## Calculus

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.

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.

Derivative
`Derivative (f,x0)`

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

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 interval `[0,L]` extended to be even on `[-L,L]` and then extended to be periodic with period `2*L`.

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 that `a@(1)` is the constant coefficient! That is, `a@(n)` refers to the term `cos(x*(n-1)*pi/L)`, while `b@(n)` refers to the term `sin(x*n*pi/L)`. Either `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.

NumericalFourierSeriesCoefficients
`NumericalFourierSeriesCoefficients (f,L,N)`

Return a vector of vectors `[a,b]` where `a` are the cosine coefficients and `b` are the sine coefficients of the Fourier series of `f` with half-period `L` (that is defined on `[-L,L]` and extended periodically) with coefficients up to `N`th harmonic computed numerically. The coefficients are computed by numerical integration using `NumericalIntegral`.

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 on `[-L,L]` and extended periodically) with coefficients up to `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`.

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 on `[0,L]` take the even periodic extension and compute the Fourier series, which only has cosine terms. The series is computed up to the `N`th harmonic. The coefficients are computed by numerical integration using `NumericalIntegral`. Note that `a@(1)` is the constant coefficient! That is, `a@(n)` refers to the term `cos(x*(n-1)*pi/L)`.

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 on `[0,L]` take the even periodic extension and compute the Fourier series, which only has cosine terms. The series is computed up to the `N`th harmonic. The coefficients are computed by numerical integration using `NumericalIntegral`.

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 on `[0,L]` take the odd periodic extension and compute the Fourier series, which only has sine terms. The series is computed up to the `N`th harmonic. The coefficients are computed by numerical integration using `NumericalIntegral`.

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 on `[0,L]` take the odd periodic extension and compute the Fourier series, which only has sine terms. The series is computed up to the `N`th harmonic. The coefficients are computed by numerical integration using `NumericalIntegral`.

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 interval `[0,L]` extended to be odd on `[-L,L]` and then extended to be periodic with period `2*L`.

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 interval `[a,b]` and has period `b-a`.

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.