function csabva(a, b)
C* returns (1. - cos(a * b) ) / a / a, or its limit as a -> 0
        term = snabva(.5 * a, b)
        csabva = .5 * term * term
        return
      end

      real function lnabva(a, b)
C* returns ln (1. + a * b) / a, or its limit as a -> 0.
        c = a * b
        if (abs(c) .gt. 0.01) then
          lnabva = log( 1. + c ) / a
        else
C* using (1. + t) / (1. - t) = 1. + c when t = c / (2. + c)
          t = c/(2. + c)
          t = t * t
          lnabva =  b / (2. + c) * (2.    + t *
     a                             (2./3. + t *
     b                             (2./5. + t *
     c                             (2./7. ))))
        endif
        return
      end

      function snabva(a, b)
C* returns sin(a * b) / a, or its limit as a -> 0.
        c = a * b
        csq = c * c
        if (csq .gt. .001) then
          snabva = sin( c )  / a
        else
          snabva = b * ( 1. - csq / 6. *
     a                 ( 1. - csq / 20. *
     b                 ( 1. - csq / 42. )))
        endif
        return
      end

C  A series of functions which may have problems in certain ranges
C  of one of their parameters.  These functions, if written normally,
C  may suffer round-off problems.  They are used in particular for
C  Lambert Conformal projections whose parameters make them almost
C  the same as a Mercator Projection.

      function xpabva(a, b)
C* returns (exp (a * b) - 1. ) / a, or its limit as a -> 0.
        c = a * b
        csq = .25 * c * c
        if ( csq  .gt. .001) then
          xpabva = (exp( c ) - 1.) / a
        else
          xpabva = exp(.5 * c) * b * (1. + csq / 6. *
     a                               (1. + csq / 20. *
     b                               (1. + csq / 42. )))
        endif
        return
      end

      function atnabv(a, b, c)
C* returns atan2(a*b,1-a*c)/a, or its limit as a -> 0
        xi = a * b
        eta = a * c
        vsq = xi * xi + eta * eta
        if ( vsq - 2. * eta + 1. .le. 0.) then
          atnabv = 0.
          return
        endif
        if (abs(xi) .gt. .01 * (1. - eta) ) then
          atnabv = atan2(xi, 1. - eta) / a
        else
          t = xi/(1. - eta)
          t = t * t
          atnabv = b / (1. - eta) * (1.    - t *
     a                              (1./3. - t *
     b                              (1./5. - t *
     c                              (1./7. ))))
        endif
        return
      end