# MATHEMATICAL METHODS OF Science and Engineering
# Lectures on Quantum Mechanics
#http://www1.uprh.edu/rbaretti
#http://www1.uprh.edu/rbaretti/Methodsoftheoreticalphysics.htm
#http://www1.uprh.edu/rbaretti/MethodsoftheoreticalphysicsPart2.htm
#http://www1.uprh.edu/rbaretti/MethodsoftheoreticalphysicsPart3.htm
#http://www1.uprh.edu/rbaretti/MethodsoftheoreticalphysicsPart4.htm
#http://www1.uprh.edu/rbaretti/MethodsoftheoreticalphysicsPart5.htm
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sage: g = Bessel(2); g
J_{2}
sage: print g
J-Bessel function of order 2
sage: g.plot(0,10)
J-Bessel function of order 2 J-Bessel function of order 2 |
a=2/3
print n(a)
0.666666666666667 0.666666666666667 |
print n(pi)
3.14159265358979 3.14159265358979 |
var('y,x');
integral(y*exp(-y),y)
-(y + 1)*e^(-y) -(y + 1)*e^(-y) |
var('x,y'); f=(1+x*y)*sin(y);
integral(f,y,0,x)
Traceback (click to the left of this block for traceback) ... Is x positive, negative, or zero? Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/1/code/2.py", line 8, in <module>
integral(f,y,_sage_const_0 ,x)
File "", line 1, in <module>
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/misc/functional.py", line 416, in integral
return x.integral(*args, **kwds)
File "expression.pyx", line 5962, in sage.symbolic.expression.Expression.integral (sage/symbolic/expression.cpp:24542)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/calculus/calculus.py", line 566, in integral
result = expression._maxima_().integrate(v, a, b)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/interfaces/maxima.py", line 2003, in integral
return I(var, min, max)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1390, in __call__
return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1298, in function_call
return self.new(s)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1094, in new
return self(code)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1029, in __call__
return cls(self, x, name=name)
File "/home/sage/sage_install/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1433, in __init__
raise TypeError, x
TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(x>0)' before integral or limit evaluation, for example):
Is x positive, negative, or zero?
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f(x)= x*(sin(x)-x*cos(x))-cos(x) +1 ;
diff(f,x)
x |--> x^2*sin(x) - x*cos(x) + 2*sin(x) x |--> x^2*sin(x) - x*cos(x) + 2*sin(x) |
var('x');
f2=x-1/2; integral(f2^2,x,0,1)
1/12 1/12 |
phi1=(12)^(1/2)*(x-1/2);integral(phi1^2,x,0,1)
1 1 |
f2=x^2; phi0=1; phi1=(12)^(1/2)*(x-1/2);
#integral(f2*phi0,x,0,1)
phi2=x^2-(1/3)*phi0-(sqrt(3)/6)*phi1;
#phi2=sqrt(540/23)*(x^2-(1/3)*phi0-(1/(6*sqrt(3)))*phi1 );
#integral(phi1*phi2,x,0,1)
phi2=sqrt(180)*(x^2-(sqrt(3)/6)*phi1-1/3);
integral(phi1*phi2,x,0,1)
0 0 |
a=1/6*sqrt(3); n(a)
0.288675134594813 0.288675134594813 |
#data g0,g1,g2 ,t0,t1,t2/8.,14.,8.,0.,50.,100./
var('t');g0=8;g1=14;g2=8;t0=0;t1=50;t2=100;
gama=g0*(t-t1)*(t-t2)/((t0-t1)*(t0-t2))+g1*(t-t0)*(t-t2)/((t1-t0)*(t1-t2))+g2*(t-t0)*(t-t1)/((t2-t0)*(t2-t1));
y=plot(gama,t,0,100)
show(y)
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var('r');
integral(4*pi*exp(-2*r)*r^2,r,0,oo)
pi pi |
s,t=var('s,t')
x(s)=exp(-s)
integral(x(s)*exp(s*t),t,0,1)
-e^(-s)/s + 1/s -e^(-s)/s + 1/s |
f(x)=(1/x)*(1 - exp(-x) ) ,p0(x) = 1; p1(x)= x; p2(x)=(1/2)*(3*x^2-1)
line 3 SyntaxError: keyword can't be an expression (, line 3) line 3 SyntaxError: keyword can't be an expression (, line 3) |
var('Z'); assume(Z>0);
phi(r)=(Z^3/pi)^(1/2)*exp(-Z*r);
integral(4*pi*r^4*phi(r)^2,r,0,oo)
3/Z^2 3/Z^2 |
var('r,theta');assume(r>0);
#f(theta)=(1/(pi*r))*(1-cos(theta))^(-1/2);
f(theta)=1/(pi*r)*(1-cos(theta))^(-1/2);
delta=pi/180;
integral(f(theta),theta,delta,pi)
integrate(1/sqrt(-cos(theta) + 1), theta, 1/180*pi, pi)/(pi*r) integrate(1/sqrt(-cos(theta) + 1), theta, 1/180*pi, pi)/(pi*r) |
var('Z,r'); assume (Z>0);
v(r)=1/r - ( Z +1/r )*exp(-2*Z*r);
phi1s(r)=(Z^3/pi)^(1/2)*exp(-Z*r);
phi2s(r)=(Z^3/(32*pi))^(1/2)*(2 - Z*r )*exp(-Z*r/2);
integral(phi1s(r)*phi2s(r)*v(r)*4*pi*r^2,r,0,oo)
4096/64827*sqrt(2)*Z 4096/64827*sqrt(2)*Z |
var('Z,r1,r2'); assume (Z>0);assume(r1>0);
phi1s(r)=(Z^3/pi)^(1/2)*exp(-Z*r);
phi2s(r)=(Z^3/(32*pi))^(1/2)*(2 - Z*r )*exp(-Z*r/2);
va=(1/r1)*integral(phi2s(r)^2*4*pi*r^2,r,0,r1)
#va
vb=integral(phi2s(r)^2*4*pi*r,r,r1,oo)
#vb
J=integral(phi1s(r1)^2*(va+vb)*4*pi*r1^2,r1,0,oo)
J
Traceback (click to the left of this block for traceback) ... Is r1 positive, negative, or zero? Traceback (most recent call last): vb=integral(phi2s(r)^2*4*pi*r,r,r1,oo)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/functional.py", line 568, in integral
return x.integral(*args, **kwds)
File "expression.pyx", line 6097, in sage.symbolic.expression.Expression.integral (sage/symbolic/expression.cpp:25344)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/calculus/calculus.py", line 637, in integral
result = expression._maxima_().integrate(v, a, b)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/maxima.py", line 2025, in integral
return I(var, min, max)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1404, in __call__
return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1312, in function_call
return self.new(s)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1096, in new
return self(code)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1031, in __call__
return cls(self, x, name=name)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1447, in __init__
raise TypeError, x
TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(r1>0)' before integral or limit evaluation, for example):
Is r1 positive, negative, or zero?
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-1/8*((Z^4*r1^4 + 4*Z^2*r1^2 + 8*Z*r1 + 8)*e^(-Z*r1)/Z^3 - 8/Z^3)*Z^3/r1 -1/8*((Z^4*r1^4 + 4*Z^2*r1^2 + 8*Z*r1 + 8)*e^(-Z*r1)/Z^3 - 8/Z^3)*Z^3/r1 |
var('Z,r1,r2'); assume (Z>0);assume(r1>0);
phi1s(r)=(Z^3/pi)^(1/2)*exp(-Z*r);
phi2s(r)=(Z^3/(32*pi))^(1/2)*(2 - Z*r )*exp(-Z*r/2);
va=-(1/8)*((Z^4*r^4 + 4*Z^2*r^2 + 8*Z*r + 8)*e^(-Z*r)/Z^3 - 8/Z^3)*Z^3/r ;
vb= (1/8)*(Z^3*r^3 - Z^2*r^2 + 2*Z*r + 2)*Z*e^(-Z*r);v=va+vb;
J=integral(phi1s(r)*phi2s(r)*v*4*pi*r^2,r,0,oo)
J
512/84375*sqrt(2)*Z 512/84375*sqrt(2)*Z |
n( 512*sqrt(2)/84375)
0.00858165740960029 0.00858165740960029 |
var('z'); assume(r1>0);assume(Z>0);
phi1s=(Z^3/pi)^(1/2)*exp(-Z*r);
phi2s=(Z^3/(32*pi))^(1/2)*(2 - Z*r )*exp(-Z*r/2);
va=(1/r1)*integral(4*pi*r^2*phi1s,r,0,r1)
va
#vb(r)= integral(4*pi*r1*phi1s(r1)^2,r1,r,oo);
#J1s1s=integral(phi1s(r)^2*(va(r)+vb(r))*4*pi*r^2,r,0,oo)
#J1s1s
va
Traceback (click to the left of this block for traceback) ... Is r1 positive, negative, or zero? Traceback (most recent call last): va
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/functional.py", line 568, in integral
return x.integral(*args, **kwds)
File "expression.pyx", line 6097, in sage.symbolic.expression.Expression.integral (sage/symbolic/expression.cpp:25344)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/calculus/calculus.py", line 637, in integral
result = expression._maxima_().integrate(v, a, b)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/maxima.py", line 2025, in integral
return I(var, min, max)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1404, in __call__
return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1312, in function_call
return self.new(s)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1096, in new
return self(code)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1031, in __call__
return cls(self, x, name=name)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1447, in __init__
raise TypeError, x
TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(r1>0)' before integral or limit evaluation, for example):
Is r1 positive, negative, or zero?
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var('z r');assume(z>0); phi2p=(z^5/(96*pi))^(1/2)*r*exp(-z*r/2);
integral(phi2p^2*4*pi*r^2,r,0,oo);
1 1 |
var('z r1 r'); assume(z>0);
phi1s=(z^3/pi)^(1/2)*exp(-z*r);phi2p=(z^5/(96*pi))^(1/2)*r*exp(-z*r/2);
phi2s=(z^3/(32*pi))^(1/2)*(2 - z*r )*exp(-z*r/2);
va=(1/r1)*integral(4*pi*r^2*phi2p^2,r,0,r1);
#vb= integral(4*pi*r*phi2p^2,r,r1,oo);
#phi1s=(z^3/pi)^(1/2)*exp(-z*r1);
#integral((va+vb)*4*pi*r1^2,r1,0,oo);
va
Traceback (click to the left of this block for traceback) ... Is r1 positive, negative, or zero? Traceback (most recent call last): #phi1s=(z^3/pi)^(1/2)*exp(-z*r1);
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/functional.py", line 568, in integral
return x.integral(*args, **kwds)
File "expression.pyx", line 6097, in sage.symbolic.expression.Expression.integral (sage/symbolic/expression.cpp:25344)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/calculus/calculus.py", line 637, in integral
result = expression._maxima_().integrate(v, a, b)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/maxima.py", line 2025, in integral
return I(var, min, max)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1404, in __call__
return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1312, in function_call
return self.new(s)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1096, in new
return self(code)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1031, in __call__
return cls(self, x, name=name)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1447, in __init__
raise TypeError, x
TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(r1>0)' before integral or limit evaluation, for example):
Is r1 positive, negative, or zero?
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var('a');
assume(a>0);
integral(exp(-x^2/a),x,0,oo)
1/2*sqrt(pi)*sqrt(a) 1/2*sqrt(pi)*sqrt(a) |
n( (1/2)*sqrt(pi))
0.886226925452758 0.886226925452758 |
(2/sqrt(pi))*integral(exp(-x^2),x,0,oo)
1 1 |
# exchange term K(1s,2s)
var('z ,r1, r');
assume(z>0);
assume(r1>0);
phi1s=(z^3/pi)^(1/2)*exp(-z*r);
phi2p=(z^5/(96*pi))^(1/2)*r*exp(-z*r/2);
phi2s=(z^3/(32*pi))^(1/2)*(2 - z*r )*exp(-z*r/2);
va=(1/r1)*integral(4*pi*r^2*phi1s*phi2s,r,0,r1)
va
#vb= integral(4*pi*r*phi1s*phi2s,r,r1,oo);
#phi1s=(z^3/pi)^(1/2)*exp(-z*r1);phi2p=(z^5/(96*pi))^(1/2)*r1*exp(-z*r1/2);
#phi2s=(z^3/(32*pi))^(1/2)*(2 - z*r1 )*exp(-z*r1/2);
#assume(r1>0);
#integral(phi1s*phi2s*(va+vb)*4*pi*r1^2,r1,0,oo)
Traceback (click to the left of this block for traceback) ... Is r1 positive, negative, or zero? Traceback (most recent call last): va
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/functional.py", line 687, in integral
return x.integral(*args, **kwds)
File "expression.pyx", line 6495, in sage.symbolic.expression.Expression.integral (sage/symbolic/expression.cpp:25927)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/calculus/calculus.py", line 824, in integral
result = expression._maxima_().integrate(v, a, b)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/maxima.py", line 2058, in integral
return I(var, min, max)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1408, in __call__
return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1316, in function_call
return self.new(s)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1097, in new
return self(code)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1032, in __call__
return cls(self, x, name=name)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1451, in __init__
raise TypeError, x
TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(r1>0)' before integral or limit evaluation, for example):
Is r1 positive, negative, or zero?
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var('a');Z=2;
E = ( a^2 /2 -Z* a) + (1/4) *( a^2 /2 -Z*a) + (17/81)* a -(16/729)* a;
y=plot(E,a,1.4,2.2)
show(y)
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n(112/2187)
0.0512117055326932 0.0512117055326932 |
var('a');z=3;
E =(9/4)* ( a^2 /2 - z* a) +(5/8)*a + 2 *(17/81)*a -(16/729)* a ;
y=plot(E,a,2.0,3.1)
show(y)
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var('a1 a2 r1 r');
#assume(a1>0);
#assume(a2>0);
assume (r>0);
assume(r1>0);
phi1s(r)=(a1^3/pi)^(1/2)*exp(-a1*r);
phi2s(r)=(a2^3/(32*pi))^(1/2)*(2 - a2*r )*exp(-a2*r/2);
va(r1)=(1/r1)*integral(4*pi*r^2*phi2s(r)^2,r,0,r1)
vb(r1)= integral(4*pi*r*phi2s(r)^2,r,r1,oo);
integral(phi1s(r)^2*(va(r)+vb(r))*4*pi*r^2,r,0,oo)
(8*a1^4*a2 + 20*a1^3*a2^2 + 12*a1^2*a2^3 + 10*a1*a2^4 + a2^5)*a1^3/(32*a1^7 + 80*a1^6*a2 + 80*a1^5*a2^2 + 40*a1^4*a2^3 + 10*a1^3*a2^4 + a1^2*a2^5) (8*a1^4*a2 + 20*a1^3*a2^2 + 12*a1^2*a2^3 + 10*a1*a2^4 + a2^5)*a1^3/(32*a1^7 + 80*a1^6*a2 + 80*a1^5*a2^2 + 40*a1^4*a2^3 + 10*a1^3*a2^4 + a1^2*a2^5) |
var(' a2 ');
z=3;
a1=z-5/16;
j1s1s=(5/8)*a1;
k1s2s=-(16/729)*a1;
j1s2s(a2)=(8*a1**4*a2 + 20*a1**3*a2**2 + 12*a1**2*a2**3 +10*a1*a2**4 +a2**5)*a1**3/(32*a1**7 + 80*a1**6*a2 + 80*a1**5*a2**2 + 40*a1**4*a2**3 +10*a1**3*a2**4 + a1**2*a2**5);
e0(a2)=2.*(a1**2/2.-z*a1)+(1./4.)*(a2**2/2.-z*a2);
E=e0(a2) +j1s1s+2.*j1s2s(a2)+k1s2s;
y=plot(E,a2,.8,2.0);
show(y)
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var('a');
Z=4;
E= (5/2)*( a^2 /2 - Z*a)+ (5/8)*a +4*(17/81)*a +(77/512)*a - 2*(16/729)*a ;
y=plot(E,a,2.6,4);
show(y)
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var('a1 a2 r1 r');
# integral of J(2s,2s)
#assume(a2>0);
assume (r>0);
assume(r1>0);
phi1s(r)=(a1^3/pi)^(1/2)*exp(-a1*r);
phi2s(r)=(a2^3/(32*pi))^(1/2)*(2 - a2*r )*exp(-a2*r/2);
va(r1)=(1/r1)*integral(4*pi*r^2*phi2s(r)^2,r,0,r1)
vb(r1)= integral(4*pi*r*phi2s(r)^2,r,r1,oo);
integral(phi1s(r)^2*(va(r)+vb(r))*4*pi*r^2,r,0,oo)
Traceback (click to the left of this block for traceback) ... Is a2 positive, negative, or zero? Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_2.py", line 14, in <module>
__tmp__=var("r1"); vb = symbolic_expression(integral(_sage_const_4 *pi*r*phi2s(r)**_sage_const_2 ,r,r1,oo)).function(r1);
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/functional.py", line 687, in integral
return x.integral(*args, **kwds)
File "expression.pyx", line 6495, in sage.symbolic.expression.Expression.integral (sage/symbolic/expression.cpp:25927)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/calculus/calculus.py", line 824, in integral
result = expression._maxima_().integrate(v, a, b)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/maxima.py", line 2058, in integral
return I(var, min, max)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1408, in __call__
return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1316, in function_call
return self.new(s)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1097, in new
return self(code)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1032, in __call__
return cls(self, x, name=name)
File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1451, in __init__
raise TypeError, x
TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(a2>0)' before integral or limit evaluation, for example):
Is a2 positive, negative, or zero?
|
var('a2');
Z=4;a1=Z-5/16;
j1s2s(a2)=(8*a1**4*a2 + 20*a1**3*a2**2 + 12*a1**2*a2**3 +10*a1*a2**4 +a2**5)*a1**3/(32*a1**7 + 80*a1**6*a2 + 80*a1**5*a2**2 + 40*a1**4*a2**3 +10*a1**3*a2**4 + a1**2*a2**5);
E= 2*( a1^2 /2 - Z*a1)+ 2*(1/4)*( a2^2 /2 - Z*a2)+(5/8)*a1 +4*j1s2s(a2) +(77/512)*a2 - 2*(16/729)*a1 ;
y=plot(E,a2,1.0,3.0);
show(y)
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|
var('a2');
Z=4;a1=Z-5/16;
j1s2s(a2)=(8*a1**4*a2 + 20*a1**3*a2**2 + 12*a1**2*a2**3 +10*a1*a2**4 +a2**5)*a1**3/(32*a1**7 + 80*a1**6*a2 + 80*a1**5*a2**2 + 40*a1**4*a2**3 +10*a1**3*a2**4 + a1**2*a2**5);
E= 2*( a1^2 /2 - Z*a1)+ (1)*(1/4)*( a2^2 /2 - Z*a2)+(5/8)*a1 +2*j1s2s(a2) +0.*(77/512)*a2 - 1*(16/729)*a1 ;
y=plot(E,a2,2,3);
show(y)
|
|
var('Z,r');
Z=2;
#V=(1/r)*(1-exp(-2*Z*r) ) - Z*exp(-2*Z*r);
V=r;
y=plot(V,r,1,4);
show(y)
Traceback (click to the left of this block for traceback) ... SyntaxError: Non-ASCII character '\xc2' in file /tmp/tmpjLDYm8/___code___.py on line 4, but no encoding declared; see http://www.python.org/peps/pep-0263.html for details Traceback (most recent call last): Z=2;
File "", line 1, in <module>
File "/tmp/tmpjLDYm8/___code___.py", line 4
SyntaxError: Non-ASCII character '\xc2' in file /tmp/tmpjLDYm8/___code___.py on line 4, but no encoding declared; see http://www.python.org/peps/pep-0263.html for details
|
var('r');Z=2;
v=(1/r)*(1-exp(-2*Z*r) ) - Z*exp(-2*Z*r);
y=plot(v,r,0.0001,5);
show(y)
|
|
var ('x');
V(x) = -0.0066*x**3 + 0.0967*x**2 - 0.4768*x + 0.975;
y=plot(V(x),x,0,5);
show(y)
|
|
var('a ,z');
assume(z>0);
assume(a>0);
f1(x)=(4*z^3)^(1/2)*x*exp(-z*x);
f2(x)=(8*z/5)^(1/2)*(1-(3*z/4)*x)*exp(-z*x/2);
#integral(f1(x)*f2(x),x,0,oo)
d2f2(x)=diff(f2(x),x,2);
#d2f1(x)=diff(f1(x),x,2);
#integral(f1(x)*(-1/2)*d2f1(x),x,0,oo)
#integral((-z/x)*f1(x)^2,x,0,oo)
integral(f2(x)*(-1/2)*d2f2(x),x,0,oo)
-11/40*z^2 -11/40*z^2 |
var('a');
assume (a>0);
f1(x)=2*x*exp(-x);
f2(x)=(1/8^(1/2))*(2-x)*x*exp(-x/2);
integral(f2(x)^2,x,0,oo)
1 1 |
k(x)=-(1/2)*diff(f1(x),x,2) -(1/x)*f1(x)
-x*e^(-x) -x*e^(-x) |
var ('x');
f1(x) = 2*x*exp(-x);
f2(x)=(1/8^(1/2))*(2-x)*x*exp(-x/2)
y=plot(f2(x),x,0,16);
show(y)
|
|
var('x,a');
assume(a>0);
#f(x)=exp(-x^2)/(1+x^2);
f(x)=exp(-x^4);
integral(f(x),x,-1,+1)
-1/2*I*gamma_incomplete(1/4, 1) + 1/2*I*gamma(1/4) -1/2*I*gamma_incomplete(1/4, 1) + 1/2*I*gamma(1/4) |
k=9E9 ;
sigma=1/k;
pia=3.1416
integral(k*sigma*2*pia*r/(1.1^2+r^2)^(1/2),r,0,10)
56.2994706006 56.2994706006 |
n(6.2832*sqrt(101) - 6.2832)
56.8621785026268 56.8621785026268 |
n(-(56.29-56.86)/.1)
5.70000000000000 5.70000000000000 |
var(' a');
assume(a>0);
f(x)=1/(1+a*x^2);
integral(f(x),x,-oo,oo);
pi/sqrt(a) pi/sqrt(a) |
var('a b c');
f(x)=x^3 +2*a*x^2+b*x +c;
d2f(x)= diff(f(x),x,2);
x=1;
d2f(x)
4*a + 6 4*a + 6 |
integrate(exp(-y^2)/(1+y^2)^(1/2),y,-oo,oo)
integrate(e^(-y^2)/sqrt(y^2 + 1), y, -Infinity, +Infinity) integrate(e^(-y^2)/sqrt(y^2 + 1), y, -Infinity, +Infinity) |
search_doc("definite integral")
|
% fortran
!f90
do i=1,5
print*,'float(i)',float(i)
enddo
Traceback (click to the left of this block for traceback) ... SyntaxError: invalid syntax Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_5.py", line 9, in <module>
open("___code___.py","w").write("# -*- coding: utf-8 -*-\n" + _support_.preparse_worksheet_cell(base64.b64decode("JSBmb3J0cmFuCiFmOTAKZG8gIGk9MSw1CnByaW50KiwnZmxvYXQoaSknLGZsb2F0KGkpCmVuZGRv"),globals())+"\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpOu6Si0/___code___.py", line 3
% fortran
^
SyntaxError: invalid syntax
|
%fortran
C FILE: FIB1.F
SUBROUTINE FIB(A,N)
C
C CALCULATE FIRST N FIBONACCI NUMBERS
C
INTEGER N
REAL*8 A(N)
DO I=1,N
IF (I.EQ.1) THEN
A(I) = 0.0D0
ELSEIF (I.EQ.2) THEN
A(I) = 1.0D0
ELSE
A(I) = A(I-1) + A(I-2)
ENDIF
ENDDO
END
C END FILE FIB1.F
Traceback (click to the left of this block for traceback) ... OSError: [Errno 2] No such file or directory: 'fortran_module_1.so' Traceback (most recent call last):
File "<string>", line 1, in <module>
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/f2py/f2py2e.py", line 557, in main
run_compile()
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/f2py/f2py2e.py", line 543, in run_compile
setup(ext_modules = [ext])
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/distutils/core.py", line 184, in setup
return old_setup(**new_attr)
File "/usr/local/sage2/local/lib/python/distutils/core.py", line 152, in setup
dist.run_commands()
File "/usr/local/sage2/local/lib/python/distutils/dist.py", line 975, in run_commands
self.run_command(cmd)
File "/usr/local/sage2/local/lib/python/distutils/dist.py", line 995, in run_command
cmd_obj.run()
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/distutils/command/build.py", line 37, in run
old_build.run(self)
File "/usr/local/sage2/local/lib/python/distutils/command/build.py", line 134, in run
self.run_command(cmd_name)
File "/usr/local/sage2/local/lib/python/distutils/cmd.py", line 333, in run_command
self.distribution.run_command(command)
File "/usr/local/sage2/local/lib/python/distutils/dist.py", line 995, in run_command
cmd_obj.run()
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/distutils/command/build_src.py", line 130, in run
self.build_sources()
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/distutils/command/build_src.py", line 147, in build_sources
self.build_extension_sources(ext)
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/distutils/command/build_src.py", line 256, in build_extension_sources
sources = self.f2py_sources(sources, ext)
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/distutils/command/build_src.py", line 514, in f2py_sources
['-m',ext_name]+f_sources)
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/f2py/f2py2e.py", line 338, in run_main
postlist=callcrackfortran(files,options)
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/f2py/f2py2e.py", line 276, in callcrackfortran
postlist=crackfortran.crackfortran(files)
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/f2py/crackfortran.py", line 2683, in crackfortran
readfortrancode(files,crackline)
File "/usr/local/sage2/local/lib/python2.6/site-packages/numpy/f2py/crackfortran.py", line 346, in readfortrancode
'this code is in fix form?\n\tline=%s' % `l`)
Exception: readfortrancode: Found non-(space,digit) char in the first column.
Are you sure that this code is in fix form?
line='SUBROUTINE FIB(A,N)'
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_6.py", line 26, in <module>
C END FILE FIB1.F''', '/sagenb/sagenb/sage_notebook.sagenb/home/reibaretti/42/cells/54')
File "/usr/local/sage2/sagenb-0.7.4/src/sagenb/sagenb/misc/support.py", line 473, in syseval
return system.eval(cmd, sage_globals, locals = sage_globals)
File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/misc/inline_fortran.py", line 92, in eval
os.unlink(name + '.so')
OSError: [Errno 2] No such file or directory: 'fortran_module_1.so'
|
f(x)=x^2*e^(-3*x^3)/(1+x)
a=0
b=infinity
integrate(f,a,b)
integrate(x^2*e^(-3*x^3)/(x + 1), x, 0, +Infinity) integrate(x^2*e^(-3*x^3)/(x + 1), x, 0, +Infinity) |
f(x)=x^2*exp(-x^2)/(1+x^2)
a=0
b=5
print integrate(f(x),x,a,b)
integrate(x^2*e^(-x^2)/(x^2 + 1), x, 0, 5) integrate(x^2*e^(-x^2)/(x^2 + 1), x, 0, 5) |
var('p, a')
assume(p>0)
assume (a>0)
f(r)=exp(-a*r)*sin(p*r)
integrate(f(r),r,0,oo)
p/(a^2 + p^2) p/(a^2 + p^2) |
#(2.**(3./2.)/pi)/(p**2+1.)**2
f(p)=(2^(3/2)/pi)/(p^2+1)^2
tk=(1/2)*p^2
integrate(4*pi*p^2*tk*f(p)^2,p,0,oo)
1/2 1/2 |
n(5/(4*pi))
0.397887357729738 0.397887357729738 |
f(p)=(2^(3/2)/pi)/(p^2+1)^2
tk=(1/2)*p^2
integrate(4*pi*p^2*f(p)^2,p,0,oo)
1 1 |
var('Z,p')
assume(Z>0)
assume(p>0)
f(r)= 4*pi*(Z^3/pi )^(1/2)*(sin(p*r) /p)* exp(-Z*r)*r
#integrate(f(r),r,0,oo)
fp(p)=8*sqrt(pi)*Z^(5/2)/(Z^4 + 2*Z^2*p^2 + p^4)
integrate(4*pi*p^2*fp(p)^2,p,0,oo)
8*pi^3 8*pi^3 |
var('Z,p')
assume(Z>0)
fp(p)=8*sqrt(pi)*Z^(5/2)/(Z^4 + 2*Z^2*p^2 + p^4)
integrate(4*pi*p^2*fp(p)^2,p,0,oo)
8*pi^3 8*pi^3 |
n(8*pi^3)
248.050213442399 248.050213442399 |
var('x,y');
v(x,y)=x^2 + 3*x*y^2;
ex(x,y)=-diff(v(x,y),x);
x=1; y=2;
ex(x,y)
-14 -14 |
f(p)=exp(-p^2);
integral(f(p)^2*4*pi*p^2,p,0,oo)
1/4*pi^(3/2)*sqrt(2) 1/4*pi^(3/2)*sqrt(2) |
f(p)= (2/pi)^(3/4)*exp(-p^2)
integral(f(p)^2*4*pi*p^2,p,0,oo)
1 1 |
g2(p)=exp(-2*p^2)
g1n(p)=(2/pi)^(3/4)*exp(-p^2)
integral(g2(p)*g1n(p)*4*pi*p^2,p,0,oo)
1/9*pi^(3/4)*2^(3/4)*sqrt(3) 1/9*pi^(3/4)*2^(3/4)*sqrt(3) |
g2(p)=exp(-2*p^2)
g1n(p)=(2/pi)^(3/4)*exp(-p^2)
u2(p)=g2(p)-(1/9)*pi^(3/4)*2^(3/4)*sqrt(3)*g1n(p)
#integral(u2(p)*g1n(p)*4*pi*p^2,p,0,oo)
integral(u2(p)^2*4*pi*p^2,p,0,oo)
1/432*(27*sqrt(2) - 32)*pi^(3/2)*sqrt(2) 1/432*(27*sqrt(2) - 32)*pi^(3/2)*sqrt(2) |
n(1/432*(27*sqrt(2) - 32)*pi^(3/2)*sqrt(2))
0.112722112725355 0.112722112725355 |
n((1/9)*pi^(3/4)*2^(3/4)*sqrt(3)/sqrt(0.112722112725355) )
2.27482744465886 2.27482744465886 |
n(1/sqrt(0.112722112725355 ))
2.97848515558980 2.97848515558980 |
u2n(p)=2.97848515558980*exp(-2*p^2) -2.27482744465886*exp(-p^2)
integral(u2n(p)^2*4*pi*p^2,p,0,oo)
1/36*((6.65353036655*sqrt(2) + 15.5245197089)*sqrt(3) - 27.1021591018*sqrt(2))*pi^(3/2)*sqrt(2)*sqrt(3) 1/36*((6.65353036655*sqrt(2) + 15.5245197089)*sqrt(3) - 27.1021591018*sqrt(2))*pi^(3/2)*sqrt(2)*sqrt(3) |
n((1/36)*((6.65353036655*sqrt(2) + 15.5245197089)*sqrt(3) -
27.1021591018*sqrt(2))*pi^(3/2)*sqrt(2)*sqrt(3))
|
|
n(1.621282/2.978485)
0.544331094499385 0.544331094499385 |
g1(p)=(2/pi)^3/4 *exp(-p^2 )
g2(p)=2.978485*exp(-2*p^2) -1.621282*exp(-p^2 )
dv(p)=4*pi*p^2
integral(g2(p)^2*dv(p),p,0,10)
|
|
var('Z');
phi(p)= (1/(8*pi^3)^(1/2))*8*sqrt(pi)*Z^(5/2)/(Z^4 + 2*Z^2*p^2 + p^4);phi(p)
2*sqrt(2)*Z^(5/2)/((Z^4 + 2*Z^2*p^2 + p^4)*pi) 2*sqrt(2)*Z^(5/2)/((Z^4 + 2*Z^2*p^2 + p^4)*pi) |
Z=1;
phi(p)= (2^(3/2)/pi)*Z^(5/2)/(p^2+Z^2)^2
integral(phi(p)^2*4*pi*p^2,p,0,oo)
1 1 |
x = var('x')
g1 = plot(cos(20*x)*exp(-2*x), 0, 1)
g2 = plot(2*exp(-30*x) - exp(-3*x), 0, 1)
show(graphics_array([g1, g2],1,1), xmin=0)
|
x=var('x')
g1=(2/pi)^(3/4)*exp(-x^2 )
g2=2.978485*exp(-2*x^2 ) -1.621282*exp(-x^2)
phi1= plot((2^(3/2) /pi) /(x^2 + 1 )^2,0,3)
phi2= plot( g1 + 8.26E-2 * g2,0,3)
show(graphics_array([phi1, phi2],1,2), xmin=0)
Traceback (click to the left of this block for traceback) ... SyntaxError: invalid syntax Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_12.py", line 9, in <module>
open("___code___.py","w").write("# -*- coding: utf-8 -*-\n" + _support_.preparse_worksheet_cell(base64.b64decode("eD12YXIoJ3gnKQpnMT0oMi9waSleKDMvNCkqZXhwKC14XjIgKQpnMj0yLjk3ODQ4NSpleHAoLTIqeF4yICkgLTEuNjIxMjgyKmV4cCgteF4yKQpwaGkxPSBwbG90KCgyXigzLzIpIC9waSkgLyh4XjIgKyAxICleMiwwLDMpIApwaGkyPSBwbG90KCBnMSArIDguMjZFLTIgKiBnMiwwLDMpwqAKc2hvdyhncmFwaGljc19hcnJheShbcGhpMSwgcGhpMl0sMSwyKSwgeG1pbj0wKQ=="),globals())+"\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpO8ZaEr/___code___.py", line 7
phi2= plot( g1 + _sage_const_8p26En2 * g2,_sage_const_0 ,_sage_const_3 )
^
SyntaxError: invalid syntax
|
var('p')
g1n(p)=(2/pi)^(3/4)*exp(-p^2 )
u2(p)= p*exp(-p^2)
dv(p)=4*pi*p^2
g2(p)=
2.11661285743602
*(u2(p)-1.11951513492025*g1n(p))
norm=integral(g2(p)^2*dv(p),p,0,oo);n(norm)
0.999999999999996 0.999999999999996 |
var('p')
g1n(p)=(2/pi)^(3/4)*exp(-p^2 )
g2n(p)= 2.116613*(p*exp(-p^2 )-1.119515*(2/pi)^(3/4)*exp(-p^2 ) )
dv(p)=4*pi*p^2
norm=integral(g2n(p)*g1n(p)*dv(p),p,0,oo); n(norm)
2.85573950264961e-7 2.85573950264961e-7 |
A=matrix([[ -0.423944443 , 0.0170494914 ],[ 0.0170494914 , 0.0841127038 ]])
A.eigenvalues()
[0.0846842114303050, -0.424515950630305] [0.0846842114303050, -0.424515950630305] |
|
|