# Asymptotics of an Integral
# 
# Maple V (Release 4) will be used to compute:
# 
#      1
#     /
#    |                    1                              Pi
#    |   ----------------------------------- dy = - 1/2 ---- + O(1)
#    |    4      3          2              2             r
#   /    y  - 2 y  + 2 I r y  - 2 I r y - r
#     -1
#
# 
# as r -> 0.
# 
# This is the function to be studied:
> ff := Int(1/(y^4-2*y^3+2*I*r*y^2-2*I*r*y-r^2),y=-1..1);
> 

                   1
                  /
                 |                    1
          ff :=  |   ----------------------------------- dy
                 |    4      3          2              2
                /    y  - 2 y  + 2 I r y  - 2 I r y - r
                  -1

# Its value is:
> f := value(ff): ff = f;

   1
  /
 |                    1                             arctan(1/r)
 |   ----------------------------------- dy = - 1/2 ----------- + 1/8
 |    4      3          2              2                 r
/    y  - 2 y  + 2 I r y  - 2 I r y - r
  -1

    /      2               1/2                          1/2
    |I ln(r  + 1) (I r - 1)    + 2 arctan(1/r) (I r - 1)
    |
    \

             2               1/2                          1/2
     - I ln(r  + 9) (I r - 1)    - 2 arctan(3/r) (I r - 1)

                       2       \   /             1/2
     - 4 I arctan(------------)|  /  (r (I r - 1)   )
                           1/2 | /
                  (I r - 1)    /

# 
# We want to know the asymptotic behavior as r -> 0.
# Now since "asympt" computes asymptotic behavior at infinity,
# substitute s=1/r:
> subs(r=1/s,f);

                        /      1                 1/2
- 1/2 s arctan(s) + 1/8 |I ln(---- + 1) (I/s - 1)
                        |       2
                        \      s

                            1/2         1                 1/2
     + 2 arctan(s) (I/s - 1)    - I ln(---- + 9) (I/s - 1)
                                         2
                                        s

                              1/2                   2       \     /
     - 2 arctan(3 s) (I/s - 1)    - 4 I arctan(------------)| s  /
                                                        1/2 |   /
                                               (I/s - 1)    /

             1/2
    (I/s - 1)

> asympt(",s,1);

   (- 1/4 Pi - 1/8 I (ln(9) - 4 I (1/2 Pi - 1/2 I ln(3)))) s + O(1)

> simplify(");

                          - 1/2 s Pi + O(1)

> ans:= subs(s=1/r,");

                                     Pi
                       ans := - 1/2 ---- + O(1)
                                     r

> ff = ans;

      1
     /
    |                    1                              Pi
    |   ----------------------------------- dy = - 1/2 ---- + O(1)
    |    4      3          2              2             r
   /    y  - 2 y  + 2 I r y  - 2 I r y - r
     -1