809 lines
22 KiB
Python
809 lines
22 KiB
Python
|
"""Fourier Series"""
|
||
|
|
|
||
|
|
from sympy.core.numbers import (oo, pi)
|
||
|
|
from sympy.core.symbol import Wild
|
||
|
|
from sympy.core.expr import Expr
|
||
|
|
from sympy.core.add import Add
|
||
|
|
from sympy.core.containers import Tuple
|
||
|
|
from sympy.core.singleton import S
|
||
|
|
from sympy.core.symbol import Dummy, Symbol
|
||
|
|
from sympy.core.sympify import sympify
|
||
|
|
from sympy.functions.elementary.trigonometric import sin, cos, sinc
|
||
|
|
from sympy.series.series_class import SeriesBase
|
||
|
|
from sympy.series.sequences import SeqFormula
|
||
|
|
from sympy.sets.sets import Interval
|
||
|
|
from sympy.utilities.iterables import is_sequence
|
||
|
|
|
||
|
|
|
||
|
|
def fourier_cos_seq(func, limits, n):
|
||
|
|
"""Returns the cos sequence in a Fourier series"""
|
||
|
|
from sympy.integrals import integrate
|
||
|
|
x, L = limits[0], limits[2] - limits[1]
|
||
|
|
cos_term = cos(2*n*pi*x / L)
|
||
|
|
formula = 2 * cos_term * integrate(func * cos_term, limits) / L
|
||
|
|
a0 = formula.subs(n, S.Zero) / 2
|
||
|
|
return a0, SeqFormula(2 * cos_term * integrate(func * cos_term, limits)
|
||
|
|
/ L, (n, 1, oo))
|
||
|
|
|
||
|
|
|
||
|
|
def fourier_sin_seq(func, limits, n):
|
||
|
|
"""Returns the sin sequence in a Fourier series"""
|
||
|
|
from sympy.integrals import integrate
|
||
|
|
x, L = limits[0], limits[2] - limits[1]
|
||
|
|
sin_term = sin(2*n*pi*x / L)
|
||
|
|
return SeqFormula(2 * sin_term * integrate(func * sin_term, limits)
|
||
|
|
/ L, (n, 1, oo))
|
||
|
|
|
||
|
|
|
||
|
|
def _process_limits(func, limits):
|
||
|
|
"""
|
||
|
|
Limits should be of the form (x, start, stop).
|
||
|
|
x should be a symbol. Both start and stop should be bounded.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
* If x is not given, x is determined from func.
|
||
|
|
* If limits is None. Limit of the form (x, -pi, pi) is returned.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.series.fourier import _process_limits as pari
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> pari(x**2, (x, -2, 2))
|
||
|
|
(x, -2, 2)
|
||
|
|
>>> pari(x**2, (-2, 2))
|
||
|
|
(x, -2, 2)
|
||
|
|
>>> pari(x**2, None)
|
||
|
|
(x, -pi, pi)
|
||
|
|
"""
|
||
|
|
def _find_x(func):
|
||
|
|
free = func.free_symbols
|
||
|
|
if len(free) == 1:
|
||
|
|
return free.pop()
|
||
|
|
elif not free:
|
||
|
|
return Dummy('k')
|
||
|
|
else:
|
||
|
|
raise ValueError(
|
||
|
|
" specify dummy variables for %s. If the function contains"
|
||
|
|
" more than one free symbol, a dummy variable should be"
|
||
|
|
" supplied explicitly e.g. FourierSeries(m*n**2, (n, -pi, pi))"
|
||
|
|
% func)
|
||
|
|
|
||
|
|
x, start, stop = None, None, None
|
||
|
|
if limits is None:
|
||
|
|
x, start, stop = _find_x(func), -pi, pi
|
||
|
|
if is_sequence(limits, Tuple):
|
||
|
|
if len(limits) == 3:
|
||
|
|
x, start, stop = limits
|
||
|
|
elif len(limits) == 2:
|
||
|
|
x = _find_x(func)
|
||
|
|
start, stop = limits
|
||
|
|
|
||
|
|
if not isinstance(x, Symbol) or start is None or stop is None:
|
||
|
|
raise ValueError('Invalid limits given: %s' % str(limits))
|
||
|
|
|
||
|
|
unbounded = [S.NegativeInfinity, S.Infinity]
|
||
|
|
if start in unbounded or stop in unbounded:
|
||
|
|
raise ValueError("Both the start and end value should be bounded")
|
||
|
|
|
||
|
|
return sympify((x, start, stop))
|
||
|
|
|
||
|
|
|
||
|
|
def finite_check(f, x, L):
|
||
|
|
|
||
|
|
def check_fx(exprs, x):
|
||
|
|
return x not in exprs.free_symbols
|
||
|
|
|
||
|
|
def check_sincos(_expr, x, L):
|
||
|
|
if isinstance(_expr, (sin, cos)):
|
||
|
|
sincos_args = _expr.args[0]
|
||
|
|
|
||
|
|
if sincos_args.match(a*(pi/L)*x + b) is not None:
|
||
|
|
return True
|
||
|
|
else:
|
||
|
|
return False
|
||
|
|
|
||
|
|
from sympy.simplify.fu import TR2, TR1, sincos_to_sum
|
||
|
|
_expr = sincos_to_sum(TR2(TR1(f)))
|
||
|
|
add_coeff = _expr.as_coeff_add()
|
||
|
|
|
||
|
|
a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k != S.Zero, ])
|
||
|
|
b = Wild('b', properties=[lambda k: x not in k.free_symbols, ])
|
||
|
|
|
||
|
|
for s in add_coeff[1]:
|
||
|
|
mul_coeffs = s.as_coeff_mul()[1]
|
||
|
|
for t in mul_coeffs:
|
||
|
|
if not (check_fx(t, x) or check_sincos(t, x, L)):
|
||
|
|
return False, f
|
||
|
|
|
||
|
|
return True, _expr
|
||
|
|
|
||
|
|
|
||
|
|
class FourierSeries(SeriesBase):
|
||
|
|
r"""Represents Fourier sine/cosine series.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
This class only represents a fourier series.
|
||
|
|
No computation is performed.
|
||
|
|
|
||
|
|
For how to compute Fourier series, see the :func:`fourier_series`
|
||
|
|
docstring.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.series.fourier.fourier_series
|
||
|
|
"""
|
||
|
|
def __new__(cls, *args):
|
||
|
|
args = map(sympify, args)
|
||
|
|
return Expr.__new__(cls, *args)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def function(self):
|
||
|
|
return self.args[0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def x(self):
|
||
|
|
return self.args[1][0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def period(self):
|
||
|
|
return (self.args[1][1], self.args[1][2])
|
||
|
|
|
||
|
|
@property
|
||
|
|
def a0(self):
|
||
|
|
return self.args[2][0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def an(self):
|
||
|
|
return self.args[2][1]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def bn(self):
|
||
|
|
return self.args[2][2]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def interval(self):
|
||
|
|
return Interval(0, oo)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def start(self):
|
||
|
|
return self.interval.inf
|
||
|
|
|
||
|
|
@property
|
||
|
|
def stop(self):
|
||
|
|
return self.interval.sup
|
||
|
|
|
||
|
|
@property
|
||
|
|
def length(self):
|
||
|
|
return oo
|
||
|
|
|
||
|
|
@property
|
||
|
|
def L(self):
|
||
|
|
return abs(self.period[1] - self.period[0]) / 2
|
||
|
|
|
||
|
|
def _eval_subs(self, old, new):
|
||
|
|
x = self.x
|
||
|
|
if old.has(x):
|
||
|
|
return self
|
||
|
|
|
||
|
|
def truncate(self, n=3):
|
||
|
|
"""
|
||
|
|
Return the first n nonzero terms of the series.
|
||
|
|
|
||
|
|
If ``n`` is None return an iterator.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
n : int or None
|
||
|
|
Amount of non-zero terms in approximation or None.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
Expr or iterator :
|
||
|
|
Approximation of function expanded into Fourier series.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> s = fourier_series(x, (x, -pi, pi))
|
||
|
|
>>> s.truncate(4)
|
||
|
|
2*sin(x) - sin(2*x) + 2*sin(3*x)/3 - sin(4*x)/2
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.series.fourier.FourierSeries.sigma_approximation
|
||
|
|
"""
|
||
|
|
if n is None:
|
||
|
|
return iter(self)
|
||
|
|
|
||
|
|
terms = []
|
||
|
|
for t in self:
|
||
|
|
if len(terms) == n:
|
||
|
|
break
|
||
|
|
if t is not S.Zero:
|
||
|
|
terms.append(t)
|
||
|
|
|
||
|
|
return Add(*terms)
|
||
|
|
|
||
|
|
def sigma_approximation(self, n=3):
|
||
|
|
r"""
|
||
|
|
Return :math:`\sigma`-approximation of Fourier series with respect
|
||
|
|
to order n.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Sigma approximation adjusts a Fourier summation to eliminate the Gibbs
|
||
|
|
phenomenon which would otherwise occur at discontinuities.
|
||
|
|
A sigma-approximated summation for a Fourier series of a T-periodical
|
||
|
|
function can be written as
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
s(\theta) = \frac{1}{2} a_0 + \sum _{k=1}^{m-1}
|
||
|
|
\operatorname{sinc} \Bigl( \frac{k}{m} \Bigr) \cdot
|
||
|
|
\left[ a_k \cos \Bigl( \frac{2\pi k}{T} \theta \Bigr)
|
||
|
|
+ b_k \sin \Bigl( \frac{2\pi k}{T} \theta \Bigr) \right],
|
||
|
|
|
||
|
|
where :math:`a_0, a_k, b_k, k=1,\ldots,{m-1}` are standard Fourier
|
||
|
|
series coefficients and
|
||
|
|
:math:`\operatorname{sinc} \Bigl( \frac{k}{m} \Bigr)` is a Lanczos
|
||
|
|
:math:`\sigma` factor (expressed in terms of normalized
|
||
|
|
:math:`\operatorname{sinc}` function).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
n : int
|
||
|
|
Highest order of the terms taken into account in approximation.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
Expr :
|
||
|
|
Sigma approximation of function expanded into Fourier series.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> s = fourier_series(x, (x, -pi, pi))
|
||
|
|
>>> s.sigma_approximation(4)
|
||
|
|
2*sin(x)*sinc(pi/4) - 2*sin(2*x)/pi + 2*sin(3*x)*sinc(3*pi/4)/3
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.series.fourier.FourierSeries.truncate
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
The behaviour of
|
||
|
|
:meth:`~sympy.series.fourier.FourierSeries.sigma_approximation`
|
||
|
|
is different from :meth:`~sympy.series.fourier.FourierSeries.truncate`
|
||
|
|
- it takes all nonzero terms of degree smaller than n, rather than
|
||
|
|
first n nonzero ones.
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Gibbs_phenomenon
|
||
|
|
.. [2] https://en.wikipedia.org/wiki/Sigma_approximation
|
||
|
|
"""
|
||
|
|
terms = [sinc(pi * i / n) * t for i, t in enumerate(self[:n])
|
||
|
|
if t is not S.Zero]
|
||
|
|
return Add(*terms)
|
||
|
|
|
||
|
|
def shift(self, s):
|
||
|
|
"""
|
||
|
|
Shift the function by a term independent of x.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
f(x) -> f(x) + s
|
||
|
|
|
||
|
|
This is fast, if Fourier series of f(x) is already
|
||
|
|
computed.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> s = fourier_series(x**2, (x, -pi, pi))
|
||
|
|
>>> s.shift(1).truncate()
|
||
|
|
-4*cos(x) + cos(2*x) + 1 + pi**2/3
|
||
|
|
"""
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
a0 = self.a0 + s
|
||
|
|
sfunc = self.function + s
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], (a0, self.an, self.bn))
|
||
|
|
|
||
|
|
def shiftx(self, s):
|
||
|
|
"""
|
||
|
|
Shift x by a term independent of x.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
f(x) -> f(x + s)
|
||
|
|
|
||
|
|
This is fast, if Fourier series of f(x) is already
|
||
|
|
computed.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> s = fourier_series(x**2, (x, -pi, pi))
|
||
|
|
>>> s.shiftx(1).truncate()
|
||
|
|
-4*cos(x + 1) + cos(2*x + 2) + pi**2/3
|
||
|
|
"""
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
an = self.an.subs(x, x + s)
|
||
|
|
bn = self.bn.subs(x, x + s)
|
||
|
|
sfunc = self.function.subs(x, x + s)
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], (self.a0, an, bn))
|
||
|
|
|
||
|
|
def scale(self, s):
|
||
|
|
"""
|
||
|
|
Scale the function by a term independent of x.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
f(x) -> s * f(x)
|
||
|
|
|
||
|
|
This is fast, if Fourier series of f(x) is already
|
||
|
|
computed.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> s = fourier_series(x**2, (x, -pi, pi))
|
||
|
|
>>> s.scale(2).truncate()
|
||
|
|
-8*cos(x) + 2*cos(2*x) + 2*pi**2/3
|
||
|
|
"""
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
an = self.an.coeff_mul(s)
|
||
|
|
bn = self.bn.coeff_mul(s)
|
||
|
|
a0 = self.a0 * s
|
||
|
|
sfunc = self.args[0] * s
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], (a0, an, bn))
|
||
|
|
|
||
|
|
def scalex(self, s):
|
||
|
|
"""
|
||
|
|
Scale x by a term independent of x.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
f(x) -> f(s*x)
|
||
|
|
|
||
|
|
This is fast, if Fourier series of f(x) is already
|
||
|
|
computed.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> s = fourier_series(x**2, (x, -pi, pi))
|
||
|
|
>>> s.scalex(2).truncate()
|
||
|
|
-4*cos(2*x) + cos(4*x) + pi**2/3
|
||
|
|
"""
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
an = self.an.subs(x, x * s)
|
||
|
|
bn = self.bn.subs(x, x * s)
|
||
|
|
sfunc = self.function.subs(x, x * s)
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], (self.a0, an, bn))
|
||
|
|
|
||
|
|
def _eval_as_leading_term(self, x, logx=None, cdir=0):
|
||
|
|
for t in self:
|
||
|
|
if t is not S.Zero:
|
||
|
|
return t
|
||
|
|
|
||
|
|
def _eval_term(self, pt):
|
||
|
|
if pt == 0:
|
||
|
|
return self.a0
|
||
|
|
return self.an.coeff(pt) + self.bn.coeff(pt)
|
||
|
|
|
||
|
|
def __neg__(self):
|
||
|
|
return self.scale(-1)
|
||
|
|
|
||
|
|
def __add__(self, other):
|
||
|
|
if isinstance(other, FourierSeries):
|
||
|
|
if self.period != other.period:
|
||
|
|
raise ValueError("Both the series should have same periods")
|
||
|
|
|
||
|
|
x, y = self.x, other.x
|
||
|
|
function = self.function + other.function.subs(y, x)
|
||
|
|
|
||
|
|
if self.x not in function.free_symbols:
|
||
|
|
return function
|
||
|
|
|
||
|
|
an = self.an + other.an
|
||
|
|
bn = self.bn + other.bn
|
||
|
|
a0 = self.a0 + other.a0
|
||
|
|
|
||
|
|
return self.func(function, self.args[1], (a0, an, bn))
|
||
|
|
|
||
|
|
return Add(self, other)
|
||
|
|
|
||
|
|
def __sub__(self, other):
|
||
|
|
return self.__add__(-other)
|
||
|
|
|
||
|
|
|
||
|
|
class FiniteFourierSeries(FourierSeries):
|
||
|
|
r"""Represents Finite Fourier sine/cosine series.
|
||
|
|
|
||
|
|
For how to compute Fourier series, see the :func:`fourier_series`
|
||
|
|
docstring.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
f : Expr
|
||
|
|
Expression for finding fourier_series
|
||
|
|
|
||
|
|
limits : ( x, start, stop)
|
||
|
|
x is the independent variable for the expression f
|
||
|
|
(start, stop) is the period of the fourier series
|
||
|
|
|
||
|
|
exprs: (a0, an, bn) or Expr
|
||
|
|
a0 is the constant term a0 of the fourier series
|
||
|
|
an is a dictionary of coefficients of cos terms
|
||
|
|
an[k] = coefficient of cos(pi*(k/L)*x)
|
||
|
|
bn is a dictionary of coefficients of sin terms
|
||
|
|
bn[k] = coefficient of sin(pi*(k/L)*x)
|
||
|
|
|
||
|
|
or exprs can be an expression to be converted to fourier form
|
||
|
|
|
||
|
|
Methods
|
||
|
|
=======
|
||
|
|
|
||
|
|
This class is an extension of FourierSeries class.
|
||
|
|
Please refer to sympy.series.fourier.FourierSeries for
|
||
|
|
further information.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.series.fourier.FourierSeries
|
||
|
|
sympy.series.fourier.fourier_series
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __new__(cls, f, limits, exprs):
|
||
|
|
f = sympify(f)
|
||
|
|
limits = sympify(limits)
|
||
|
|
exprs = sympify(exprs)
|
||
|
|
|
||
|
|
if not (isinstance(exprs, Tuple) and len(exprs) == 3): # exprs is not of form (a0, an, bn)
|
||
|
|
# Converts the expression to fourier form
|
||
|
|
c, e = exprs.as_coeff_add()
|
||
|
|
from sympy.simplify.fu import TR10
|
||
|
|
rexpr = c + Add(*[TR10(i) for i in e])
|
||
|
|
a0, exp_ls = rexpr.expand(trig=False, power_base=False, power_exp=False, log=False).as_coeff_add()
|
||
|
|
|
||
|
|
x = limits[0]
|
||
|
|
L = abs(limits[2] - limits[1]) / 2
|
||
|
|
|
||
|
|
a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k is not S.Zero, ])
|
||
|
|
b = Wild('b', properties=[lambda k: x not in k.free_symbols, ])
|
||
|
|
|
||
|
|
an = {}
|
||
|
|
bn = {}
|
||
|
|
|
||
|
|
# separates the coefficients of sin and cos terms in dictionaries an, and bn
|
||
|
|
for p in exp_ls:
|
||
|
|
t = p.match(b * cos(a * (pi / L) * x))
|
||
|
|
q = p.match(b * sin(a * (pi / L) * x))
|
||
|
|
if t:
|
||
|
|
an[t[a]] = t[b] + an.get(t[a], S.Zero)
|
||
|
|
elif q:
|
||
|
|
bn[q[a]] = q[b] + bn.get(q[a], S.Zero)
|
||
|
|
else:
|
||
|
|
a0 += p
|
||
|
|
|
||
|
|
exprs = Tuple(a0, an, bn)
|
||
|
|
|
||
|
|
return Expr.__new__(cls, f, limits, exprs)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def interval(self):
|
||
|
|
_length = 1 if self.a0 else 0
|
||
|
|
_length += max(set(self.an.keys()).union(set(self.bn.keys()))) + 1
|
||
|
|
return Interval(0, _length)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def length(self):
|
||
|
|
return self.stop - self.start
|
||
|
|
|
||
|
|
def shiftx(self, s):
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
_expr = self.truncate().subs(x, x + s)
|
||
|
|
sfunc = self.function.subs(x, x + s)
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], _expr)
|
||
|
|
|
||
|
|
def scale(self, s):
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
_expr = self.truncate() * s
|
||
|
|
sfunc = self.function * s
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], _expr)
|
||
|
|
|
||
|
|
def scalex(self, s):
|
||
|
|
s, x = sympify(s), self.x
|
||
|
|
|
||
|
|
if x in s.free_symbols:
|
||
|
|
raise ValueError("'%s' should be independent of %s" % (s, x))
|
||
|
|
|
||
|
|
_expr = self.truncate().subs(x, x * s)
|
||
|
|
sfunc = self.function.subs(x, x * s)
|
||
|
|
|
||
|
|
return self.func(sfunc, self.args[1], _expr)
|
||
|
|
|
||
|
|
def _eval_term(self, pt):
|
||
|
|
if pt == 0:
|
||
|
|
return self.a0
|
||
|
|
|
||
|
|
_term = self.an.get(pt, S.Zero) * cos(pt * (pi / self.L) * self.x) \
|
||
|
|
+ self.bn.get(pt, S.Zero) * sin(pt * (pi / self.L) * self.x)
|
||
|
|
return _term
|
||
|
|
|
||
|
|
def __add__(self, other):
|
||
|
|
if isinstance(other, FourierSeries):
|
||
|
|
return other.__add__(fourier_series(self.function, self.args[1],\
|
||
|
|
finite=False))
|
||
|
|
elif isinstance(other, FiniteFourierSeries):
|
||
|
|
if self.period != other.period:
|
||
|
|
raise ValueError("Both the series should have same periods")
|
||
|
|
|
||
|
|
x, y = self.x, other.x
|
||
|
|
function = self.function + other.function.subs(y, x)
|
||
|
|
|
||
|
|
if self.x not in function.free_symbols:
|
||
|
|
return function
|
||
|
|
|
||
|
|
return fourier_series(function, limits=self.args[1])
|
||
|
|
|
||
|
|
|
||
|
|
def fourier_series(f, limits=None, finite=True):
|
||
|
|
r"""Computes the Fourier trigonometric series expansion.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Fourier trigonometric series of $f(x)$ over the interval $(a, b)$
|
||
|
|
is defined as:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
\frac{a_0}{2} + \sum_{n=1}^{\infty}
|
||
|
|
(a_n \cos(\frac{2n \pi x}{L}) + b_n \sin(\frac{2n \pi x}{L}))
|
||
|
|
|
||
|
|
where the coefficients are:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
L = b - a
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
a_0 = \frac{2}{L} \int_{a}^{b}{f(x) dx}
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
a_n = \frac{2}{L} \int_{a}^{b}{f(x) \cos(\frac{2n \pi x}{L}) dx}
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
b_n = \frac{2}{L} \int_{a}^{b}{f(x) \sin(\frac{2n \pi x}{L}) dx}
|
||
|
|
|
||
|
|
The condition whether the function $f(x)$ given should be periodic
|
||
|
|
or not is more than necessary, because it is sufficient to consider
|
||
|
|
the series to be converging to $f(x)$ only in the given interval,
|
||
|
|
not throughout the whole real line.
|
||
|
|
|
||
|
|
This also brings a lot of ease for the computation because
|
||
|
|
you do not have to make $f(x)$ artificially periodic by
|
||
|
|
wrapping it with piecewise, modulo operations,
|
||
|
|
but you can shape the function to look like the desired periodic
|
||
|
|
function only in the interval $(a, b)$, and the computed series will
|
||
|
|
automatically become the series of the periodic version of $f(x)$.
|
||
|
|
|
||
|
|
This property is illustrated in the examples section below.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
limits : (sym, start, end), optional
|
||
|
|
*sym* denotes the symbol the series is computed with respect to.
|
||
|
|
|
||
|
|
*start* and *end* denotes the start and the end of the interval
|
||
|
|
where the fourier series converges to the given function.
|
||
|
|
|
||
|
|
Default range is specified as $-\pi$ and $\pi$.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
FourierSeries
|
||
|
|
A symbolic object representing the Fourier trigonometric series.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Computing the Fourier series of $f(x) = x^2$:
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = x**2
|
||
|
|
>>> s = fourier_series(f, (x, -pi, pi))
|
||
|
|
>>> s1 = s.truncate(n=3)
|
||
|
|
>>> s1
|
||
|
|
-4*cos(x) + cos(2*x) + pi**2/3
|
||
|
|
|
||
|
|
Shifting of the Fourier series:
|
||
|
|
|
||
|
|
>>> s.shift(1).truncate()
|
||
|
|
-4*cos(x) + cos(2*x) + 1 + pi**2/3
|
||
|
|
>>> s.shiftx(1).truncate()
|
||
|
|
-4*cos(x + 1) + cos(2*x + 2) + pi**2/3
|
||
|
|
|
||
|
|
Scaling of the Fourier series:
|
||
|
|
|
||
|
|
>>> s.scale(2).truncate()
|
||
|
|
-8*cos(x) + 2*cos(2*x) + 2*pi**2/3
|
||
|
|
>>> s.scalex(2).truncate()
|
||
|
|
-4*cos(2*x) + cos(4*x) + pi**2/3
|
||
|
|
|
||
|
|
Computing the Fourier series of $f(x) = x$:
|
||
|
|
|
||
|
|
This illustrates how truncating to the higher order gives better
|
||
|
|
convergence.
|
||
|
|
|
||
|
|
.. plot::
|
||
|
|
:context: reset
|
||
|
|
:format: doctest
|
||
|
|
:include-source: True
|
||
|
|
|
||
|
|
>>> from sympy import fourier_series, pi, plot
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> f = x
|
||
|
|
>>> s = fourier_series(f, (x, -pi, pi))
|
||
|
|
>>> s1 = s.truncate(n = 3)
|
||
|
|
>>> s2 = s.truncate(n = 5)
|
||
|
|
>>> s3 = s.truncate(n = 7)
|
||
|
|
>>> p = plot(f, s1, s2, s3, (x, -pi, pi), show=False, legend=True)
|
||
|
|
|
||
|
|
>>> p[0].line_color = (0, 0, 0)
|
||
|
|
>>> p[0].label = 'x'
|
||
|
|
>>> p[1].line_color = (0.7, 0.7, 0.7)
|
||
|
|
>>> p[1].label = 'n=3'
|
||
|
|
>>> p[2].line_color = (0.5, 0.5, 0.5)
|
||
|
|
>>> p[2].label = 'n=5'
|
||
|
|
>>> p[3].line_color = (0.3, 0.3, 0.3)
|
||
|
|
>>> p[3].label = 'n=7'
|
||
|
|
|
||
|
|
>>> p.show()
|
||
|
|
|
||
|
|
This illustrates how the series converges to different sawtooth
|
||
|
|
waves if the different ranges are specified.
|
||
|
|
|
||
|
|
.. plot::
|
||
|
|
:context: close-figs
|
||
|
|
:format: doctest
|
||
|
|
:include-source: True
|
||
|
|
|
||
|
|
>>> s1 = fourier_series(x, (x, -1, 1)).truncate(10)
|
||
|
|
>>> s2 = fourier_series(x, (x, -pi, pi)).truncate(10)
|
||
|
|
>>> s3 = fourier_series(x, (x, 0, 1)).truncate(10)
|
||
|
|
>>> p = plot(x, s1, s2, s3, (x, -5, 5), show=False, legend=True)
|
||
|
|
|
||
|
|
>>> p[0].line_color = (0, 0, 0)
|
||
|
|
>>> p[0].label = 'x'
|
||
|
|
>>> p[1].line_color = (0.7, 0.7, 0.7)
|
||
|
|
>>> p[1].label = '[-1, 1]'
|
||
|
|
>>> p[2].line_color = (0.5, 0.5, 0.5)
|
||
|
|
>>> p[2].label = '[-pi, pi]'
|
||
|
|
>>> p[3].line_color = (0.3, 0.3, 0.3)
|
||
|
|
>>> p[3].label = '[0, 1]'
|
||
|
|
|
||
|
|
>>> p.show()
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
Computing Fourier series can be slow
|
||
|
|
due to the integration required in computing
|
||
|
|
an, bn.
|
||
|
|
|
||
|
|
It is faster to compute Fourier series of a function
|
||
|
|
by using shifting and scaling on an already
|
||
|
|
computed Fourier series rather than computing
|
||
|
|
again.
|
||
|
|
|
||
|
|
e.g. If the Fourier series of ``x**2`` is known
|
||
|
|
the Fourier series of ``x**2 - 1`` can be found by shifting by ``-1``.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.series.fourier.FourierSeries
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://mathworld.wolfram.com/FourierSeries.html
|
||
|
|
"""
|
||
|
|
f = sympify(f)
|
||
|
|
|
||
|
|
limits = _process_limits(f, limits)
|
||
|
|
x = limits[0]
|
||
|
|
|
||
|
|
if x not in f.free_symbols:
|
||
|
|
return f
|
||
|
|
|
||
|
|
if finite:
|
||
|
|
L = abs(limits[2] - limits[1]) / 2
|
||
|
|
is_finite, res_f = finite_check(f, x, L)
|
||
|
|
if is_finite:
|
||
|
|
return FiniteFourierSeries(f, limits, res_f)
|
||
|
|
|
||
|
|
n = Dummy('n')
|
||
|
|
center = (limits[1] + limits[2]) / 2
|
||
|
|
if center.is_zero:
|
||
|
|
neg_f = f.subs(x, -x)
|
||
|
|
if f == neg_f:
|
||
|
|
a0, an = fourier_cos_seq(f, limits, n)
|
||
|
|
bn = SeqFormula(0, (1, oo))
|
||
|
|
return FourierSeries(f, limits, (a0, an, bn))
|
||
|
|
elif f == -neg_f:
|
||
|
|
a0 = S.Zero
|
||
|
|
an = SeqFormula(0, (1, oo))
|
||
|
|
bn = fourier_sin_seq(f, limits, n)
|
||
|
|
return FourierSeries(f, limits, (a0, an, bn))
|
||
|
|
a0, an = fourier_cos_seq(f, limits, n)
|
||
|
|
bn = fourier_sin_seq(f, limits, n)
|
||
|
|
return FourierSeries(f, limits, (a0, an, bn))
|