xxxxxxxxxx
var('x, n, i')
f(x) = 1
def b(n,l):
coeff = 2/l*integral(f(x)*sin(n*pi*x/l), (x,0,l))
return coeff
def FS(n,l):
return sum(b(i,l)*sin(i*pi*x/l), i, 1, n)
# l = 1 로 놓음
q=sum([plot(FS(n,1), (x,0,1), color=hue((n+1)/7.0), legend_label='n='+str(n))
for n in range(1,9,2)])
p=plot(f(x), (x, 0, 1), color='black', legend_label='f(x)')
(p+q).show()
xxxxxxxxxx
var('x, n, i')
f(x) = x
def a(n,l):
coeff = 2/l*integral(f(x)*cos(n*pi*x/l), (x,0,l))
return coeff
def FS(n,l):
return a(0,1)/2 + sum(a(i,l)*cos(i*pi*x/l), i, 1, n)
# l = 1 로 놓음
q=sum([plot(FS(n,1), (x,0,1), color=hue((n+1)/7.0), legend_label='n='+str(n))
for n in range(1,9,2)])
p=plot(f(x), (x, 0, 1), color='black', legend_label='f(x)')
(p+q).show()
xxxxxxxxxx
var('x, n, i')
def b(n,l):
coeff = 2/l*integral(f(x)*sin(n*pi*x/l), (x,0,l))
return coeff
def FS(n,l):
return sum(b(i,l)*sin(i*pi*x/l), i, 1, n)
f(x) = x^3
print FS(5, 1)
xxxxxxxxxx
var('x, n, i')
def a(n,l):
coeff = 2/l*integral(f(x)*cos(n*pi*x/l), (x,0,l))
return coeff
def FC(n,l):
return 1/l*integral(f(x), (x, 0, 1))+sum(a(i,l)*cos(i*pi*x/l), i, 1, n)
f(x) = x^3
print FC(5, 1)
xxxxxxxxxx
var('x, n, j')
phi(x)=x^2
a=-10;b=10
m=floor((b+1)/2)
def b(n,l):
coeff = 2/l*integral(phi(x)*sin(n*pi*x/l), (x,0,l))
return coeff
def FS(n,l):
return sum(b(j,l)*sin(j*pi*x/l), j, 1, n)
f(x)=FS(5, 1)
p=Graphics()
for i in xrange(-m+1, m):
p += plot(Piecewise([[(2*i-1, 2*i), -phi(-(x-2*i)) ],[(2*i, 2*i+1), phi((x-2*i))]]))
q=Graphics()
for i in xrange(-m+1, m):
q += plot(Piecewise([[(2*i-1, 2*i), -f(-(x-2*i)) ],[(2*i, 2*i+1), f((x-2*i))]]), color=hue(6/7.0))
(p+q).show()
xxxxxxxxxx
var('x, n, j')
phi(x)=exp(-x)
a=-10;b=10
m=floor((b+1)/2)
def b(n,l):
coeff = 2/l*integral(phi(x)*sin(n*pi*x/l), (x,0,l))
return coeff
def FS(n,l):
return sum(b(j,l)*sin(j*pi*x/l), j, 1, n)
f(x)=FS(5, 1)
p=Graphics()
for i in xrange(-m+1, m):
p += plot(Piecewise([[(2*i-1, 2*i), -phi(-(x-2*i)) ],[(2*i, 2*i+1), phi((x-2*i))]]))
q=Graphics()
for i in xrange(-m+1, m):
q += plot(Piecewise([[(2*i-1, 2*i), -f(-(x-2*i)) ],[(2*i, 2*i+1), f((x-2*i))]]), color=hue(6/7.0))
(p+q).show()
xxxxxxxxxx
var('x, n, j')
phi(x)= x/2
a=-10;b=10
m=floor((b+1)/2)
def b(n,l):
coeff = 2/l*integral(phi(x)*sin(n*pi*x/l), (x,0,2))
return coeff
def FS(n,l):
return sum(b(j,l)*sin(j*pi*x/l), j, 1, n)
f(x)=FS(5, 1)
p=Graphics()
for i in xrange(-m+1, m):
p += plot(Piecewise([[(2*i-1, 2*i), -phi(-(x-2*i)) ],[(2*i, 2*i+1), phi((x-2*i))]]))
q=Graphics()
for i in xrange(-m+1, m):
q += plot(Piecewise([[(2*i-1, 2*i), -f(-(x-2*i)) ],[(2*i, 2*i+1), f((x-2*i))]]), color=hue(6/7.0))
(p+q).show()
xxxxxxxxxx
P.<x> = QQ[]
P_0(x)=legendre_P(0,x);P_1(x)=legendre_P(1,x)
P_2(x)=legendre_P(2,x);P_3(x)=legendre_P(3,x)
print "P_0(x)=", P_0(x)
print "P_1(x)=", P_1(x)
print "P_2(x)=", P_2(x)
print "P_3(x)=", P_3(x)
print integral(P_0(x)*P_1(x), x, -1, 1)
print integral(P_0(x)*P_2(x), x, -1, 1)
print integral(P_0(x)*P_3(x), x, -1, 1)
print integral(P_1(x)*P_2(x), x, -1, 1)
print integral(P_1(x)*P_3(x), x, -1, 1)
print integral(P_2(x)*P_3(x), x, -1, 1)
xxxxxxxxxx
P.<x> = QQ[]
H_0(x)=hermite(0,x);H_1(x)=hermite(1,x)
H_2(x)=hermite(2,x);H_3(x)=hermite(3,x)
print "H_0(x)=", H_0(x)
print "H_1(x)=", H_1(x)
print "H_2(x)=", H_2(x)
print "H_3(x)=", H_3(x)
print integral(H_0(x)*H_1(x)*exp(-x^2), x, -oo, +oo)
print integral(H_0(x)*H_2(x)*exp(-x^2), x, -oo, +oo)
print integral(H_0(x)*H_3(x)*exp(-x^2), x, -oo, +oo)
print integral(H_1(x)*H_2(x)*exp(-x^2), x, -oo, +oo)
print integral(H_1(x)*H_3(x)*exp(-x^2), x, -oo, +oo)
print integral(H_2(x)*H_3(x)*exp(-x^2), x, -oo, +oo)