chore: import upstream snapshot with attribution
This commit is contained in:
wehub-resource-sync
@@ -0,0 +1,108 @@
|
||||
Bessel functions and related functions
|
||||
--------------------------------------
|
||||
|
||||
The functions in this section arise as solutions to various differential
|
||||
equations in physics, typically describing wavelike oscillatory behavior or a
|
||||
combination of oscillation and exponential decay or growth. Mathematically,
|
||||
they are special cases of the confluent hypergeometric functions `\,_0F_1`,
|
||||
`\,_1F_1` and `\,_1F_2` (see :doc:`hypergeometric`).
|
||||
|
||||
|
||||
Bessel functions
|
||||
................
|
||||
|
||||
.. autofunction:: mpmath.besselj
|
||||
.. autofunction:: mpmath.j0
|
||||
.. autofunction:: mpmath.j1
|
||||
.. autofunction:: mpmath.bessely
|
||||
.. autofunction:: mpmath.besseli
|
||||
.. autofunction:: mpmath.besselk
|
||||
|
||||
|
||||
Bessel function zeros
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.besseljzero
|
||||
.. autofunction:: mpmath.besselyzero
|
||||
|
||||
|
||||
Hankel functions
|
||||
................
|
||||
|
||||
.. autofunction:: mpmath.hankel1
|
||||
.. autofunction:: mpmath.hankel2
|
||||
|
||||
|
||||
Spherical Bessel functions
|
||||
..........................
|
||||
|
||||
.. autofunction:: mpmath.spherical_jn
|
||||
.. autofunction:: mpmath.spherical_yn
|
||||
.. autofunction:: mpmath.spherical_in
|
||||
.. autofunction:: mpmath.spherical_kn
|
||||
|
||||
|
||||
Kelvin functions
|
||||
................
|
||||
|
||||
.. autofunction:: mpmath.ber
|
||||
.. autofunction:: mpmath.bei
|
||||
.. autofunction:: mpmath.ker
|
||||
.. autofunction:: mpmath.kei
|
||||
|
||||
|
||||
Struve functions
|
||||
................
|
||||
|
||||
.. autofunction:: mpmath.struveh
|
||||
.. autofunction:: mpmath.struvel
|
||||
|
||||
|
||||
Anger-Weber functions
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.angerj
|
||||
.. autofunction:: mpmath.webere
|
||||
|
||||
|
||||
Lommel functions
|
||||
................
|
||||
|
||||
.. autofunction:: mpmath.lommels1
|
||||
.. autofunction:: mpmath.lommels2
|
||||
|
||||
|
||||
Airy and Scorer functions
|
||||
.........................
|
||||
|
||||
.. autofunction:: mpmath.airyai
|
||||
.. autofunction:: mpmath.airybi
|
||||
.. autofunction:: mpmath.airyaizero
|
||||
.. autofunction:: mpmath.airybizero
|
||||
.. autofunction:: mpmath.scorergi
|
||||
.. autofunction:: mpmath.scorerhi
|
||||
|
||||
|
||||
Coulomb wave functions
|
||||
......................
|
||||
|
||||
.. autofunction:: mpmath.coulombf
|
||||
.. autofunction:: mpmath.coulombg
|
||||
.. autofunction:: mpmath.coulombc
|
||||
|
||||
|
||||
Confluent U and Whittaker functions
|
||||
...................................
|
||||
|
||||
.. autofunction:: mpmath.hyperu(a, b, z)
|
||||
.. autofunction:: mpmath.whitm(k,m,z)
|
||||
.. autofunction:: mpmath.whitw(k,m,z)
|
||||
|
||||
|
||||
Parabolic cylinder functions
|
||||
............................
|
||||
|
||||
.. autofunction:: mpmath.pcfd
|
||||
.. autofunction:: mpmath.pcfu
|
||||
.. autofunction:: mpmath.pcfv
|
||||
.. autofunction:: mpmath.pcfw
|
||||
@@ -0,0 +1,45 @@
|
||||
Mathematical constants
|
||||
----------------------
|
||||
|
||||
Mpmath supports arbitrary-precision computation of various common (and less
|
||||
common) mathematical constants. These constants are implemented as lazy
|
||||
objects that can evaluate to any precision. Whenever the objects are used as
|
||||
function arguments or as operands in arithmetic operations, they automagically
|
||||
evaluate to the current working precision. A lazy number can be converted to a
|
||||
regular ``mpf`` using the unary ``+`` operator, or by calling it as a
|
||||
function::
|
||||
|
||||
>>> from mpmath import pi, mp
|
||||
>>> pi
|
||||
<pi: 3.14159~>
|
||||
>>> 2*pi
|
||||
mpf('6.2831853071795862')
|
||||
>>> +pi
|
||||
mpf('3.1415926535897931')
|
||||
>>> pi()
|
||||
mpf('3.1415926535897931')
|
||||
>>> mp.dps = 40
|
||||
>>> pi
|
||||
<pi: 3.14159~>
|
||||
>>> 2*pi
|
||||
mpf('6.28318530717958647692528676655900576839434')
|
||||
>>> +pi
|
||||
mpf('3.14159265358979323846264338327950288419717')
|
||||
>>> pi()
|
||||
mpf('3.14159265358979323846264338327950288419717')
|
||||
|
||||
The predefined objects ``j`` (imaginary unit), ``inf`` (positive infinity) and
|
||||
``nan`` (not-a-number) are shortcuts to ``mpc`` and ``mpf`` instances with
|
||||
these fixed values.
|
||||
|
||||
.. autofunction:: mpmath.mp.pi
|
||||
.. autoattribute:: mpmath.mp.degree
|
||||
.. autoattribute:: mpmath.mp.e
|
||||
.. autoattribute:: mpmath.mp.phi
|
||||
.. autofunction:: mpmath.mp.euler
|
||||
.. autoattribute:: mpmath.mp.catalan
|
||||
.. autoattribute:: mpmath.mp.apery
|
||||
.. autoattribute:: mpmath.mp.khinchin
|
||||
.. autoattribute:: mpmath.mp.glaisher
|
||||
.. autoattribute:: mpmath.mp.mertens
|
||||
.. autoattribute:: mpmath.mp.twinprime
|
||||
@@ -0,0 +1,65 @@
|
||||
Elliptic functions
|
||||
------------------
|
||||
|
||||
.. automodule:: mpmath.functions.elliptic
|
||||
:no-index:
|
||||
|
||||
|
||||
Elliptic arguments
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.qfrom
|
||||
.. autofunction:: mpmath.qbarfrom
|
||||
.. autofunction:: mpmath.mfrom
|
||||
.. autofunction:: mpmath.kfrom
|
||||
.. autofunction:: mpmath.taufrom
|
||||
|
||||
|
||||
Legendre elliptic integrals
|
||||
...........................
|
||||
|
||||
.. autofunction:: mpmath.ellipk
|
||||
.. autofunction:: mpmath.ellipf
|
||||
.. autofunction:: mpmath.ellipe
|
||||
.. autofunction:: mpmath.ellippi
|
||||
|
||||
|
||||
Carlson symmetric elliptic integrals
|
||||
....................................
|
||||
|
||||
.. autofunction:: mpmath.elliprf
|
||||
.. autofunction:: mpmath.elliprc
|
||||
.. autofunction:: mpmath.elliprj
|
||||
.. autofunction:: mpmath.elliprd
|
||||
.. autofunction:: mpmath.elliprg
|
||||
|
||||
|
||||
Jacobi theta functions
|
||||
......................
|
||||
|
||||
.. autofunction:: mpmath.jtheta
|
||||
|
||||
|
||||
Jacobi elliptic functions
|
||||
.........................
|
||||
|
||||
.. autofunction:: mpmath.ellipfun
|
||||
|
||||
|
||||
Weierstrass elliptic functions
|
||||
..............................
|
||||
|
||||
.. autofunction:: mpmath.weierinvariants
|
||||
.. autofunction:: mpmath.weierhalfperiods
|
||||
.. autofunction:: mpmath.weierp
|
||||
.. autofunction:: mpmath.weierpprime
|
||||
.. autofunction:: mpmath.weiersigma
|
||||
.. autofunction:: mpmath.weierzeta
|
||||
.. autofunction:: mpmath.weierpinv
|
||||
|
||||
|
||||
Modular functions
|
||||
.................
|
||||
|
||||
.. autofunction:: mpmath.eta
|
||||
.. autofunction:: mpmath.kleinj
|
||||
@@ -0,0 +1,70 @@
|
||||
Exponential integrals and error functions
|
||||
-----------------------------------------
|
||||
|
||||
Exponential integrals give closed-form solutions to a large class of commonly
|
||||
occurring transcendental integrals that cannot be evaluated using elementary
|
||||
functions. Integrals of this type include those with an integrand of the form
|
||||
`t^a e^{t}` or `e^{-x^2}`, the latter giving rise to the Gaussian (or normal)
|
||||
probability distribution.
|
||||
|
||||
The most general function in this section is the incomplete gamma function, to
|
||||
which all others can be reduced. The incomplete gamma function, in turn, can
|
||||
be expressed using hypergeometric functions (see :doc:`hypergeometric`).
|
||||
|
||||
Incomplete gamma functions
|
||||
..........................
|
||||
|
||||
.. autofunction:: mpmath.gammainc
|
||||
.. autofunction:: mpmath.lower_gamma
|
||||
.. autofunction:: mpmath.upper_gamma
|
||||
|
||||
|
||||
Exponential integrals
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.ei
|
||||
.. autofunction:: mpmath.e1
|
||||
.. autofunction:: mpmath.expint
|
||||
|
||||
|
||||
Logarithmic integral
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.li
|
||||
|
||||
|
||||
Trigonometric integrals
|
||||
.......................
|
||||
|
||||
.. autofunction:: mpmath.ci
|
||||
.. autofunction:: mpmath.si
|
||||
|
||||
|
||||
Hyperbolic integrals
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.chi
|
||||
.. autofunction:: mpmath.shi
|
||||
|
||||
|
||||
Error functions
|
||||
...............
|
||||
|
||||
.. autofunction:: mpmath.erf
|
||||
.. autofunction:: mpmath.erfc
|
||||
.. autofunction:: mpmath.erfi
|
||||
.. autofunction:: mpmath.erfinv
|
||||
|
||||
|
||||
The normal distribution
|
||||
.......................
|
||||
|
||||
.. autofunction:: mpmath.npdf
|
||||
.. autofunction:: mpmath.ncdf
|
||||
|
||||
|
||||
Fresnel integrals
|
||||
.................
|
||||
|
||||
.. autofunction:: mpmath.fresnels
|
||||
.. autofunction:: mpmath.fresnelc
|
||||
@@ -0,0 +1,79 @@
|
||||
Factorials and gamma functions
|
||||
------------------------------
|
||||
|
||||
Factorials and factorial-like sums and products are basic tools of
|
||||
combinatorics and number theory. Much like the exponential function is
|
||||
fundamental to differential equations and analysis in general, the factorial
|
||||
function (and its extension to complex numbers, the gamma function) is
|
||||
fundamental to difference equations and functional equations.
|
||||
|
||||
A large selection of factorial-like functions is implemented in mpmath. All
|
||||
functions support complex arguments, and arguments may be arbitrarily large.
|
||||
Results are numerical approximations, so to compute *exact* values a high
|
||||
enough precision must be set manually::
|
||||
|
||||
>>> from mpmath import mp, fac
|
||||
>>> mp.dps = 15
|
||||
>>> mp.pretty = True
|
||||
>>> fac(100)
|
||||
9.33262154439442e+157
|
||||
>>> print(int(_)) # most digits are wrong
|
||||
93326215443944150965646704795953882578400970373184098831012889540582227238570431295066113089288327277825849664006524270554535976289719382852181865895959724032
|
||||
>>> mp.dps = 160
|
||||
>>> fac(100)
|
||||
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000.0
|
||||
|
||||
The gamma and polygamma functions are closely related to :doc:`zeta`. See also
|
||||
:doc:`qfunctions` for q-analogs of factorial-like functions.
|
||||
|
||||
|
||||
Factorials
|
||||
..........
|
||||
|
||||
.. autofunction:: mpmath.factorial
|
||||
.. autofunction:: mpmath.fac2
|
||||
|
||||
|
||||
Binomial coefficients
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.binomial
|
||||
|
||||
|
||||
Gamma function
|
||||
..............
|
||||
|
||||
.. autofunction:: mpmath.gamma
|
||||
.. autofunction:: mpmath.rgamma
|
||||
.. autofunction:: mpmath.gammaprod
|
||||
.. autofunction:: mpmath.loggamma
|
||||
|
||||
|
||||
Rising and falling factorials
|
||||
.............................
|
||||
|
||||
.. autofunction:: mpmath.rf
|
||||
.. autofunction:: mpmath.ff
|
||||
|
||||
|
||||
Beta function
|
||||
.............
|
||||
|
||||
.. autofunction:: mpmath.beta
|
||||
.. autofunction:: mpmath.betainc
|
||||
|
||||
|
||||
Super- and hyperfactorials
|
||||
..........................
|
||||
|
||||
.. autofunction:: mpmath.superfac
|
||||
.. autofunction:: mpmath.hyperfac
|
||||
.. autofunction:: mpmath.barnesg
|
||||
|
||||
|
||||
Polygamma functions and harmonic numbers
|
||||
........................................
|
||||
|
||||
.. autofunction:: mpmath.psi
|
||||
.. autofunction:: mpmath.digamma
|
||||
.. autofunction:: mpmath.harmonic
|
||||
@@ -0,0 +1,23 @@
|
||||
Hyperbolic functions
|
||||
--------------------
|
||||
|
||||
Hyperbolic functions
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.cosh
|
||||
.. autofunction:: mpmath.sinh
|
||||
.. autofunction:: mpmath.tanh
|
||||
.. autofunction:: mpmath.sech
|
||||
.. autofunction:: mpmath.csch
|
||||
.. autofunction:: mpmath.coth
|
||||
|
||||
|
||||
Inverse hyperbolic functions
|
||||
............................
|
||||
|
||||
.. autofunction:: mpmath.acosh
|
||||
.. autofunction:: mpmath.asinh
|
||||
.. autofunction:: mpmath.atanh
|
||||
.. autofunction:: mpmath.asech
|
||||
.. autofunction:: mpmath.acsch
|
||||
.. autofunction:: mpmath.acoth
|
||||
@@ -0,0 +1,74 @@
|
||||
Hypergeometric functions
|
||||
------------------------
|
||||
|
||||
The functions listed in :doc:`expintegrals`, :doc:`bessel` and
|
||||
:doc:`orthogonal`, and many other functions as well, are merely particular
|
||||
instances of the generalized hypergeometric function `\,_pF_q`. The functions
|
||||
listed in the following section enable efficient direct evaluation of the
|
||||
underlying hypergeometric series, as well as linear combinations, limits with
|
||||
respect to parameters, and analytic continuations thereof. Extensions to
|
||||
twodimensional series are also provided. See also the basic or q-analog of the
|
||||
hypergeometric series in :doc:`qfunctions`.
|
||||
|
||||
For convenience, most of the hypergeometric series of low order are provided as
|
||||
standalone functions. They can equivalently be evaluated using
|
||||
:func:`~mpmath.hyper`. As will be demonstrated in the respective docstrings,
|
||||
all the ``hyp#f#`` functions implement analytic continuations and/or asymptotic
|
||||
expansions with respect to the argument `z`, thereby permitting evaluation for
|
||||
`z` anywhere in the complex plane. Functions of higher degree can be computed
|
||||
via :func:`~mpmath.hyper`, but generally only in rapidly convergent instances.
|
||||
|
||||
Most hypergeometric and hypergeometric-derived functions accept optional
|
||||
keyword arguments to specify options for :func:`~mpmath.hypercomb` or
|
||||
:func:`~mpmath.hyper`. Some useful options are *maxprec*, *maxterms*,
|
||||
*zeroprec*, *accurate_small*, *hmag*, *force_series*, *asymp_tol* and
|
||||
*eliminate*. These options give control over what to do in case of slow
|
||||
convergence, extreme loss of accuracy or evaluation at zeros (these two cases
|
||||
cannot generally be distinguished from each other automatically), and singular
|
||||
parameter combinations.
|
||||
|
||||
Common hypergeometric series
|
||||
............................
|
||||
|
||||
.. autofunction:: mpmath.hyp0f1
|
||||
.. autofunction:: mpmath.hyp1f1
|
||||
.. autofunction:: mpmath.hyp1f2
|
||||
.. autofunction:: mpmath.hyp2f0
|
||||
.. autofunction:: mpmath.hyp2f1
|
||||
.. autofunction:: mpmath.hyp2f2
|
||||
.. autofunction:: mpmath.hyp2f3
|
||||
.. autofunction:: mpmath.hyp3f2
|
||||
|
||||
|
||||
Generalized hypergeometric functions
|
||||
....................................
|
||||
|
||||
.. autofunction:: mpmath.hyper
|
||||
.. autofunction:: mpmath.hypercomb
|
||||
|
||||
|
||||
Meijer G-function
|
||||
.................
|
||||
|
||||
.. autofunction:: mpmath.meijerg
|
||||
|
||||
Fox H-function
|
||||
.................
|
||||
|
||||
.. autofunction:: mpmath.foxh
|
||||
|
||||
|
||||
Bilateral hypergeometric series
|
||||
...............................
|
||||
|
||||
.. autofunction:: mpmath.bihyper
|
||||
|
||||
|
||||
Hypergeometric functions of two variables
|
||||
.........................................
|
||||
|
||||
.. autofunction:: mpmath.hyper2d
|
||||
.. autofunction:: mpmath.appellf1
|
||||
.. autofunction:: mpmath.appellf2
|
||||
.. autofunction:: mpmath.appellf3
|
||||
.. autofunction:: mpmath.appellf4
|
||||
@@ -0,0 +1,22 @@
|
||||
Mathematical functions
|
||||
======================
|
||||
|
||||
Mpmath implements the standard functions from Python's ``math`` and ``cmath`` modules, for both real and complex numbers and with arbitrary precision. Many other functions are also available in mpmath, including commonly-used variants of standard functions (such as the alternative trigonometric functions sec, csc, cot), but also a large number of "special functions" such as the gamma function, the Riemann zeta function, error functions, Bessel functions, etc.
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
constants
|
||||
powers
|
||||
trigonometric
|
||||
hyperbolic
|
||||
signals
|
||||
gamma
|
||||
expintegrals
|
||||
bessel
|
||||
orthogonal
|
||||
hypergeometric
|
||||
elliptic
|
||||
zeta
|
||||
numtheory
|
||||
qfunctions
|
||||
@@ -0,0 +1,58 @@
|
||||
Number-theoretical, combinatorial and integer functions
|
||||
-------------------------------------------------------
|
||||
|
||||
For factorial-type functions, including binomial coefficients, double
|
||||
factorials, etc, see the separate section :doc:`gamma`.
|
||||
|
||||
Fibonacci numbers
|
||||
.................
|
||||
|
||||
.. autofunction:: mpmath.fibonacci
|
||||
|
||||
|
||||
Bernoulli numbers and polynomials
|
||||
.................................
|
||||
|
||||
.. autofunction:: mpmath.bernoulli
|
||||
.. autofunction:: mpmath.bernfrac
|
||||
.. autofunction:: mpmath.bernpoly
|
||||
|
||||
|
||||
Euler numbers and polynomials
|
||||
.............................
|
||||
|
||||
.. autofunction:: mpmath.eulernum
|
||||
.. autofunction:: mpmath.eulerpoly
|
||||
|
||||
|
||||
Bell numbers and polynomials
|
||||
............................
|
||||
|
||||
.. autofunction:: mpmath.bell
|
||||
|
||||
|
||||
Stirling numbers
|
||||
................
|
||||
|
||||
.. autofunction:: mpmath.stirling1
|
||||
.. autofunction:: mpmath.stirling2
|
||||
|
||||
|
||||
Prime counting functions
|
||||
........................
|
||||
|
||||
.. autofunction:: mpmath.primepi
|
||||
.. autofunction:: mpmath.primepi2
|
||||
.. autofunction:: mpmath.riemannr
|
||||
|
||||
|
||||
Cyclotomic polynomials
|
||||
......................
|
||||
|
||||
.. autofunction:: mpmath.cyclotomic
|
||||
|
||||
|
||||
Arithmetic functions
|
||||
......................
|
||||
|
||||
.. autofunction:: mpmath.mangoldt
|
||||
@@ -0,0 +1,77 @@
|
||||
Orthogonal polynomials
|
||||
----------------------
|
||||
|
||||
An orthogonal polynomial sequence is a sequence of polynomials `P_0(x), P_1(x),
|
||||
\ldots` of degree `0, 1, \ldots`, which are mutually orthogonal in the sense
|
||||
that
|
||||
|
||||
.. math ::
|
||||
|
||||
\int_S P_n(x) P_m(x) w(x) dx =
|
||||
\begin{cases}
|
||||
c_n \ne 0 & \text{if $m = n$} \\
|
||||
0 & \text{if $m \ne n$}
|
||||
\end{cases}
|
||||
|
||||
where `S` is some domain (e.g. an interval `[a,b] \in \mathbb{R}`) and `w(x)`
|
||||
is a fixed *weight function*. A sequence of orthogonal polynomials is
|
||||
determined completely by `w`, `S`, and a normalization convention (e.g. `c_n =
|
||||
1`). Applications of orthogonal polynomials include function approximation and
|
||||
solution of differential equations.
|
||||
|
||||
Orthogonal polynomials are sometimes defined using the differential equations
|
||||
they satisfy (as functions of `x`) or the recurrence relations they satisfy
|
||||
with respect to the order `n`. Other ways of defining orthogonal polynomials
|
||||
include differentiation formulas and generating functions. The standard
|
||||
orthogonal polynomials can also be represented as hypergeometric series (see
|
||||
:doc:`hypergeometric`), more specifically using the Gauss hypergeometric
|
||||
function `\,_2F_1` in most cases. The following functions are generally
|
||||
implemented using hypergeometric functions since this is computationally
|
||||
efficient and easily generalizes.
|
||||
|
||||
For more information, see the `Wikipedia article on orthogonal polynomials
|
||||
<http://en.wikipedia.org/wiki/Orthogonal_polynomials>`_.
|
||||
|
||||
Legendre functions
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.legendre
|
||||
.. autofunction:: mpmath.legenp
|
||||
.. autofunction:: mpmath.legenq
|
||||
|
||||
|
||||
Chebyshev polynomials
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.chebyt
|
||||
.. autofunction:: mpmath.chebyu
|
||||
|
||||
|
||||
Jacobi polynomials
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.jacobi
|
||||
|
||||
|
||||
Gegenbauer polynomials
|
||||
......................
|
||||
|
||||
.. autofunction:: mpmath.gegenbauer
|
||||
|
||||
|
||||
Hermite polynomials
|
||||
...................
|
||||
|
||||
.. autofunction:: mpmath.hermite
|
||||
|
||||
|
||||
Laguerre polynomials
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.laguerre
|
||||
|
||||
|
||||
Spherical harmonics
|
||||
...................
|
||||
|
||||
.. autofunction:: mpmath.spherharm
|
||||
@@ -0,0 +1,45 @@
|
||||
Powers and logarithms
|
||||
---------------------
|
||||
|
||||
Nth roots
|
||||
.........
|
||||
|
||||
.. autofunction:: mpmath.sqrt
|
||||
.. autofunction:: mpmath.hypot
|
||||
.. autofunction:: mpmath.cbrt
|
||||
.. autofunction:: mpmath.root
|
||||
.. autofunction:: mpmath.unitroots
|
||||
|
||||
|
||||
Exponentiation
|
||||
..............
|
||||
|
||||
.. autofunction:: mpmath.exp
|
||||
.. autofunction:: mpmath.exp2
|
||||
.. autofunction:: mpmath.power
|
||||
.. autofunction:: mpmath.expj
|
||||
.. autofunction:: mpmath.expjpi
|
||||
.. autofunction:: mpmath.expm1(x)
|
||||
.. autofunction:: mpmath.powm1(x, y)
|
||||
|
||||
|
||||
Logarithms
|
||||
..........
|
||||
|
||||
.. autofunction:: mpmath.log
|
||||
.. autofunction:: mpmath.ln
|
||||
.. autofunction:: mpmath.log2
|
||||
.. autofunction:: mpmath.log10
|
||||
.. autofunction:: mpmath.log1p(x)
|
||||
|
||||
|
||||
Lambert W function
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.lambertw
|
||||
|
||||
|
||||
Arithmetic-geometric mean
|
||||
.........................
|
||||
|
||||
.. autofunction:: mpmath.agm
|
||||
@@ -0,0 +1,20 @@
|
||||
q-functions
|
||||
-----------
|
||||
|
||||
q-Pochhammer symbol
|
||||
...................
|
||||
|
||||
.. autofunction:: mpmath.qp
|
||||
|
||||
|
||||
q-gamma and factorial
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.qgamma
|
||||
.. autofunction:: mpmath.qfac
|
||||
|
||||
|
||||
Hypergeometric q-series
|
||||
.......................
|
||||
|
||||
.. autofunction:: mpmath.qhyper
|
||||
@@ -0,0 +1,34 @@
|
||||
Signal functions
|
||||
----------------
|
||||
|
||||
The functions in this section describe non-sinusoidal waveforms, which are
|
||||
often used in signal processing and electronics.
|
||||
|
||||
Square wave signal
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.squarew
|
||||
|
||||
|
||||
Triangle wave signal
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.trianglew
|
||||
|
||||
|
||||
Sawtooth wave signal
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.sawtoothw
|
||||
|
||||
|
||||
Unit triangle signal
|
||||
....................
|
||||
|
||||
.. autofunction:: mpmath.unit_triangle
|
||||
|
||||
|
||||
Sigmoid wave signal
|
||||
.....................
|
||||
|
||||
.. autofunction:: mpmath.sigmoid
|
||||
@@ -0,0 +1,62 @@
|
||||
Trigonometric functions
|
||||
-----------------------
|
||||
|
||||
Except where otherwise noted, the trigonometric functions take a radian angle
|
||||
as input and the inverse trigonometric functions return radian angles.
|
||||
|
||||
The ordinary trigonometric functions are single-valued functions defined
|
||||
everywhere in the complex plane (except at the poles of tan, sec, csc, and
|
||||
cot). They are defined generally via the exponential function, e.g.
|
||||
|
||||
.. math ::
|
||||
|
||||
\cos(x) = \frac{e^{ix} + e^{-ix}}{2}.
|
||||
|
||||
The inverse trigonometric functions are multivalued, thus requiring branch
|
||||
cuts, and are generally real-valued only on a part of the real line.
|
||||
Definitions and branch cuts are given in the documentation of each function.
|
||||
The branch cut conventions used by mpmath are essentially the same as those
|
||||
found in most standard mathematical software, such as Mathematica and Python's
|
||||
own ``cmath`` libary.
|
||||
|
||||
Degree-radian conversion
|
||||
........................
|
||||
|
||||
.. autofunction:: mpmath.degrees
|
||||
.. autofunction:: mpmath.radians
|
||||
|
||||
Trigonometric functions
|
||||
.......................
|
||||
|
||||
.. autofunction:: mpmath.cos
|
||||
.. autofunction:: mpmath.sin
|
||||
.. autofunction:: mpmath.tan
|
||||
.. autofunction:: mpmath.sec
|
||||
.. autofunction:: mpmath.csc
|
||||
.. autofunction:: mpmath.cot
|
||||
|
||||
|
||||
Trigonometric functions with modified argument
|
||||
..............................................
|
||||
|
||||
.. autofunction:: mpmath.cospi
|
||||
.. autofunction:: mpmath.sinpi
|
||||
|
||||
|
||||
Inverse trigonometric functions
|
||||
...............................
|
||||
|
||||
.. autofunction:: mpmath.acos
|
||||
.. autofunction:: mpmath.asin
|
||||
.. autofunction:: mpmath.atan
|
||||
.. autofunction:: mpmath.atan2
|
||||
.. autofunction:: mpmath.asec
|
||||
.. autofunction:: mpmath.acsc
|
||||
.. autofunction:: mpmath.acot
|
||||
|
||||
|
||||
Sinc function
|
||||
.............
|
||||
|
||||
.. autofunction:: mpmath.sinc
|
||||
.. autofunction:: mpmath.sincpi
|
||||
@@ -0,0 +1,60 @@
|
||||
Zeta functions, L-series and polylogarithms
|
||||
-------------------------------------------
|
||||
|
||||
This section includes the Riemann zeta functions and associated functions
|
||||
pertaining to analytic number theory.
|
||||
|
||||
|
||||
Riemann and Hurwitz zeta functions
|
||||
..................................
|
||||
|
||||
.. autofunction:: mpmath.zeta
|
||||
|
||||
|
||||
Dirichlet L-series
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.altzeta
|
||||
.. autofunction:: mpmath.dirichlet
|
||||
|
||||
|
||||
Stieltjes constants
|
||||
...................
|
||||
|
||||
.. autofunction:: mpmath.stieltjes
|
||||
|
||||
|
||||
Zeta function zeros
|
||||
...................
|
||||
|
||||
These functions are used for the study of the Riemann zeta function in the
|
||||
critical strip.
|
||||
|
||||
.. autofunction:: mpmath.zetazero
|
||||
.. autofunction:: mpmath.nzeros
|
||||
.. autofunction:: mpmath.siegelz
|
||||
.. autofunction:: mpmath.siegeltheta
|
||||
.. autofunction:: mpmath.grampoint
|
||||
.. autofunction:: mpmath.backlunds
|
||||
|
||||
|
||||
Lerch transcendent
|
||||
..................
|
||||
|
||||
.. autofunction:: mpmath.lerchphi
|
||||
|
||||
|
||||
Polylogarithms and Clausen functions
|
||||
....................................
|
||||
|
||||
.. autofunction:: mpmath.polylog
|
||||
.. autofunction:: mpmath.clsin
|
||||
.. autofunction:: mpmath.clcos
|
||||
.. autofunction:: mpmath.polyexp
|
||||
|
||||
|
||||
Zeta function variants
|
||||
......................
|
||||
|
||||
.. autofunction:: mpmath.primezeta
|
||||
.. autofunction:: mpmath.secondzeta
|
||||
Reference in New Issue
Block a user