526.sagews

# Mathematical Methods
# Hundreds of solved problems with numerical codes
# Lectures on Quantum Mechanics
# www1.uprh.edu/rbaretti
# www1.uprh.edu/rbaretti/LQMIntro.htm
#www1.uprh.edu/rbaretti/LQMch1.htm
#www1.uprh.edu/rbaretti/LQMch2.htm
#www1.uprh.edu/rbaretti/LQMch3.htm
#www1.uprh.edu/rbaretti/LQMch4.htm
#www1.uprh.edu/rbaretti/LQMch5.htm
# www.geocities.com/serienumerica4
# www.geocities.com/serienumerica
# www.geocities.com/serienumerica2
# www.geocities.com/serienumerica3
# www.geocities.com/mathematicalmethods
# PLOTs
y=plot(100*exp(-3*x^2),-2,2)
show(y)

# www.geocities.com/serienumerica4
# PLOTs
#y=plot(100*exp(-3*x^2),-2,2)
#show(y)

var('gama,pi,h,c,k,T')
pi=3.1416 ; h=6.63e-34 ,c=3e8 ,k=1.38e-23;T=2000;
#rho=(8*pi/c^3)* (h*gama^3)/( exp((h*gama)/(k*T))-1);
gamaf=3*k*T/h;
y=plot( (8*pi/c^3)*(h*gama^3)/( exp((h*gama)/(k*T))-1),0,gamaf)
show(y)

Traceback (most recent call last): gamaf=3*k*T/h; TypeError: 'sage.rings.real_mpfr.RealLiteral' object is not iterable

# black body radiation curve
var('gama')
pi=3.1416 ;
y=plot(gama^3/(exp(gama)-1),0,10)
show(y)

var('u');
f=(u^3)/exp(u);
integral(f,u,0,3)

6 - 78*e^-3

var('u')
f=1/(exp(u)-1)
y=plot(f,.05,2)
show(y)

y=plot(u^3/(exp(u)-1),0,15)
show(y)

integral(u^3/(exp(u)-1),0.0001,10)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/39/code/11.py", line 7, in <module> exec compile(ur'integral(u**_sage_const_3 /(exp(u)-_sage_const_1 ),_sage_const_0p0001 ,_sage_const_10 )' + '\n', '', 'single') File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/SQLAlchemy-0.4.6-py2.5.egg/", line 1, in <module> File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/misc/functional.py", line 437, in integral return x.integral(*args, **kwds) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 3083, in integral return self.parent()(self._maxima_().integrate(v, a, b)) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 508, in __call__ return self._coerce_impl(x) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 532, in _coerce_impl return symbolic_expression_from_maxima_element(x) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 10283, in symbolic_expression_from_maxima_element return symbolic_expression_from_maxima_string(x.name()) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 10261, in symbolic_expression_from_maxima_string raise TypeError, "unable to make sense of Maxima expression '%s' in Sage"%s TypeError: unable to make sense of Maxima expression '6*li[4](e^10)-60*polylog3(e^10)+300*polylog2(e^10)+1000*log(1-e^10)-6*li[4](1.000100005000167)-0.0000000000031415926428244910*I-2499.999278716501' in Sage

var('u')
y=plot(u^5/(exp(u)-1),0,15)
show(y)

integral(sin(x),x,0,pi)

2

integral((2)^(1/2)*sin(pi*x)*(2)^(1/2)*sin(2*pi*x),x,0,1)

0

integral((2)^(1/2)*sin(pi*x)*x^2*(2)^(1/2)*sin(pi*x),x,0,1)

(2*pi^2 - 3)/(6*pi^2)

var('h')
integral((2)^(1/2)*sin(pi*x)*(h^2*pi^2)*(2)^(1/2)*sin(pi*x),x,0,1)

pi^2*h^2

var('r , Z ')
Z>0
psi=(Z^3/pi)^(1/2)*exp(-Z*r)
integral(psi^2*4*pi*r^2,r,0,20/Z)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/39/code/12.py", line 10, in <module> exec compile(ur'integral(psi**_sage_const_2 *_sage_const_4 *pi*r**_sage_const_2 ,r,_sage_const_0 ,_sage_const_20 /Z)' + '\n', '', 'single') File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/SQLAlchemy-0.4.6-py2.5.egg/", line 1, in <module> File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/misc/functional.py", line 437, in integral return x.integral(*args, **kwds) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 3083, in integral return self.parent()(self._maxima_().integrate(v, a, b)) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/interfaces/maxima.py", line 1996, in integral return I(var, min, max) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 1355, in __call__ return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 1275, in function_call ['%s=%s'%(key,value.name()) for key, value in kwds.items()]))) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 1088, in new return self(code) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 1023, in __call__ return cls(self, x, name=name) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 1398, in __init__ raise TypeError, x TypeError: Computation failed since Maxima requested additional constraints (try the command 'assume(Z>0)' before integral or limit evaluation, for example): Is Z positive or negative?

var('u ')
f1=u^2 ;  f2=(u^3)/(exp(u)-1);
y1=integral(f1,u,0,.01) ; y2=integral(f2,u,0.01,10);

y1+y2

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/39/code/26.py", line 9, in <module> y1=integral(f1,u,_sage_const_0 ,_sage_const_p01 ) ; y2=integral(f2,u,_sage_const_0p01 ,_sage_const_10 ); File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/misc/functional.py", line 437, in integral return x.integral(*args, **kwds) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 3083, in integral return self.parent()(self._maxima_().integrate(v, a, b)) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 508, in __call__ return self._coerce_impl(x) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 532, in _coerce_impl return symbolic_expression_from_maxima_element(x) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 10283, in symbolic_expression_from_maxima_element return symbolic_expression_from_maxima_string(x.name()) File "/home/sage/sage_install/sage-a/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 10261, in symbolic_expression_from_maxima_string raise TypeError, "unable to make sense of Maxima expression '%s' in Sage"%s TypeError: unable to make sense of Maxima expression '6*li[4](e^10)-60*polylog3(e^10)+300*polylog2(e^10)+1000*log(1-e^10)-6*li[4](1.010050167084168)-0.0000031415926428203120*I-2499.92737700061' in Sage

var('u')
f=u^3/(exp(u)-1)
y=plot(f,0,14)
show(y)

h=6.63e-34 ; k=1.38e-23 ; c=3.00e8;
#factor= 8*pi*(k^4/(h*c)^3)
factor=2*4