$ceil \left( 1.75\right) = 2.0$$ $$ceil \left( 2\right) = 2.0$$

The most positive (closest to positive infinity) integer value less than or equal to the argument

$$floor \left( 1\right) = 1.0$$ $$floor \left( 1.25\right) = 1.0$$

$$floor \left( 1.75\right) = 1.0$$ $$floor \left( 2\right) = 2.0$$

Rounds the real number x to n places

$$round \left( {\pi},\, 2\right) = 3.14$$

Returns the integer part of a real number by removing the fractional part

$$trunc \left( {\pi}\right) = 3.0$$

Compute the signum of a number: -1 for negative numbers, +1 for positive numbers and 0 otherwise:

$$sign \left( -5\right) = -1.0$$ $$sign \left( 0\right) = 0.0$$ $$sign \left( e\right) = 1.0$$

7 Processing of the sequence

Summation $$\displaystyle\sum_{k=1}^{10} k = 55.0$$

Product $$\displaystyle\prod_{k=1}^{10} k = 3628800.0$$

8 Derivative of the function

$$f1(x) := \frac{d}{dx} sin \left( x\right) $$ $$f1 \left( 0\right) = 1.0$$

$$f2(x) := \frac{d}{dx} \frac{d}{dx} sin \left( x\right) $$ $$f2 \left( 0\right) = 0.0$$

9 The definite integral of a function

$$\displaystyle\int_{1}^{e}\frac{1}{0.5 \cdot x}\, dx = 2.0$$

$$G({\varphi},{\rho}) := \frac{3 \cdot cos \left( {\varphi}\right) - 2 \cdot sin \left( {\varphi}\right) }{{\rho}}$$

$$\displaystyle\int_{-{\pi} / 2}^{0}\displaystyle\int_{2}^{3}{\rho} \cdot G \left( {\varphi},\, {\rho}\right) \, d{\rho}\, d{\varphi} = 5.0$$

10 IF-function and logical operators

Select a term to be used depending on comparison result for two expressions

$$true := 1$$ $$false := 0$$

$$if \left( 1 = 2,\, true,\, false\right) = 0.0$$

$$if \left( 1 \neq 2,\, true,\, false\right) = 1.0$$

$$if \left( 1 > 2,\, true,\, false\right) = 0.0$$

$$if \left( 1 \ge 2,\, true,\, false\right) = 0.0$$

$$if \left( 1 < 2,\, true,\, false\right) = 1.0$$

$$if \left( 1 \le 2,\, true,\, false\right) = 1.0$$

$$if \left( \left( 1 = 2 \right) or \left( 1 \neq 2 \right),\, true,\, false\right) = 1.0$$

$$if \left( \left( 1 = 2 \right) and \left( 1 \neq 2 \right),\, true,\, false\right) = 0.0$$

11 Function plots

The first x-value: $$x1 := -10$$

The last x-value: $$x2 := 10$$

The sampling step: $$dx := 0.1$$

X-values, interval (":")

$$x := \left[ x1,\, x1 + dx \,..\, x2 \right]$$

Y-values, a function of X

$$f(x) := {e}^{-x / 10} \cdot \displaystyle\sum_{n=0}^{20} \frac{{ \left( -1\right) }^{n}}{\left( 2 \cdot n \right)! } \cdot {x}^{2 \cdot n}$$

A plot with automatically calculated boundaries:

Image

A plot with manually defined boundaries:

Image

12 Arrays

Define index range

$$N := 100$$ $$n := \left[ 1,\, 2 \,..\, N \right]$$

Define 1-D array (enter "[")

$$z_{n} := x1 + \frac{x2 - x1}{N - 1} \cdot \left( n - 1\right) $$ $$G_{n} := f \left( z_{n} \right) $$

Show elements of 1-D array

$$z_{n} = \begin{bmatrix}-10.0\\-9.79798\\-9.59596\\-9.39394\\-9.19192\\-8.9899\\\dots\\10.0\\\end{bmatrix}$$ $$G_{n} = \begin{bmatrix}-2.28083\\-2.48055\\-2.57248\\-2.55721\\-2.43959\\-2.22841\\\dots\\-0.308677\\\end{bmatrix}$$

Assign value by index

$$z_{5} := -10$$

Show values of single elements of array

$$z_{5} = -10.0$$ $$G_{5} = -2.43959$$

Define 2-D array

$$M := 200$$ $$m := \left[ 1,\, 2 \,..\, M \right]$$

$$G2_{n,\, m} := round \left( sin \left( 2 \cdot pi \cdot \frac{n + m}{N + M}\right) ,\, 3\right) $$

$$G2_{n,\, m} = \begin{bmatrix}0.042&0.063&0.084&0.105&0.125&0.146&\dots&-0.876\\0.063&0.084&0.105&0.125&0.146&0.167&\dots&-0.886\\0.084&0.105&0.125&0.146&0.167&0.187&\dots&-0.896\\0.105&0.125&0.146&0.167&0.187&0.208&\dots&-0.905\\0.125&0.146&0.167&0.187&0.208&0.228&\dots&-0.914\\0.146&0.167&0.187&0.208&0.228&0.249&\dots&-0.922\\\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\0.855&0.844&0.833&0.821&0.809&0.797&\dots&0.0\\\end{bmatrix}$$

Get count of rows and columns

$$rows \left( G2\right) = 101.0$$ $$cols \left( G2\right) = 201.0$$

Define an array as vector or matrix, use object properties to change the array size:

$$E := \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}$$ $$E = \begin{bmatrix}1.0&0.0&0.0\\0.0&1.0&0.0\\0.0&0.0&1.0\\\end{bmatrix}$$

13 Output radix

Using result property dialog, it is possible to change the radix for integer results, for example use binary, octal or hexadecimal bases:

$$M := 254$$

$$M = 11111110$$ $$M = 376$$ $$M = fe$$