# 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