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Fourier Analysis
Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
Examples
Square wave
Slider(n, 1, 30, 1, 6) Plot(SquareWave(x), color=orange) Plot(FourierSeries([[-1, -@pi..0], [1, 0..@pi]], x, n), color=red)
Sawtooth wave
Slider(n, 1, 30, 1, 6) Plot(2@pi*SawToothWave(x-@pi)-@pi, color=cyan) Plot(FourierTrigSeries(x, x, n), color=blue)
Triangle wave
Slider(n, 1, 30, 1, 1) Plot(@pi/2*TriangleWave(x-@pi/2)+@pi/2, color=magenta) Plot(FourierSeries([[-x, -@pi..0], [x, 0..@pi]], x, n), color=purple)
Fourier series
FourierTrigSeries(x, x, 5)
FourierExpSeries(x, x, 5)
Fourier cosine transform
FourierCos(sin(x), x, w)
Fourier sine transform
FourierSin(sin(x)^2, x, w)
References
https://en.wikipedia.org/wiki/Fourier_analysis
Related Functions
FourierCos
,
FourierExpSeries
,
FourierSin
,
FourierTrigSeries
,
FourierSeries
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Using MathStudio
The Basics
1.
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2.
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3.
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4.
Lists
5.
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6.
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7.
Calculus
8.
Fourier Analysis
9.
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10.
Base Conversions
11.
Regression Analysis
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Natural Language
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Constants
14.
Dynamic Variables
Operators
1.
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2.
Double Power Operator
3.
Comparison Operators
4.
Logical Operators
5.
Compound Assignment Operators
6.
Type Comparison Operators
7.
Size Operator
8.
Range Operator
9.
Transformation Operator
10.
Index Operator
11.
Union Operator
12.
Intersection Operator
13.
Difference Operator
14.
Double Modulo Operator
15.
Matrix Operator
16.
Matrix Size Operator
17.
Angle Operator
18.
Choose Operator
19.
String Concatenation Operator
20.
Conditional Operator
21.
Append Operator
22.
Element Operator
Graphing
1.
2D Graphing
2.
Plot Styles
3.
Min, Max and Zero Values
4.
3D Graphing
5.
Multiple Plots
6.
Color Names
7.
Time Graphing
Scripting Language
Variables
1.
Creating Variables
2.
Unicode Characters
3.
Variable Functions
4.
Strings
5.
Wildcards
Scripts
1.
Creating Scripts
2.
Semicolons
3.
Comments
4.
Variable Controls
5.
Conditional Statements
6.
For-In Loops
7.
While Loops
8.
Timing
9.
Objects
10.
Working with JSON
Functions
A
abs
(
z
)
AiryAi
(
z
)
AiryBi
(
z
)
AlternatingSeries
(
n, [terms]
)
Angle
(
A, B
)
Animate
(
variable, start, end, [step], [value], [fps]
)
Apart
(
f(x), x
)
Append
(
expression1, expression2
)
Arg
(
z
)
B
BarPlot
(
list1, frequency1, ...
)
Bernoulli
(
n, [z]
)
BesselI
(
v, z
)
BesselJ
(
v, z
)
BesselK
(
v, z
)
BesselY
(
v, z
)
Beta
(
m, n
)
Binomial
(
n, r
)
BinomialCDF
(
n, p, x
)
BinomialPDF
(
n, p, x
)
BodePlot
(
function, variable, minimum, maximum, mode
)
BoxPlot
(
x, [frequency1], [x2], [frequency2], ...
)
Button
(
name, [list], [value]
)
C
Call
(
function, parameter1, parameter2, ...
)
Caps
(
string, index, [mode]
)
Catalan
(
n
)
Ceil
(
z
)
cFrac
(
n, [terms]
)
Char
(
x
)
ChebyshevT
(
n, z
)
ChebyshevU
(
n, z
)
CheckBox
(
variable, [value]
)
Chi
(
z
)
ChiSquareCDF
(
lower, upper, df
)
ChiSquarePDF
(
x, df
)
Cholesky
(
A
)
Choose
(
test1, value1, test2, value2, ...
)
Ci
(
z
)
Clear
(
variable, ...
)
clip
(
x, [min], [max]
)
Coefficient
(
f(x), x, degree
)
coFactor
(
matrix, i, j
)
Command
(
setting
)
Conj
(
z
)
ContourPlot
(
expression, ...
)
Convergents
(
n, [mode=0], [terms]
)
Cross
(
A, B
)
CubicFit
(
[x], y
)
CubicFitModel
(
[x], y
)
CubicFitPlot
(
[x], y
)
Curl
(
vector, [varlist], [mode]
)
CylindricalPlot3D
(
expression, ...
)
D
D
(
f(x), x, [n]
)
Date
(
letter
)
Dawson
(
z
)
Degree
(
expression
)
DEGtoDMS
(
angle, [digits]
)
Delete
(
expression, position, [length]
)
Denominator
(
expression
)
Det
(
A
)
Diff
(
f(x), x
)
DiGamma
(
z
)
DiLog
(
z
)
Dirichlet_Eta
(
z
)
Dirichlet_Lambda
(
z
)
Divergence
(
[varlist], vector, [mode]
)
Divisors
(
n
)
DivisorSigma
(
n
)
Dot
(
A, B
)
DSolve
(
equation, dependent(independent), values, mode=1
)
Duf
(
function, [varlist], [point], [direction]
)
dynamic
(
name, value
)
E
Ei
(
z
)
Eigenvalues
(
A
)
Eigenvectors
(
A
)
Else If
(
condition
)
Erf
(
z
)
Erfc
(
z
)
Error
(
string
)
Euler
(
n, [z]
)
Eulerian
(
n, k
)
Eval
(
function, variable, value
)
Exp
(
z
)
Expand
(
expression
)
ExpConvert
(
expression
)
ExpFit
(
[x], y
)
ExpFitModel
(
[x], y
)
ExpFitPlot
(
[x], y
)
ExponentialFit
(
[x], y
)
ExponentialFitModel
(
[x], y
)
ExponentialFitPlot
(
[x], y
)
Extract
(
expression, position, [length]
)
F
Factor
(
expression
)
Factorial
(
n
)
Fcdf
(
lower, upperbound, numerator, denominator
)
fDiff
(
f(x,y,z,..), (x,y,z,...)
)
Fibonacci
(
n, [z]
)
Finance
(
PV, FV, Pmt, i, n, [mode=0]
)
FindFit
(
function, variable, [x], y
)
FindFitModel
(
function, variable, [x], y
)
FindFitPlot
(
function, variable, [x], y
)
Flatten
(
matrix
)
Floor
(
z
)
FourierCos
(
f(x), x, w
)
FourierExpSeries
(
f(x), x, n
)
FourierSeries
(
f(x), x, [n], [mode]
)
FourierSin
(
f(x), x, w
)
FourierTrigSeries
(
f(x), x, n
)
fPart
(
z
)
Fpdf
(
x, d1, d2
)
FractalPlot
(
expression, ...
)
FresnelCos
(
z
)
FresnelSin
(
z
)
FullRectSineWave
(
x, [T]
)
Function
(
function, parameter1, parameter2, ...
)
G
Gamma
(
z
)
GCD
(
n1, n2
)
GegenbauerC
(
n, a, z
)
GeoCDF
(
p, x
)
GeoPDF
(
p, x
)
Gradient
(
function, [varlist], [mode]
)
Gudermannian
(
z
)
H
HalfRectSineWave
(
x, [T]
)
HankelH1
(
v, z
)
HankelH2
(
v, z
)
Harmonic
(
z
)
Heaviside
(
x
)
HermiteH
(
n, z
)
Hessian
(
function, [varlist], [mode]
)
HRStoHMS
(
hours, [digits]
)
http
(
url, [filter]
)
Hypergeom_2F1
(
a, b, c, z, [mode]
)
I
Identity
(
n
)
iDiff
(
f(x,y,z,..), dependant_variables, independant_variables
)
If
(
condition
)
iLaplace
(
F(s), s
)
Im
(
z
)
ImagePlot
(
data
)
ImaginaryPart
(
expression
)
ImplicitPlot
(
expression, ...
)
Include
(
filename
)
Insert
(
expression, item, position
)
Integrate
(
f(x), x, [a], [b]
)
Inverse
(
A
)
InverseNormal
(
p, [mu], [sigma]
)
invGudermannian
(
z
)
iPart
(
z
)
IsList
(
expression
)
IsMatrix
(
expression
)
IsNumber
(
expression
)
IsPoly
(
expression, variable
)
IsPrime
(
n
)
J
Jacobian
(
function, [varlist], [point]
)
json
(
data, [filter]
)
JuliaPlot
(
expression, ...
)
K
KelvinBei
(
v, z
)
KelvinBer
(
v, z
)
KelvinKei
(
v, z
)
KelvinKer
(
v, z
)
L
LaguerreL
(
n, z
)
LambertW
(
z
)
Laplace
(
f(t), t
)
Laplacian
(
function, [varlist], [mode]
)
LCM
(
n1, n2
)
Left
(
expression
)
LegendreP
(
n, z
)
LegendreQ
(
n, z
)
Length
(
expression
)
Li
(
x
)
Limit
(
function, variable, point, [direction]
)
LinearFit
(
[x], y
)
LinearFitModel
(
[x], y
)
LinearFitPlot
(
[x], y
)
List
(
length
)
ListMap
(
list, name, [value]
)
ListPlot
(
x, [y], [x2], [y2]
)
ListPlot3D
(
x, y, z, [x2], [y2], [z2], ...
)
Ln
(
z
)
LnFit
(
[x], y
)
LnFitModel
(
[x], y
)
LnFitPlot
(
[x], y
)
LnGamma
(
z
)
local
(
name, value
)
Log
(
[n], z
)
LogisticFit
(
[x], y
)
LogisticFitModel
(
[x], y
)
LogisticFitPlot
(
[x], y
)
Lucas
(
n, [z]
)
M
Matrix
(
rows, [columns]
)
MatrixLU
(
matrix, [mode]
)
MatrixMap
(
matrix, name, [value]
)
MatrixQR
(
A
)
MatrixSVD
(
A
)
Max
(
number, ...
)
Mean
(
list
)
MedianFit
(
[x], y
)
MedianFitModel
(
[x], y
)
MedianFitPlot
(
[x], y
)
Message
(
string
)
Min
(
list
)
Mod
(
m, n
)
Multinomial
(
nList
)
MultiPlot
(
plot1, plot2, ...
)
MultiPlot3D
(
expression, ...
)
N
nCr
(
n, r
)
NIntegrate
(
f(x), x, a, b
)
Norm
(
z
)
NormalCDF
(
lower, upper, [mu], [sigma]
)
NormalPDF
(
x, mu, sigma
)
nPr
(
n, r
)
nPrimes
(
n
)
nRoot
(
z, n, [mode]
)
P
ParametricPlot
(
expression, ...
)
ParametricPlot3D
(
expression, ...
)
Part
(
expression, n
)
pDiff
(
f(x,y,z,..), dependent_variables, dependent_equations, independent_variables
)
Pi_Digits
(
n
)
Plot
(
expression, ...
)
Plot3D
(
expression, ...
)
Pochhammer
(
n, k
)
PoissonCDF
(
mu, x
)
PoissonPDF
(
mu, x
)
PolarPlot
(
expression, ...
)
PolyDivide
(
f(z), g(z)
)
PolyFit
(
pList, var
)
PolyGamma
(
n, z
)
PolyGCD
(
polynomial1, polynomial2
)
PolyLCM
(
polynomial1, polynomial2, [mode=1]
)
PolyLog
(
n, z
)
PowerExpand
(
expression
)
PowerFit
(
[x], y
)
PowerFitModel
(
[x], y
)
PowerFitPlot
(
[x], y
)
Prime
(
n
)
Primorial
(
n
)
Product
(
f(n), n, start, end
)
Psi
(
z
)
Q
QuadraticFit
(
[x], y
)
QuadraticFitModel
(
[x], y
)
QuadraticFitPlot
(
[x], y
)
QuarticFit
(
[x], y
)
QuarticFitModel
(
[x], y
)
QuarticFitPlot
(
[x], y
)
Quotient
(
f(x), g(x)
)
R
Random
(
min, max, [size1], [size2]
)
Re
(
z
)
RealPart
(
expression
)
Remainder
(
f(x), g(x)
)
Replace
(
expression, old, new
)
Reshape
(
matrix
)
Return
(
expression
)
Reverse
(
list
)
Right
(
expression
)
RK4
(
function, t, y, a, b, y0, N=10
)
RK45
(
function, t, y, a, b, y0, Tol=10E-5, hmax=0.25, hmin=10E-6, N=20
)
Round
(
number, digits
)
RowReduce
(
A
)
S
SawToothWave
(
x, [T]
)
Sequence
(
f(x), x, start, end, [step], [start term], [number of terms]
)
Series
(
f(z), z, n
)
Shi
(
z
)
Si
(
z
)
Sign
(
z
)
SimplifyFunction
(
expression
)
SimplifyPoly
(
f(x)/g(x), [variable]
)
sin
(
z
)
SinFit
(
[x], y
)
SinFitModel
(
[x], y
)
SinFitPlot
(
[x], y
)
Size
(
list
)
Slider
(
variable, start, end, [step], [initial value]
)
Solve
(
f(x), x, [guess]
)
SolveSystem
(
equations, [variables], [guesses]
)
Sort
(
list
)
SphericalPlot3D
(
expression, ...
)
Sqrt
(
z
)
SquareWave
(
x, [T]
)
StaircaseWave
(
x, [T]
)
StandardDeviation
(
list
)
String
(
expression1, expression2, ...
)
StudentTCDF
(
x, df
)
StudentTPDF
(
x, df
)
Sum
(
f(n), n, start, end
)
SurfaceNormal
(
function, [varlist], [point]
)
T
Together
(
expression
)
Trace
(
variable
)
Transpose
(
A
)
TriangleWave
(
x, [T]
)
TrigConvert
(
expression
)
TrigExpand
(
expression
)
TrigReduce
(
expression
)
V
Variables
(
expression
)
Variance
(
list
)
VectorPlot
(
list, ...
)
VectorPlot3D
(
list, ...
)
W
While
(
condition
)
Z
Zeta
(
z, [a]
)
Computing
