Analysis
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## Analysis

AdaptiveGaussKronrod

      AdaptiveGaussKronrod (f,a,b)

  Find the integral of `f` over the interval [a,b] using an adaptive algorithm using Gauss-Kronrod rule G7 K15. It will subdivide adaptively until the relative error is less than `AdaptiveGaussKronrodRelativeTolerance` or the absolute error is within `AdaptiveGaussKronrodAbsoluteTolerance`. The subinterval with the largest error is subdivided into two until we get a small enough error or until we hit `AdaptiveGaussKronrodMaxIterations` iterations.

  If an estimate within the given range is not achieved within the iteration limit, then `null` is returned and error is printed.

  See Wikipedia for more information.

  Version 1.0.28 onwards.

AdaptiveGaussKronrodWithTolerance

      AdaptiveGaussKronrod (f,a,b,abstol,reltol)

  Find the integral of `f` over the interval [a,b] using an adaptive algorithm using Gauss-Kronrod rule G7 K15. It will subdivide adaptively until the relative error is less than `abstol` or the absolute error is within `reltol`. The subinterval with the largest error is subdivided into two until we get a small enough error or until we hit `AdaptiveGaussKronrodMaxIterations` iterations.

  If an estimate within the given range is not achieved within the iteration limit, then `null` is returned and error is printed.

  This function is useful if different precision than the defaults is needed and one does not want to change global parameters. For example, if less precision is needed and speed is paramount. Otherwise just use `AdaptiveGaussKronrod` or `NumericalIntegral`

  See Wikipedia for more information.

  Version 1.0.28 onwards.

CompositeSimpsonsRule

      CompositeSimpsonsRule (f,a,b,n)

  Integration of `f` by Composite 1/3 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. If the given `n` is odd, then 1 is added to make it even.

  It is the 1/3 variant of the rule that is used, that is, if the x0,x1,x2,...,xn are the points, then the rule is ((b-a)/n) * (f(x0) + 4*f(x1) + 2*f(x2) + ... + 4*f(x(n-1)) + f(xn)).

  The `n` argument is optional. If it is not given the value of `NumericalIntegralSteps` is used, which is by default 1000.

  See Wikipedia or Planetmath for more information.

CompositeSimpsonsRuleData

      CompositeSimpsonsRule (f,len)

  Integration using Composite Simpson's rule of a function given by a vector of values `f` given at equal subintervals. The integration interval is taken to be of length `len`, that is, if the interval is [a,b], then `len` should be b-a. The vector `f` should have at least 3 values (representing 2 subintervals). Normally the 1/3 rule is used. If there is an odd number of subintervals, the 3/8 rule is used on the last 3 subintervals.

  See Wikipedia or Planetmath for more information.

  Version 1.0.28 onwards.

CompositeSimpsonsRuleTolerance

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

  Integration of f by Composite 1/3 Simpson's Rule on the interval [a,b] with the number of steps calculated by the fourth derivative bound and the desired tolerance.

  See Wikipedia or 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.

GaussKronrodRule

      GaussKronrodRule (f,a,b)

  A single shot Gauss-Kronrod rule G7 K15 over the interval [a,b]. It returns a vector where the first element is the approximate integral and the second is the approximate error obtained by subtracting the G7 and K15 approximates. This is already quite good, but often it is better to call it from within the `AdaptiveGaussKronrod` function, which is the default for `NumericalIntegral`.

  See Wikipedia for more information.

  Version 1.0.28 onwards.

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.

LeftHandRule

      LeftHandRule (f,a,b,n)

  Integration by left hand rule on the interval [a,b] with n subintervals.

  The `n` argument is optional. If it is not given the value of `NumericalIntegralSteps` is used, which is by default 1000.

  Version 1.0.28 onwards.

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 on the interval [a,b] with n subintervals.

  The `n` argument is optional. If it is not given the value of `NumericalIntegralSteps` is used, which is by default 1000.

  The `n` is optional for version 1.0.28 onwards.

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. By default NumericalIntegralFunction is the `AdaptiveGaussKronrod`, which implements an adaptive algorithm based on the Gauss-Kronrod G7 K15 rule. It is possible that `null` is returned if the algorithm cannot find an approximation within tolerance in a tunable maximum number of iterations.

  Gauss-Kronrod is the default algorithm since version 1.0.28 onwards.

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.

RightHandRule

      RightHandRule (f,a,b,n)

  Integration by right hand rule on the interval [a,b] with n subintervals.

  The `n` argument is optional. If it is not given the value of `NumericalIntegralSteps` is used, which is by default 1000.

  Version 1.0.28 onwards.

RightLimit

      RightLimit (f,x0)

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

TrapezoidRule

      TrapezoidRule (f,a,b,n)

  Integration by trapezoid rule on the interval [a,b] with `n` subintervals.

  The `n` argument is optional. If it is not given the value of `NumericalIntegralSteps` is used, which is by default 1000.

  Version 1.0.28 onwards.

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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