type-a index cannot be "9".
![[Graphics:../Images/index_gr_521.gif]](../Images/index_gr_521.gif)
![[Graphics:../Images/index_gr_522.gif]](../Images/index_gr_522.gif)
using the above fact.
![[Graphics:../Images/index_gr_523.gif]](../Images/index_gr_523.gif)
raising -9 to 9 for a unique expression.
![[Graphics:../Images/index_gr_524.gif]](../Images/index_gr_524.gif)
further simplification
![[Graphics:../Images/index_gr_525.gif]](../Images/index_gr_525.gif)
Ignore terms for scalar
![[Graphics:../Images/index_gr_527.gif]](../Images/index_gr_527.gif)
Ignore terms for tensor.
![[Graphics:../Images/index_gr_529.gif]](../Images/index_gr_529.gif)
Series expansion for scalar ignoring
![[Graphics:../Images/index_gr_531.gif]](../Images/index_gr_531.gif)
Series expansion for tensor ignoring
![[Graphics:../Images/index_gr_533.gif]](../Images/index_gr_533.gif)
![[Graphics:../Images/index_gr_534.gif]](../Images/index_gr_534.gif)
![[Graphics:../Images/index_gr_535.gif]](../Images/index_gr_535.gif)
Define an operator to separate the regular index into the type-a index and dim.
![[Graphics:../Images/index_gr_536.gif]](../Images/index_gr_536.gif)
Define differential operator (for type-a index)
![[Graphics:../Images/index_gr_537.gif]](../Images/index_gr_537.gif)