A Fourier series, after Joseph Fourier (1768-1830), is the series expansion of a periodic, sectionally continuous function into a function series of sine and cosine functions.

The calculator can be used to perform a Fourier series expansion on any measured value or, alternatively, on a function.

Scale:

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Fourier series:

Number of Fourier components:

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With the Fourier expansion of a function the integration range can be specified (interval). When points are specified, linear interpolation is performed between the points and the integration range extends from the first to the last specified point.

Points:

Polygon:

An alternative input is possible with load data from file. The values may be separated comma or space or semicolon. The values must be given pairwise x_{1},y_{1},x_{2},y_{2}...

Measured values and functions can be approximated by the periodic functions. The procedure for this is the development of a Fourier series. The elements of the Fourier series are sine and cosine functions. The development takes place in ascending order of frequencies.

The Fourier series is:

$${s}_{n}(x)=\frac{{a}_{0}}{2}+\sum _{k=1}^{n}({a}_{k}cos(k\omega x)+{b}_{k}sin(k\omega x))$$

with the Fourier coefficients a_{k} und b_{k} and ω = 2π/T. This is the period T = b - a with the initial interval a and the end of interval b.

The Fourier coefficients a_{k} und b_{k} satisfy the least squares condition for the associated sine or cosine function. The coefficients are calculated as follows.

$${a}_{k}=\frac{2}{l}\underset{a}{\overset{\mathrm{b}}{\int}}f(x)cos(k\omega x)dx$$

$${b}_{k}=\frac{2}{l}\underset{a}{\overset{\mathrm{b}}{\int}}f(x)sin(k\omega x)dx$$

**Constants**

Name | Description |
---|---|

LN2 | Natural logarithm of 2 |

LN10 | Natural logarithm of 10 |

LOG2E | Base 2 logarithm of EULER |

LOG10E | Base 10 logarithm of EULER |

PI | Ratio of the circumference of a circle to its diameter |

SQRT1_2 | Square root of 1/2 |

SQRT2 | Square root of 2 |

**Trigonometric Functions**

Function | Description |
---|---|

sin(x) | sine of x |

cos(x) | Cosine of x |

tan(x) | Tangent of x |

asin(x) | arcsine |

acos(x) | arccosine of x |

atan(x) | arctangent of x |

atan2(y, x) | Returns the arctangent of the quotient of its arguments. |

cosh(x) | Hyperbolic cosine of x |

sinh(x) | Hyperbolic sine of x |

**Logarithm and Exponential**

Function | Description |
---|---|

pow(x, y) | x^{y} |

sqrt(x) | Square root of x |

exp(x) | e^{x} |

log(x), ln(x) | Natural logarithm |

log(x, b) | Logarithm to base b |

log2(x), lb(x) | Logarithm to base 2 |

log10(x), ld(x) | Logarithm to base 10 |

**More functions**

Function | Description |
---|---|

ceil(x) | Get smallest integer n with n > x. |

abs(x) | Absolute value of x |

max(a, b, c, ...) | Maximum value of all given values. |

min(a, b, c, ...) | Minimum value of all given values. |

random(max = 1) | Generate a random number between 0 and max. |

round(v) | Returns the value of a number rounded to the nearest integer. |

floor(x) | Returns the biggest integer n with n < x. |

factorial(n) | Calculates n! |

trunc(v, p = 0) | Truncate v after the p-th decimal. |

V(s) | Returns the value of the given element, e.g. sliders and angles. |

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Here is a list of of further useful sites:

Index Curve fitting for: linear line, power function, polynomial, normal distribution Newton Interpolation Horners Method Trigonometric calculations Taylor series calculator Normal Distribution Plot NxN Gauss method Derivation rules ODE first order