#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=pi*t/180 else: angle=rad*180/pi# Should be t*180/pi return angle
convert_angle(90,0)
1/2*pi
######################## #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) fplot=plot(f,-11,11,color='red',thickness=3) top=plot(-2*cos(.15*pi*t),-11,11,color='blue',thickness=2) bottom=plot(2*cos(.15*pi*t),-11,11,color='blue',thickness=2) (fplot+top+bottom).show()
reset
<function reset at 0x7f4cd92ae0c8>
######################## #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*cos(k*t),t,t0,T) for k in range(n+1) ] return [c/pi for c in L]
#integral?
#My example f(t)=t^2 acoef(f,6,-pi,pi)
[2/3*pi^2, -4, 1, -4/9, 1/4, -4/25, 1/9]
#Your example: g(t)=t for 0<t<=2*pi ##Incorrect example f(t)=t^2 acoef(f,2,-pi,pi)
[2/3*pi^2, -4, 1]
####################### #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. f(t)=abs(t) acoef(f,6,-pi,pi)
[pi, -4/pi, 0, -4/9/pi, 0, -4/25/pi, 0]
#(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).
####################### #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.
#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(?,?,?,pi);A
[2/pi, 0, -2/3/pi, 0]
B=bcoef(?);B
[0, 1/2, 0, 0]
S5(t)=?
s5_plot=plot(?,(t,?,?), linestyle="--")
(f_plot+s5_plot).show()
## Exercises 8 and 10 not done; Exercises 6: incorrect example (-0.5); Exercise 1: conversion from radians to degrees incorrect (-0.5). Total: 4/6 ︠6a4d89f6-85a6-48e7-ae52-2c97af795e46︠