> 
# Unbounded integrands
# 
# Here we consider examples in which the integrands are unbounded. The justifications for the choices of gauges are found in the file  unbounded.pdf 
> restart:
> with(plots):
Warning, the name changecoords has been redefined
> read("khtools.m");
# Example 1.
# 
> f1:=proc(x)
> local f:
> if x=0 then f:=0;
> else f:=-ln(abs(x));
> end if:
> end:
> d1:=proc(x)
> local d1:
> if x=0 then d1:=epsilon:
> else d1:=abs(x)*(exp(epsilon)-1)/(exp(epsilon)+1):
> end if:
> end:
> epsilon:=0.12;
                        epsilon := 0.12
> khbox(f1,-1,1,d1,y=0..5);

The Riemann sum is   1.250355

> 
# Example 2.
# 
> f2:=proc(x)
> local f2:
>   if (x=0) then f2:=0;
>   else f2:=1/(abs(x))^(1/2):
>   end if:
> end:

> d2:=proc(x)
> local d2:
>    if (x=0) then d2:=epsilon2:
>    else d2:=epsilon2*abs(x):
>    end if:
> end:
> epsilon2:=0.1:
> khbox(f2,0,2,d2,y=0..4);

The Riemann sum is   2.226162

> khbox(f2,-1/2,1/2,d2,y=0..4);

The Riemann sum is   1.557272

# Example 3.
> 
> f3:=x->piecewise(x<=-1,0.1,x<1,1/sqrt(1-x^2),0.1);
                      /                          1           \
  f3 := x -> piecewise|x <= -1, 0.1, x < 1, ------------, 0.1|
                      |                         /     2\     |
                      \                     sqrt\1 - x /     /
> d3:=proc(x)
> local d3;
> if abs(1-abs(x))>0 then d3:=epsilon3*abs(1-abs(x));
> else d3:=epsilon;
> end if;
> end:
> epsilon3:=0.09;
                        epsilon3 := 0.09
> khbox(f3,-1,1,d3,y=0..2);

The Riemann sum is   2.177344

> 
# Example 4.
# 
> f4:=x->piecewise(x<=-1,0,x<0,2*(1+x)^(-1/2)-(-x)^(-1/2),x=0,0,x<1,(1-x)^(-1/2)-x^(-1/2),x=1,0);
                    /                        2             1                
f4 := x -> piecewise|x <= -1, 0, x < 0, ------------ - ---------, x = 0, 0, 
                    |                          (1/2)       (1/2)            
                    \                   (1 + x)        (-x)                 

              1           1             \
  x < 1, ------------ - ------, x = 1, 0|
                (1/2)    (1/2)          |
         (1 - x)        x               /
> 
> epsilon4:=0.1;
                        epsilon4 := 0.1
> 
> d4:=x->epsilon4*(min(abs(x),1-abs(x))+floor(x+1)-ceil(x));
d4 := x -> epsilon4 (min(|x|, 1 - |x|) + floor(1 + x) - ceil(x))
> khbox(f4,-1,1,d4,y=-2.5..5.5);

The Riemann sum is   1.280994

> 
>