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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.
EvenPeriodicExtension (f,L) |
Return a function which is 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.
FourierSeriesFunction (a,b,L) |
Return a function which 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.
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.
NumericalFourierSeriesFunction (f,L,N) |
Return a function which 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.
NumericalFourierCosineSeriesCoefficients (f,L,N) |
Return a vector of coefficients of the
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 sine 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).
NumericalFourierCosineSeriesFunction (f,L,N) |
Return a function which 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.
NumericalFourierSineSeriesCoefficients (f,L,N) |
Return a vector of coefficients of the
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.
NumericalFourierSineSeriesFunction (f,L,N) |
Return a function which 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.
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 which is 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.
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 which is the periodic extension of f defined on the interval [a,b] and has period b-a.
See also OddPeriodicExtension and EvenPeriodicExtension.
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.
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