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c integration Complex  variables     R. V.  Churchill ,Integral
c   of p.185-186
      complex  sum1,sum2 ,f ,z,z1,z2,dz ,rooti
      f(z) = exp(rooti*z)/(z**2+1.)**2
c      f(z) =1./(z**2+1.)
c      f(z)=z/(2.*z+1.)
      pi=2.*asin(1.)
      rooti=(0.,1.)
      r=2.
      thetai =0.
      thetaf=pi
      sum1=0.
      sum2=0.
      nstep=4000
      dtheta=(thetaf-thetai)/float(nstep)
      do 10 i=1, nstep
      theta=thetai+float(i)*dtheta
      z1=r*exp(rooti*theta)
      z2=r*exp(rooti*(theta-dtheta))
      dz=r*exp(rooti*theta)*rooti*dtheta
      sum1=sum1 + (dz/2.)*(f(z1) + f(z2))
10    continue
      sum2=0.
      dz=(2.*r)/float(nstep)
      x1=-r
      do 20 i=1,nstep
      z=x1+dz*float(i)
      sum2=sum2+(dz/2.)*(f(z) +f(z-dz))
20    continue
      print*,'real(sum1), aimag(sum1)=',real(sum1) , aimag(sum1)
      print*,'real(sum2), aimag(sum2)=',real(sum2) , aimag(sum2)
      aintr=real(sum1) + real(sum2)
      aintimg=aimag(sum1) +aimag(sum2)
      print*,'aintr,aintimg,pi/exp(1.)=', aintr,aintimg,pi/exp(1.)
      stop
      end

c      exact value is pi/exp(1.)=  1.15572739
c Our numerical estimate over a semicircle of radius =2.
c                               is 1.15572751

      RUN
       real(sum1), aimag(sum1)= -0.0395202562 -1.55354192E-005
 real(sum2), aimag(sum2)=  1.19524777 -3.3179024E-008
 aintr,aintimg,pi/exp(1.)=  1.15572751 -1.55685975E-005  1.15572739