def test_sawtooth_wave():
s = fourier_series(x, (x, 0, pi))
assert s.truncate(4) == \
pi/2 - sin(2*x) - sin(4*x)/2 - sin(6*x)/3
s = fourier_series(x, (x, 0, 1))
assert s.truncate(4) == \
S.Half - sin(2*pi*x)/pi - sin(4*pi*x)/(2*pi) - sin(6*pi*x)/(3*pi)
def test_FourierSeries_finite():
assert fourier_series(sin(x)).truncate(1) == sin(x)
# assert type(fourier_series(sin(x)*log(x))).truncate() == FourierSeries
# assert type(fourier_series(sin(x**2+6))).truncate() == FourierSeries
assert fourier_series(sin(x) * log(y) * exp(z), (x, pi, -pi)).truncate() == sin(
x
) * log(y) * exp(z)
assert fourier_series(sin(x) ** 6).truncate(oo) == -15 * cos(2 * x) / 32 + 3 * cos(
4 * x
) / 16 - cos(6 * x) / 32 + Rational(5, 16)
assert fourier_series(sin(x) ** 6).truncate() == -15 * cos(2 * x) / 32 + 3 * cos(
4 * x
) / 16 + Rational(5, 16)
assert fourier_series(sin(4 * x + 3) + cos(3 * x + 4)).truncate(oo) == -sin(
4
) * sin(3 * x) + sin(4 * x) * cos(3) + cos(4) * cos(3 * x) + sin(3) * cos(4 * x)
assert fourier_series(sin(x) + cos(x) * tan(x)).truncate(oo) == 2 * sin(x)
assert fourier_series(cos(pi * x), (x, -1, 1)).truncate(oo) == cos(pi * x)
assert fourier_series(
cos(3 * pi * x + 4) - sin(4 * pi * x) * log(pi * y), (x, -1, 1)
).truncate(oo) == -log(pi * y) * sin(4 * pi * x) - sin(4) * sin(3 * pi * x) + cos(
4
) * cos(
3 * pi * x
)
def test_FourierSeries_2():
p = Piecewise((0, x < 0), (x, True))
f = fourier_series(p, (x, -2, 2))
assert f.term(3) == (2 * sin(3 * pi * x / 2) / (3 * pi) -
4 * cos(3 * pi * x / 2) / (9 * pi**2))
assert f.truncate() == (2 * sin(pi * x / 2) / pi - sin(pi * x) / pi -
4 * cos(pi * x / 2) / pi**2 + Rational(1, 2))
def test_FourierSeries_2():
p = Piecewise((0, x < 0), (x, True))
f = fourier_series(p, (x, -2, 2))
assert f.term(3) == (2*sin(3*pi*x / 2) / (3*pi) -
4*cos(3*pi*x / 2) / (9*pi**2))
assert f.truncate() == (2*sin(pi*x / 2) / pi - sin(pi*x) / pi -
4*cos(pi*x / 2) / pi**2 + Rational(1, 2))
def test_FourierSeries():
fo, fe, fp = _get_examples()
assert fourier_series(1, (-pi, pi)) == 1
assert (
Piecewise((0, x < 0), (pi, True)).fourier_series((x, -pi, pi)).truncate()
) == fp.truncate()
assert isinstance(fo, FourierSeries)
assert fo.function == x
assert fo.x == x
assert fo.period == (-pi, pi)
assert fo.term(3) == 2 * sin(3 * x) / 3
assert fe.term(3) == -4 * cos(3 * x) / 9
assert fp.term(3) == 2 * sin(3 * x) / 3
assert fo.as_leading_term(x) == 2 * sin(x)
assert fe.as_leading_term(x) == pi ** 2 / 3
assert fp.as_leading_term(x) == pi / 2
assert fo.truncate() == 2 * sin(x) - sin(2 * x) + (2 * sin(3 * x) / 3)
assert fe.truncate() == -4 * cos(x) + cos(2 * x) + pi ** 2 / 3
assert fp.truncate() == 2 * sin(x) + (2 * sin(3 * x) / 3) + pi / 2
fot = fo.truncate(n=None)
s = [0, 2 * sin(x), -sin(2 * x)]
for i, t in enumerate(fot):
if i == 3:
break
assert s[i] == t
def _check_iter(f, i):
for ind, t in enumerate(f):
assert t == f[ind]
if ind == i:
break
_check_iter(fo, 3)
_check_iter(fe, 3)
_check_iter(fp, 3)
assert fo.subs(x, x ** 2) == fo
raises(ValueError, lambda: fourier_series(x, (0, 1, 2)))
raises(ValueError, lambda: fourier_series(x, (x, 0, oo)))
raises(ValueError, lambda: fourier_series(x * y, (0, oo)))
def test_FourierSeries():
fo, fe, fp = _get_examples()
assert fourier_series(1, (-pi, pi)) == 1
assert (Piecewise((0, x < 0), (pi, True)).
fourier_series((x, -pi, pi)).truncate()) == fp.truncate()
assert isinstance(fo, FourierSeries)
assert fo.function == x
assert fo.x == x
assert fo.period == (-pi, pi)
assert fo.term(3) == 2*sin(3*x) / 3
assert fe.term(3) == -4*cos(3*x) / 9
assert fp.term(3) == 2*sin(3*x) / 3
assert fo.as_leading_term(x) == 2*sin(x)
assert fe.as_leading_term(x) == pi**2 / 3
assert fp.as_leading_term(x) == pi / 2
assert fo.truncate() == 2*sin(x) - sin(2*x) + (2*sin(3*x) / 3)
assert fe.truncate() == -4*cos(x) + cos(2*x) + pi**2 / 3
assert fp.truncate() == 2*sin(x) + (2*sin(3*x) / 3) + pi / 2
fot = fo.truncate(n=None)
s = [0, 2*sin(x), -sin(2*x)]
for i, t in enumerate(fot):
if i == 3:
break
assert s[i] == t
def _check_iter(f, i):
for ind, t in enumerate(f):
assert t == f[ind]
if ind == i:
break
_check_iter(fo, 3)
_check_iter(fe, 3)
_check_iter(fp, 3)
assert fo.subs(x, x**2) == fo
raises(ValueError, lambda: fourier_series(x, (0, 1, 2)))
raises(ValueError, lambda: fourier_series(x, (x, 0, oo)))
raises(ValueError, lambda: fourier_series(x*y, (0, oo)))
def test_fourier_series_square_wave():
"""Test if fourier_series approximates discontinuous function correctly."""
square_wave = Piecewise((1, x < pi), (-1, True))
s = fourier_series(square_wave, (x, 0, 2 * pi))
assert s.truncate(3) == 4 / pi * sin(x) + 4 / (3 * pi) * sin(3 * x) + \
4 / (5 * pi) * sin(5 * x)
assert s.sigma_approximation(4) == 4 / pi * sin(x) * sinc(pi / 4) + \
4 / (3 * pi) * sin(3 * x) * sinc(3 * pi / 4)
def test_fourier_series_square_wave():
"""Test if fourier_series approximates discontinuous function correctly."""
square_wave = Piecewise((1, x < pi), (-1, True))
s = fourier_series(square_wave, (x, 0, 2*pi))
assert s.truncate(3) == 4 / pi * sin(x) + 4 / (3 * pi) * sin(3 * x) + \
4 / (5 * pi) * sin(5 * x)
assert s.sigma_approximation(4) == 4 / pi * sin(x) * sinc(pi / 4) + \
4 / (3 * pi) * sin(3 * x) * sinc(3 * pi / 4)
def test_FourierSeries__add__sub():
assert fo + fo == fo.scale(2)
assert fo - fo == 0
assert -fe - fe == fe.scale(-2)
assert (fo + fe).truncate() == 2*sin(x) - sin(2*x) - 4*cos(x) + cos(2*x) \
+ pi**2 / 3
assert (fo - fe).truncate() == 2*sin(x) - sin(2*x) + 4*cos(x) - cos(2*x) \
- pi**2 / 3
assert isinstance(fo + 1, Add)
raises(ValueError, lambda: fo + fourier_series(x, (x, 0, 2)))
def test_FourierSeries__add__sub():
assert fo + fo == fo.scale(2)
assert fo - fo == 0
assert -fe - fe == fe.scale(-2)
assert (fo + fe).truncate() == 2*sin(x) - sin(2*x) - 4*cos(x) + cos(2*x) \
+ pi**2 / 3
assert (fo - fe).truncate() == 2*sin(x) - sin(2*x) + 4*cos(x) - cos(2*x) \
- pi**2 / 3
assert isinstance(fo + 1, Add)
raises(ValueError, lambda: fo + fourier_series(x, (x, 0, 2)))
def test_FourierSeries_finite():
assert fourier_series(sin(x)).truncate(1) == sin(x)
# assert type(fourier_series(sin(x)*log(x))).truncate() == FourierSeries
# assert type(fourier_series(sin(x**2+6))).truncate() == FourierSeries
assert fourier_series(sin(x)*log(y)*exp(z),(x,pi,-pi)).truncate() == sin(x)*log(y)*exp(z)
assert fourier_series(sin(x)**6).truncate(oo) == -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 \
+ Rational(5, 16)
assert fourier_series(sin(x) ** 6).truncate() == -15 * cos(2 * x) / 32 + 3 * cos(4 * x) / 16 \
+ Rational(5, 16)
assert fourier_series(sin(4*x+3) + cos(3*x+4)).truncate(oo) == -sin(4)*sin(3*x) + sin(4*x)*cos(3) \
+ cos(4)*cos(3*x) + sin(3)*cos(4*x)
assert fourier_series(sin(x)+cos(x)*tan(x)).truncate(oo) == 2*sin(x)
assert fourier_series(cos(pi*x), (x, -1, 1)).truncate(oo) == cos(pi*x)
assert fourier_series(cos(3*pi*x + 4) - sin(4*pi*x)*log(pi*y) , (x, -1, 1)).truncate(oo) == -log(pi*y)*sin(4*pi*x)\
- sin(4)*sin(3*pi*x) + cos(4)*cos(3*pi*x)
def constructTrajectory_FourierSeriesY():
dc = 0.8
ht = 3.5
T = 1.0
t_current = symbols('t_current')
fy1 = -18
fy2 = -18 + 0 + 16*ht/(T-dc)**2*(t_current-dc)**2 - 32*ht/(T-dc)**3*(t_current-dc)**3 + 16*ht/(T-dc)**4*(t_current-dc)**4
print("Calculating Piecewise Fourier Series for Y")
FourierY = Piecewise((fy1,t_current <= dc),(fy2, t_current<=1))
print("\n")
print("Fourier piecewise for Y is",FourierY)
FourierYSeries = fourier_series(FourierY,(t_current,0,1))
q = FourierYSeries.truncate(100)
print("FourierY Series calculated: ",q)
lq = np.array(range( 0 , 100))
for i in lq:
currentY = q.subs(t_current,i/100)
plt.scatter( i/100 , currentY )
plt.show()
from sympy import fourier_series, pi, cos from sympy.abc import x from sympy.utilities.lambdify import lambdify, implemented_function from sympy import Function from sympy import * s = fourier_series(x**2, (x, -pi, pi)) res = s.truncate(n=9) print(res) lam_f = lambdify(x, res) print(lam_f(4)) plot(res)
def constructTrajectory_FourierSeriesX():
dc = 0.8
dt = 0.025
T = 1.0
stroke = 12.0
ht= 3.5
a3 = (-4.0*(stroke/2.0+stroke/dc*dt)-2.0*stroke/dc*(T-dc-2.0*dt))/(T-dc-2.0*dt)**3.0;
a2 = -3.0/2.0*a3*(T-dc-2.0*dt)
a1 = -stroke/dc
a0 = -(stroke/2.0+stroke/dc*dt)
t_current = symbols('t_current')
fx1 = (stroke/2.0 + stroke/dc*(-1* t_current))
fx2 = a0 + a1*(t_current-(dc+dt)) + a2*(t_current-(dc+dt))**2.0 + a3*(t_current-(dc+dt))**3.0; #stride
fx3 = (stroke/2.0 + stroke/dc*(-1* t_current))
print("Calculating Piecewise Fourier Series for X")
FourierX = Piecewise( (fx1,t_current<=dc+dt) , (fx2 ,t_current<=T-dt ) , (fx3 , t_current<=1) )
print("\n")
print("Fourier piecewise for X is ",FourierX)
print("\n")
FourierXSeries = fourier_series( FourierX , (t_current,0,1) )
p = FourierXSeries.truncate(1000)
print("FourierX Series Calculated: ", p)
# lp = np.array(range( 0 , 100))
# for i in lp:
# currentX = p.subs(t_current,i/100)
# plt.scatter( i/100 , currentX )
# plt.show()
fy1 = -18
fy2 = -18 + 0 + 16*ht/(T-dc)**2*(t_current-dc)**2 - 32*ht/(T-dc)**3*(t_current-dc)**3 + 16*ht/(T-dc)**4*(t_current-dc)**4
print("Calculating Piecewise Fourier Series for Y")
FourierY = Piecewise((fy1,t_current <= dc),(fy2, t_current<=1))
print("\n")
print("Fourier piecewise for Y is",FourierY)
FourierYSeries = fourier_series(FourierY,(t_current,0,1))
q = FourierYSeries.truncate(1000)
print("FourierY Series calculated: ",q)
lq = np.array(range( 0 , 100))
for i in lq:
a=0
currentX = p.subs(t_current,i/100)
currentY = q.subs(t_current,i/100)
plt.scatter( currentX , currentY )
print("loop = ",i)
plt.show()
def _get_examples():
fo = fourier_series(x, (x, -pi, pi))
fe = fourier_series(x ** 2, (-pi, pi))
fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi))
return fo, fe, fp
##def sin(x): ## return sin(x) #t_list=[] #for x in xrange: # print(x) # fx = f(x) # fx_sum=0 # for n in count: # if n == 0: # print(f(x)) # a0=integrate.quad(f,a=-pi,b=pi,args=x)[0]*(1/pi) # # fx_sum += a0*cos(n*x) # else: # a=integrate.quad(f*cos(n*x),-pi,pi,args=x)*(1/pi) # asum=a*cos(n) # b=integrate.quad(f*sin(n*x),-pi,pi,args=x)*(1/pi) # fx_sum +=b*sin(x) # t_list.append(fx_sum) # #print(t_list) from sympy import fourier_series, pi from sympy.abc import x from sympy.plotting import plot s = fourier_series(pi + x, (x, -pi, pi)) print(type(s)) s.scale(2).truncate() sp = plot(s) #plt.show() print(s)
# Fprma de calcular series de fourier truncadas de orde n , ampliadas y trasladadas. from sympy import fourier_series, pi, plot from sympy.abc import x f = x**2 sn = fourier_series(f, (x, -pi, pi)) sn.shift(5).truncate(7) sn.shiftx(3).truncate(7) sn.scale(5).truncate(7) sn.scalex(7).truncate(7) # Series de fourier y graficadoras import numpy as np from sympy import fourier_series, pi, plot from sympy.abc import x f = x fser = fourier_series(f,(x,-pi, pi)) fser1 = fser.truncate(3) fser2 = fser.truncate(7) fser3 = fser.truncate(14) p = plot(f,fser1, fser2, fser3, (x,-pi, pi), show = False, legend = True) p[0].line_color = (0,0,0) p[0].label = 'x' p[1].line_color = (1,0,0) p[1].label = 'n=3' p[2].label = 'n=7' p[3].label = 'n=14'
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()
def _get_examples():
fo = fourier_series(x, (x, -pi, pi))
fe = fourier_series(x**2, (-pi, pi))
fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi))
return fo, fe, fp
from sympy import (symbols, pi, Piecewise, sin, cos, Rational, oo,
fourier_series, Add)
from sympy.series.fourier import FourierSeries
from sympy.utilities.pytest import raises
x, y, z = symbols('x y z')
fo = fourier_series(x, (x, -pi, pi))
fe = fourier_series(x**2, (-pi, pi))
fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi))
def test_FourierSeries():
assert fourier_series(1, (-pi, pi)) == 1
assert (Piecewise((0, x < 0), (pi, True)).fourier_series(
(x, -pi, pi)).truncate()) == fp.truncate()
assert isinstance(fo, FourierSeries)
assert fo.function == x
assert fo.x == x
assert fo.period == (-pi, pi)
assert fo.term(3) == 2 * sin(3 * x) / 3
assert fe.term(3) == -4 * cos(3 * x) / 9
assert fp.term(3) == 2 * sin(3 * x) / 3
assert fo.as_leading_term(x) == 2 * sin(x)
assert fe.as_leading_term(x) == pi**2 / 3
assert fp.as_leading_term(x) == pi / 2
assert fo.truncate() == 2 * sin(x) - sin(2 * x) + (2 * sin(3 * x) / 3)
assert fe.truncate() == -4 * cos(x) + cos(2 * x) + pi**2 / 3
"Enter 1 to enter interval in terms of pi or it's multiples,Otherwise enter 0"
))
if try1 == 1:
interval1 = int(
input(
"Enter the multiple of pi that you want,Eg:2 represents 2*pi"))
interval2 = int(
input(
"Enter the multiple of pi that you want,Eg:2 represents 2*pi"))
interval1 = interval1 * pi
interval2 = interval2 * pi
else:
interval1 = float(input("Enter the Starting point of the interval"))
interval2 = float(input("Enter the Ending point of the interval"))
s = fourier_series(expr1, (x, interval1, interval2))
truncval = int(
input("enter the number of terms desired in the fourier series"))
v = s.truncate(truncval)
print("FOURIER SERIES OF THE ENTERED FUNCTION\n\n\n")
print(v)
x1 = [i for i in np.arange(interval1, interval2, 0.02)]
y1 = [(v.evalf(subs={x: j}))
for j in np.arange(interval1, interval2, 0.02)]
x2 = [i for i in np.arange(interval1, interval2, 0.0001)]
y2 = [(expr1.evalf(subs={x: j}))
from sympy import fourier_series, pi, Piecewise
from sympy.abc import t
from sympy.plotting import plot
f = Piecewise((0, t < -pi), (-1, t < -pi / 2), (1, t < pi / 2), (-1, t < pi),
(0, True))
T = 2 * pi
s = fourier_series(f, (t, -T / 2, T / 2))
F = s.truncate(5)
print(F)
n = [10, 50, 200]
p = plot(s.truncate(n[0]), (t, -10, 10),
title="For 10 terms",
line_color='firebrick',
show=False)
p.xlabel = '$t$'
p.ylabel = '$F(t)$'
p.show()
p = plot(s.truncate(n[1]), (t, -10, 10),
title="For 50 terms",
line_color='darkblue',
show=False)
p.xlabel = '$t$'
p.ylabel = '$F(t)$'
p.show()
p = plot(s.truncate(n[2]), (t, -10, 10),
title="For 200 terms",
line_color='green',
show=False)
from sympy import (symbols, pi, Piecewise, sin, cos, sinc, Rational,
oo, fourier_series, Add)
from sympy.series.fourier import FourierSeries
from sympy.utilities.pytest import raises
x, y, z = symbols('x y z')
fo = fourier_series(x, (x, -pi, pi))
fe = fourier_series(x**2, (-pi, pi))
fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi))
def test_FourierSeries():
assert fourier_series(1, (-pi, pi)) == 1
assert (Piecewise((0, x < 0), (pi, True)).
fourier_series((x, -pi, pi)).truncate()) == fp.truncate()
assert isinstance(fo, FourierSeries)
assert fo.function == x
assert fo.x == x
assert fo.period == (-pi, pi)
assert fo.term(3) == 2*sin(3*x) / 3
assert fe.term(3) == -4*cos(3*x) / 9
assert fp.term(3) == 2*sin(3*x) / 3
assert fo.as_leading_term(x) == 2*sin(x)
assert fe.as_leading_term(x) == pi**2 / 3
assert fp.as_leading_term(x) == pi / 2
assert fo.truncate() == 2*sin(x) - sin(2*x) + (2*sin(3*x) / 3)
assert fe.truncate() == -4*cos(x) + cos(2*x) + pi**2 / 3