Section 14.3 Planck's Distribution Law, Wien's Displacement Law and the Stefan-Boltzmann Law. Example 14.3-1

>  restart;
>  qbe:=int(qbl(lambda),lambda=0...infinity);
>  qbl:=lambda->2*Pi*c^2*h/(lambda^5*(exp(c*h/
(lambda*kappa*T))-1)); eq. 14.3-7: Planck distribution law.
>  qbe;
>  simplify(%,assume=positive);

Error, (in limit) invalid directional argument
It is obvious that Maple can not perform the integration directly 
to give a simple solution like we see in 14.3-8. So we have to solve 
the integration by changing the variable of integration.
let x=c*h/(lambda*kappa*T)
>  lambda:=c*h/(x*kappa*T);
>  qbl(lambda);  substitute the new variable x to qbl
>  dlam:=diff(lambda,x);  express dlamda in terms of dx
>  qbe:=int(qbl(lambda)*dlam,x=infinity....0);  perform the 
integration in terms of x. Note that as lambda goes from 0 to infinity, x goes from infinity to 0.
This agrees with eq. 14.3-8
>  kappa:=1.3805e-16*erg/K; h:=6.624e-27*erg*s; 
c:=2.99793e10*cm/s;
>  sigma:=evalf(qbe)/T^4; The Stefan-Boltzmann constant is given 
as 1.355e-12cal/(s*cm*cm*K*K*K*K) in Appx. C.2 of BS&L
>  erg:=0.23901E-07*cal; sigma; This checks.

>  eq:=diff(qbl(lambda),x)=0;
>  x:=solve(eq,x); Solving for the maximum value of x. BS&L 
gives 4.9651.

>  lammax:=c*h/(T*kappa*x[1]); Eq. 14.3-11 gives 0.2884 for the constant in this expression. It gives Wien's displacement Law.

>  Tsun:=solve(0.5e-4*cm=lammax,T); Estimating the surface temperature of the sun as in Example 14.3-1.

>  qbe:=sigma*Tsun^4; Estimating the emitted flux from the sun as in the same example.
>  cal:=0.396887E-02*Btu; cm:=0.032808*ft; s:=hr/3600; qbe;

Example 14.3-1 gives 2.0E07 Btu/ft^2/hr.