dist_tests.mac

(kill(all),0)
0$

%test heaviside

heaviside(-1);
0$

heaviside(1);
1$

heaviside(0);
1$

heaviside(3*x);
heaviside(x)$

%test delta

delta(-1);
0$

delta(1);
0$

delta(3*x);
delta(x)/3$

0*delta(x);
0$

delta(-3*x);
delta(x)/3$

delta(0): error
delta(0*x): error

%test doublet

doublet(-1);
0$

doublet(1);
0$

0*doublet(x);
0$

doublet(3*x);
doublet(x)/9$

doublet(-3*x);
-doublet(x)/9$

doublet(0): error
doublet(0*x): error
%note: make a few tests involving assumptions, such as doublet(a*x) with a
%assumed to be a real number

%derivative tests

diff(delta(t),t);
doublet(t)$

diff(delta(-3*t+2),t);
-3*doublet(-3*t+2)$

diff(heaviside(t),t);
delta(t)$

diff(heaviside(-3*t+2),t);
-3*delta(-3*t+2)$

diff(t^2*delta(2*t),t);
2*t*delta(2*t) + 2*t^2*doublet(2*t)$

diff(t^2*heaviside(2*t),t);
2*t*heaviside(2*t) + 2*t^2*delta(2*t)$

%indefinite integral tests

integrate(doublet(x),x);
delta(x)$

integrate(delta(x),x);
heaviside(x)$

integrate(heaviside(x),x);
x*heaviside(x)$

%definite integral tests
integrate(heaviside(x-2)*f(x),x,1,3);
integrate(f(x),x,2,3)$

integrate(heaviside(-x+4)*x^2,x,0,5);
64/3$

integrate(heaviside(x+2)*x,x,1,3);
0$

%note: make a few tests involving assumptions, such as int(h(t-c),t,a,b) with
%various assumptions on a,b,c

integrate(delta(x)*f(x),x,-1,1);
f(0)$

integrate(delta(x)*exp(2*x),x,-2,3);
1$

integrate(delta(x)*f(x),x,-2,-1);
0$

integrate(delta(x)*f(x),x,1,2);
0$

integrate(delta(2*x)*cos(x),x,-4,4);
1/2$

integrate(delta(2*x-2)*(x^2),x,-4,4);
1/2$

integrate(delta(2*x-2)*x^2,x,-3,1/2);
0$

integrate(delta(2*x-2)*x^2,x,0,1)
%see above note.  Also, here we will have to add a fifth argument to the
%integrate command to indicate a closed or open interval, and we have to
%decide which is the default.

integrate(doublet(x)*f(x),x,-1,1);
f'(0)$

integrate(doublet(x)*exp(2*x),x,-2,3);
2$

integrate(doublet(x)*f(x),x,-2,-1);
0$

integrate(doublet(x)*f(x),x,1,2);
0$

integrate(doublet(2*x)*cos(x),x,-4,4): 
integrate(doublet(2*x-2)*(x^2),x,-4-4):
 
integrate(doublet(2*x-2)*x^2,x,-3,1/2);
0$

integrate(doublet(2*x-2)*x^2,x,0,1)
%see notes about assumptions and open/closed interval

%Verify abstract linearity
laplace(f(t)+g(t),t,s) : laplace(f(t),t,s) + laplace(g(t),t,s)
laplace(a*f(t),t,s): a*laplace(f(t),t,s)
laplace(a*f(t)+b*g(t),t,s): a*laplace(f(t),t,s) + b*laplace(g(t),t,s)

%Verify "naked" functions laplace transforms
sin(5*t)
sin(a*t)
sin(5*t+1)
sin(a*t+b)
sinh(5*t)
sinh(a*t)
sinh(5*t+1)
sinh(a*t+b)
cos(6*t)
cos(a*t)
cos(5*t+1)
cos(a*t+b)
cosh(5*t)
cosh(a*t)
cosh(5*t+1)
cosh(a*t+b)
exp(-2*t)
exp(a*t)
log(3*t)
log(a*t)
erf(2*t)
erf(a*t)
erf(2*t+3) (question: do we want a*t+b, or just a*t?)
erf(a*t+b)
bessel_j(3,4*t)
bessel_j(n,a*t)
heaviside(3*t)
heaviside(2*t+4)
heaviside(-2*t+4)
heaviside(a*t+b) %for various assumptions
delta(3*t)
delta(2*t+4)
delta(-2*t+4)
delta(a*t+b) %for various assumptions
doublet(3*t)
doublet(2*t+4)
doublet(-2*t+4)
doublet(a*t+b) %for various assumptions

%think up more laplace tests involving products, exponentials, derivatives,
%integrals and invlaps

%inverse laplace tests needed

%desolve tests needed