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 (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 (f,x0)
Attempt to calculate derivative by trying first symbolically and then numerically.
See Wikipedia for more information.
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 (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 (func,start,inc)
Try to calculate an infinite product for a single parameter function.
InfiniteProduct2 (func,arg,start,inc)
Try to calculate an infinite product for a double parameter function with func(arg,n).
InfiniteSum (func,start,inc)
Try to calculate an infinite sum for a single parameter function.
InfiniteSum2 (func,arg,start,inc)
Try to calculate an infinite sum for a double parameter function with func(arg,n).
IsContinuous (f,x0)
Try and see if a real-valued function is continuous at x0 by calculating the limit there.
IsDifferentiable (f,x0)
Test for differentiability by approximating the left and right limits and comparing.
LeftLimit (f,x0)
Calculate the left limit of a real-valued function at x0.
Limit (f,x0)
Calculate the limit of a real-valued function at x0. Tries to calculate both left and right limits.
MidpointRule (f,a,b,n)
Integration by midpoint rule.
NumericalDerivative (f,x0)
Aliases: NDerivative
Attempt to calculate numerical derivative.
See Wikipedia for more information.
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 Nth 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 (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 Nth 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 (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
Nth 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 (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
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
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
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
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
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalIntegral (f,a,b)
Integration by rule set in NumericalIntegralFunction of f from a to b using NumericalIntegralSteps steps.
NumericalLeftDerivative (f,x0)
Attempt to calculate numerical left derivative.
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 (f,x0)
Attempt to calculate numerical right derivative.
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 (f,x0,h)
Compute one-sided derivative using five point formula.
OneSidedThreePointFormula (f,x0,h)
Compute one-sided derivative using three-point formula.
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 (f,x0)
Calculate the right limit of a real-valued function at x0.
TwoSidedFivePointFormula (f,x0,h)
Compute two-sided derivative using five-point formula.
TwoSidedThreePointFormula (f,x0,h)
Compute two-sided derivative using three-point formula.