Fourier Series¶
Provides methods to compute Fourier series.
-
class
sympy.series.fourier.
FourierSeries
[source]¶ Represents Fourier sine/cosine series.
This class only represents a fourier series. No computation is performed.
For how to compute Fourier series, see the
fourier_series()
docstring.See also
-
scale
(s)[source]¶ Scale the function by a term independent of x.
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
-
scalex
(s)[source]¶ Scale x by a term independent of x.
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
-
shift
(s)[source]¶ Shift the function by a term independent of x.
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
-
shiftx
(s)[source]¶ Shift x by a term independent of x.
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
-
-
sympy.series.fourier.
fourier_series
(f, limits=None)[source]¶ Computes Fourier sine/cosine series expansion.
Returns a
FourierSeries
object.See also
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 ofx**2 - 1
can be found by shifting by-1
.References
[R432] mathworld.wolfram.com/FourierSeries.html Examples
>>> from sympy import fourier_series, pi, cos >>> from sympy.abc import x
>>> s = fourier_series(x**2, (x, -pi, pi)) >>> s.truncate(n=3) -4*cos(x) + cos(2*x) + pi**2/3
Shifting
>>> 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
>>> 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