- CompositeSimpsonsRule
CompositeSimpsonsRule (f,a,b,n)

Integration of f by Composite Simpson's Rule on the interval [a,b] with n subintervals with error of max(f'''')*h^4*(b-a)/180, note that n should be even.

See Planetmath for more information.

- CompositeSimpsonsRuleTolerance
CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)

Integration of f by Composite Simpson's Rule on the interval [a,b] with the number of steps calculated by the fourth derivative bound and the desired tolerance.

See Planetmath for more information.

- Derivative
Derivative (f,x0)

Attempt to calculate derivative by trying first symbolically and then numerically.

See Wikipedia for more information.

- EvenPeriodicExtension
EvenPeriodicExtension (f,L)

Return a function that is the even periodic extension of

`f`

with half period`L`

. That is a function defined on the intervalextended to be even on`[0,L]`

and then extended to be periodic with period`[-L,L]`

.`2*L`

See also OddPeriodicExtension and PeriodicExtension.

Version 1.0.7 onwards.

- FourierSeriesFunction
FourierSeriesFunction (a,b,L)

Return a function that is a Fourier series with the coefficients given by the vectors

`a`

(sines) and`b`

(cosines). Note thatis the constant coefficient! That is,`a@(1)`

refers to the term`a@(n)`

, while`cos(x*(n-1)*pi/L)`

refers to the term`b@(n)`

. Either`sin(x*n*pi/L)`

`a`

or`b`

can be`null`

.- InfiniteProduct
InfiniteProduct (func,start,inc)

Try to calculate an infinite product for a single parameter function.

- InfiniteProduct2
InfiniteProduct2 (func,arg,start,inc)

Try to calculate an infinite product for a double parameter function with func(arg,n).

- InfiniteSum
InfiniteSum (func,start,inc)

Try to calculate an infinite sum for a single parameter function.

- InfiniteSum2
InfiniteSum2 (func,arg,start,inc)

Try to calculate an infinite sum for a double parameter function with func(arg,n).

- IsContinuous
IsContinuous (f,x0)

Try and see if a real-valued function is continuous at x0 by calculating the limit there.

- IsDifferentiable
IsDifferentiable (f,x0)

Test for differentiability by approximating the left and right limits and comparing.

- LeftLimit
LeftLimit (f,x0)

Calculate the left limit of a real-valued function at x0.

- Limit
Limit (f,x0)

Calculate the limit of a real-valued function at x0. Tries to calculate both left and right limits.

- MidpointRule
MidpointRule (f,a,b,n)

Integration by midpoint rule.

- NumericalDerivative
NumericalDerivative (f,x0)

Aliases:

`NDerivative`

Attempt to calculate numerical derivative.

See Wikipedia for more information.

- NumericalFourierSeriesCoefficients
NumericalFourierSeriesCoefficients (f,L,N)

Return a vector of vectors

where`[a,b]`

`a`

are the cosine coefficients and`b`

are the sine coefficients of the Fourier series of`f`

with half-period`L`

(that is defined onand extended periodically) with coefficients up to`[-L,L]`

`N`

th harmonic computed numerically. The coefficients are computed by numerical integration using`NumericalIntegral`

.See Wikipedia or Mathworld for more information.

Version 1.0.7 onwards.

- NumericalFourierSeriesFunction
NumericalFourierSeriesFunction (f,L,N)

Return a function that is the Fourier series of

`f`

with half-period`L`

(that is defined onand extended periodically) with coefficients up to`[-L,L]`

`N`

th harmonic computed numerically. This is the trigonometric real series composed of sines and cosines. The coefficients are computed by numerical integration using`NumericalIntegral`

.See Wikipedia or Mathworld for more information.

Version 1.0.7 onwards.

- NumericalFourierCosineSeriesCoefficients
NumericalFourierCosineSeriesCoefficients (f,L,N)

Return a vector of coefficients of the cosine Fourier series of

`f`

with half-period`L`

. That is, we take`f`

defined ontake the even periodic extension and compute the Fourier series, which only has cosine terms. The series is computed up to the`[0,L]`

`N`

th harmonic. The coefficients are computed by numerical integration using`NumericalIntegral`

. Note thatis the constant coefficient! That is,`a@(1)`

refers to the term`a@(n)`

.`cos(x*(n-1)*pi/L)`

See Wikipedia or Mathworld for more information.

Version 1.0.7 onwards.

- NumericalFourierCosineSeriesFunction
NumericalFourierCosineSeriesFunction (f,L,N)

Return a function that is the cosine Fourier series of

`f`

with half-period`L`

. That is, we take`f`

defined ontake the even periodic extension and compute the Fourier series, which only has cosine terms. The series is computed up to the`[0,L]`

`N`

th harmonic. The coefficients are computed by numerical integration using`NumericalIntegral`

.See Wikipedia or Mathworld for more information.

Version 1.0.7 onwards.

- NumericalFourierSineSeriesCoefficients
NumericalFourierSineSeriesCoefficients (f,L,N)

Return a vector of coefficients of the sine Fourier series of

`f`

with half-period`L`

. That is, we take`f`

defined ontake the odd periodic extension and compute the Fourier series, which only has sine terms. The series is computed up to the`[0,L]`

`N`

th harmonic. The coefficients are computed by numerical integration using`NumericalIntegral`

.See Wikipedia or Mathworld for more information.

Version 1.0.7 onwards.

- NumericalFourierSineSeriesFunction
NumericalFourierSineSeriesFunction (f,L,N)

Return a function that is the sine Fourier series of

`f`

with half-period`L`

. That is, we take`f`

defined ontake the odd periodic extension and compute the Fourier series, which only has sine terms. The series is computed up to the`[0,L]`

`N`

th harmonic. The coefficients are computed by numerical integration using`NumericalIntegral`

.See Wikipedia or Mathworld for more information.

Version 1.0.7 onwards.

- NumericalIntegral
NumericalIntegral (f,a,b)

Integration by rule set in NumericalIntegralFunction of f from a to b using NumericalIntegralSteps steps.

- NumericalLeftDerivative
NumericalLeftDerivative (f,x0)

Attempt to calculate numerical left derivative.

- NumericalLimitAtInfinity
NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)

Attempt to calculate the limit of f(step_fun(i)) as i goes from 1 to N.

- NumericalRightDerivative
NumericalRightDerivative (f,x0)

Attempt to calculate numerical right derivative.

- OddPeriodicExtension
OddPeriodicExtension (f,L)

Return a function that is the odd periodic extension of

`f`

with half period`L`

. That is a function defined on the intervalextended to be odd on`[0,L]`

and then extended to be periodic with period`[-L,L]`

.`2*L`

See also EvenPeriodicExtension and PeriodicExtension.

Version 1.0.7 onwards.

- OneSidedFivePointFormula
OneSidedFivePointFormula (f,x0,h)

Compute one-sided derivative using five point formula.

- OneSidedThreePointFormula
OneSidedThreePointFormula (f,x0,h)

Compute one-sided derivative using three-point formula.

- PeriodicExtension
PeriodicExtension (f,a,b)

Return a function that is the periodic extension of

`f`

defined on the intervaland has period`[a,b]`

.`b-a`

See also OddPeriodicExtension and EvenPeriodicExtension.

Version 1.0.7 onwards.

- RightLimit
RightLimit (f,x0)

Calculate the right limit of a real-valued function at x0.

- TwoSidedFivePointFormula
TwoSidedFivePointFormula (f,x0,h)

Compute two-sided derivative using five-point formula.

- TwoSidedThreePointFormula
TwoSidedThreePointFormula (f,x0,h)

Compute two-sided derivative using three-point formula.