pbsetCh10student_version.sagews

reset
#Exercise 1. Make a function convert_angle(t, p) that takes an angle t and parameter p.  Zero value of p indicates that the angle t is in degrees, and the function should return the value of the angle in radians; value p = 1  indicates that the angle t is in radians, and the function should return the measure of this angle in degrees.
def convert_angle(t,p):
    if p==0:
        angle=t*(pi/180)
    else:
        angle=t*(180/pi)
    return angle

<function reset at 0x7f38ea66e0c8>

res=convert_angle(90,0);res.n()

1.57079632679490

convert_angle(pi,1)

180

########################
#Exercise 4. Make a figure similar to Fig. 10.4 in Chapter 10 for the sum cos(3 Pi*t)+cos(2.7 Pi*t).
#The exercise is similar to the example about beats in Chapter 10.

f(t)=2*cos(5.7*pi*t/2)*cos(.15*pi*t)
g(t)=2*cos(.15*pi*t)
p=plot(f(t),-10,10,figsize=[6,5])
q=plot(g(t),-10,10,figsize=[6,5],color='red')
r=plot(-g(t),-10,10,figsize=[6,5],color='red');p+q+r

reset

<function reset at 0x7f88094c20c8>

########################
#Exercise 6.  Make a CAS function acoef(f(x), n) that takes two inputs, a piecewise differentiable 2π-periodic function f and an integer n and returns a list of n+1 Fourier coefficients a[k],  k=0, 1, ..., n, of the function f.
#Use the formula for a[k] calculation in Chapter 10.
def acoef(f,n,t0,T):
    L=[integral(f(t)*cos(k*t),t,-pi,pi) for k in range(n) ]
    return [N(c/pi) for c in L]

f(t)=t^2
acoef(f,6,-pi,pi)

[6.57973626739290, -4.00000000000000, 1.00000000000000, -0.444444444444444, 0.250000000000000, -0.160000000000000]

#My example
f(t)=t^2
acoef(f,6,-pi,pi)

[6.57973626739290, -4.00000000000000, 1.00000000000000, -0.444444444444444, 0.250000000000000, -0.160000000000000]

#Your example: g(t)=t for 0<t<=2*pi
acoef(t,4,0,2*pi)

[0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000]

#######################
#Exercise 7.  Make a CAS function bcoef(f(x), n) that takes two arguments, a piecewise differentiable 2π-periodic function and an integer n and returns a list of n Fourier coefficients b[k],  k=1, 2, ..., n, of the function f.
#Use the formula for b[k] calculation in Chapter 10.
#Similar to Exercise 6.
def bcoef(f,n,t0,T):
    L=[integral(f*sin(k*t),t,t0,T) for k in range(n+1) ]
    return [l/pi for l in L]

g(t)=t
bcoef(g,6,-pi,pi)

[0, 2, -1, 2/3, -1/2, 2/5, -1/3]

#######################
#Exercise 8.  Consider the 2π-periodic function f(t) = abs(t), t in [-Pi, Pi]  (the triangular wave function).
#(a) Find three partial sums S__n,  n=3, 4, 6, of the Fourier series for this function.
#(b) Make three figures, each with the pairs of graphs f and one of the partial sums S__n, n=3, 4, 6, over three periods.
#Be careful, the formula for the function for t beyond the interval [-pi,pi] is different from abs(t).

#a

f(t)=abs(t)
f1(t)=abs(t-2*pi)
f2(t)=abs(t-4*pi)
L=acoef(f,7,-pi,pi)
S_3(t)=L[0]/2+sum(L[k]*cos(k*t) for k in range(1,4))
S_4(t)=L[0]/2+sum(L[k]*cos(k*t) for k in range(1,5))
S_6(t)=L[0]/2+sum(L[k]*cos(k*t) for k in range(1,7))

L

[3.14159265358979, -1.27323954473516, 0.000000000000000, -0.141471060526129, 0.000000000000000, -0.0509295817894065, 0.000000000000000]

#b
s=plot(f,-pi,pi,color='red',figsize=[4,3])
#s_p3=piecewise(-pi<=t<=pi,f(t),pi<t<=3*pi,f(t-2*pi),3*pi<t<=5*pi,f(t-4*pi))
s1=plot(f1,pi,3*pi,color='red',figsize=[4,3])
s2=plot(f2,3*pi,5*pi,color='red',figsize=[4,3])
p=plot(S_3,-pi,5*pi,figsize=[4,3])
q=plot(S_4,-pi,5*pi,figsize=[4,3])
r=plot(S_6,-pi,5*pi,figsize=[4,3])

s+s1+s2+p
s+s1+s2+q
s+s1+s2+r

#######################
#Exercise 10.
#(a) Construct the Fourier series for the function g(t) = sin(t) for 0<=t<pi, g(t)=0 for pi<= t < 2*Pi.
#Directions: Use your functions in Exercise 6 and Exercise 7 to find several Fourier coefficients a[k] and b[k]. By visual inspection, determine the pattern and write the Fourier series.
#(b) Plot the function and one of its partial Fourier series in one figure.

reset

<function reset at 0x7f88094c20c8>

def acoef(f,n,t0,T):
    L=[integral(f(t)*cos(k*t),t,-pi,pi) for k in range(n) ]
    return [N(c/pi) for c in L]

g(t)=

A=acoef(g,3,0,pi);A

[0.000000000000000, 0.000000000000000, 0.000000000000000]

B=

#piecewise?
var('t')
s=piecewise([([0,pi],sin(t)),((pi,2*pi),0)]) #Just for plotting

t

f_plot=plot(s,(t,0,2*pi),color='red',figsize=[4,2]);f_plot

# Integration of a piecewise function creates a problem:
#test=acoef(s,3,0,2*pi);test

# To overcome the problem, I used the specific feature of the function: it is zero for pi<=t<=2*pi.
A=acoef(s,3,0,2*pi);A

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1191, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> File "", line 2, in acoef File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/misc/functional.py", line 753, in integral return x.integral(*args, **kwds) File "sage/symbolic/expression.pyx", line 12680, in sage.symbolic.expression._eval_on_operands.new_f (build/cythonized/sage/symbolic/expression.cpp:66926) return f(ex, *new_args, **kwds) File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/functions/piecewise.py", line 822, in integral return F(b) - F(a) File "sage/symbolic/expression.pyx", line 12680, in sage.symbolic.expression._eval_on_operands.new_f (build/cythonized/sage/symbolic/expression.cpp:66926) return f(ex, *new_args, **kwds) File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/functions/piecewise.py", line 473, in __call__ return self.subs(substitution) File "sage/symbolic/expression.pyx", line 5321, in sage.symbolic.expression.Expression.substitute (build/cythonized/sage/symbolic/expression.cpp:30443) res = self._gobj.subs_map(smap, 0) File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/functions/piecewise.py", line 227, in _subs_ point = point.pyobject() File "sage/symbolic/expression.pyx", line 374, in sage.symbolic.expression.Expression.pyobject (build/cythonized/sage/symbolic/expression.cpp:6104) cpdef object pyobject(self): File "sage/symbolic/expression.pyx", line 430, in sage.symbolic.expression.Expression.pyobject (build/cythonized/sage/symbolic/expression.cpp:6030) raise TypeError("self must be a numeric expression") TypeError: self must be a numeric expression

B=bcoef(?);B

[0, 1/2, 0, 0]

S5(t)=?

s5_plot=plot(?,(t,?,?), linestyle="--")

(f_plot+s5_plot).show()