Analysis
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## Analysis

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

  See Wikipedia or Mathworld for more information.

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 `[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`.

  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 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`.

  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 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)`.

  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 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`.

  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 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`.

  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 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`.

  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 interval `[0,L]` extended to be odd on `[-L,L]` and then extended to be periodic with period `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 interval `[a,b]` and has period `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.

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