convolution theorema:f*g (Laplace Transform)
peter.vlasschaert@gmail.com, 28/09/2017

kill(all)$

definition :
 =>     Laplace Transform. ( = L{f(t)}) = F(s))

'laplace(f(t),t,s) = 'integrate(f(t)*exp(-s*t),t,0,inf);

 ? Find F(s),G(s)

q1:f(u)*exp(-s*u);
q11:part(q1,2);
q2:g(v)*exp(-s*v);
q22:part(q2,2);
q3:part(q1,1)*part(q2,1);
q4: factor(part(q3,2));
q5:part(q3,1)^q4;

p1:laplace(f(u),u,s) ='integrate(q1,u,0,inf);
p2:F(s) = lhs(p1);
p3:laplace(g(v),v,s) ='integrate(q2,v,0,inf);
p4:G(s) = lhs(p3);

find product : F(s)*G(s)

p5:lhs(p2) = rhs(p1);
p6:lhs(p4) = rhs(p3);
p7:p5*p6;

double integral : construction. ( to find u:u1,u2,t:t1,t2 )

p8:lhs(p7) ='integrate(integrate(q1*q2,u,0,inf),v,0,inf);

uv plane   => u,t coordinates

convert :  du*dv    =>     du*dt   : du*dv =det( jacobean([u,v],[u,t]))*du*dt

p9:t=u+v;
p10: u = u;
p11: v = t-u;
p12: jacobian([rhs(p10),rhs(p11)],[u,t]);
p13:determinant(p12);

p14:ev(q5,u+v=t);
p15:ev(q11*q22,p11);

p16:lhs(p7)='integrate(integrate(p13*p14*p15,u,u1,u2),t,t1,t2);

how to find : u1 ,u2 ,t = fixed
=> u1=0 to u2=t
how: uv plane take u+v = t , 'line =u+v = fixed' in the uv plane
line  t : point 1 to point 2
point 1 : along  v axis : (u=0 )=> u+v = t => v=t
point 2 : along  u axis:  v=0 => u+v = t => (u=t)

u1:0;
u2:t;

assume(t>0);

p17:lhs(p7)='integrate(integrate(p13*p14*p15,u,u1,u2),t,t1,t2);

how to find : t1,t2, u=fixed
=> t1=0 to t2=infinity
how: all lines t,takes t all values to fill uv plane
from t=0 to t=infinity

t1:0;
t2:inf;
p18:lhs(p7)='integrate(integrate(p13*p14*p15,u,u1,u2),t,t1,t2);

definition : convolution (laplace transform)

p19:FstarG='integrate(p13*p15,u,u1,u2);