Mathematical functions, formulas and equations
Trigonmetrical Functions
Trigonometrical functions, sin, cos, tan and others are widely used in the design, development, and analysis of radio and electronics circuits, as well as in other applications such as antennas, and RF components.
These trignometric relationships are widely used in the design and development of many RF and other radio circuits. In particualr sin, cos and tan are the most widely used.
The fact that their use is embedded inth e very basics of electronics and radio technology is demonstrated by the fact that a single frequency repetitive waveform is known as a sine wave because of the fact that its waveform follows the values mapped out by the sine function.
Hyperbolic Functions
Hyperbolic functions are used in a number of mathematical calculations associated with te design and development of radio and electronics circuits.
While these hyperbolic functions may not be widely used in everyday arithmetic, they can be very useful when addressing some more mathematical concepts.
The three hyperbolic formulae for sinh, cosh, and tanh are given below:
Fourier Series Function and Formula
The Fourier series and its resulting Fourier analysis is used in mathematics and also within electronics engineering to analyse waveforms and process them in a process known as digital signal processing (DSP).
A Fourier series uses mathematical processes and decomposes a periodic function into a sum of simple oscillating functions, i.e. sines and cosines. By manipulating these series using mathematical processes it is possible to analyse and process waveforms using computer techniques. Often this is done in real time to enable complex processing to replace analogue circuitry. This has the advantage that waveforms can be processed more exactly to give very high levels of performance.
Basic Fourier series formula and function
For a continuous-time, T-periodic signal x(t), the N-harmonic Fourier series approximation can be written as the following function or formula:
| x(t) = | a0 + a1 cos (wot + q1) + a2 cos (2 wot + q2) + ... |
| + ... + aN cos (N wot + qN) |
where
the fundamental frequency wo is 2pi /T rad/sec;
the amplitude coefficients a1, ..., aN are non-negative
the radian phase angles satisfy 0 � q1 , ..., qN < 2pi
the fundamental frequency wo is 2pi /T rad/sec;
the amplitude coefficients a1, ..., aN are non-negative
the radian phase angles satisfy 0 � q1 , ..., qN < 2pi
Mathematical Constants, e, pi, Euler's constant and the golden ratio
Within mathematics, math, or maths there are four major constants that appear in a number of situations: Pi (π), the natural log base or Euler's number (e), Euler's constant often called the Euler Mascheroni constant (g) and finally the Golden ratio (f).
These four mathematical constants appear in a number of maths formulae, and can be seen as forming some very basic cornerstones in mathematical calculations.
In view of this, these constants are widely seen within the mathematical arena.
Table of the mathematical constants
| CONSTANT | LETTER | VALUE |
|---|---|---|
| Pi | π | 3.14159265358979323846264 |
| Natural log base / Euler's number | e | 2.71828 18284 59045 23536.... |
| Euler's constant Euler-Mascheroni constant | g | 0.57721 56649 01532 86060 65120 90082... |
| Golden ratio | f | 1.618033988749894... |
Pi, π
Pi represented by the Greek letter p is a mathematical constant. It is the ratio between the circumference of a circle to its diameter as well as being the ratio between the area of a circle to the square of its radius.
Pi is an irrational number, i.e. it cannot be expressed as a fraction of two integers. The commonly used fraction 22/7 which is often used is only a rough approximation, although sufficient for many basic calculations where only accuracy is not required. In addition to this the decimal representation of Pi never ends or repeats. Beyond being irrational, Pi is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can ever produce it exactly.
Pi can be given in several representations:
| NOTATION | NUMBER |
|---|---|
| Decimal | 3.14159265358979323846 |
| Hexadecimal | 3.243F6A8885A308D31319 |
| Binary | 11.00100100001111110110 |
Natural log base
The mathematical number e, also known as Euler's number (not to be confused with the Euler-Mascheroni constant, sometimes called simply Euler's constant) is the unique real number that has the mathematical property that the function ex has the same value as the slope of the tangent line, for all values of x.
Euler's number, e, is transcendental, i.e. it is a number that does not arise from an ordinary algebraic expression. As a result, Euler's number, e is also irrational; its value cannot be given exactly as a finite or eventually repeating decimal.
Euler's constant
constant is also sometimes called the Euler-Mascheroni constant, and denoted by the Greek letter gamma. It is a less well known mathematical constant than pi or e, but it is still a very important one.
Euler's constant is defined as the limit, as n tends to infinity, of the sum of 1 + 1/2 + 1/3 + ... up to 1/n, minus the natural logarithm of n
Golden ratio
The golden ratio is an unusual number which exists in mathematics. It is a ratio that is said to have perfect dimensions and as a result it has also been used in artistic endeavour as well as mathematical calculations. Also, mathematicians down the years have studied the golden ratio because of its unique and interesting properties.
Two quantities are said to have the golden ratio if the ratio between the sum of the two quantities and the larger one is the same as the ratio between the larger one and the smaller.
The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter (phi). The figure of a golden section illustrates the geometric relationship that defines this constant.
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