Next, we separate the angular variables (O, 4) in hST"ra (0, 4) and decompose it
into the components of the Legendre polynomial expansions


 h0race (6, H) Z ,(4) P,(cos ). (5-157)
 n
Then, taking the same steps as in Eqs. (5-66)-(5-70), we obtain


 o ,aCe (H (4E ))) (5-158)

 From Eqs. (5-157), (5-154) and (5-155) we find our /-mode singular fields

STrace, S e and h sd,, respectively,

 hSTrace (7 ()) (5-159)

 h~ ev (f()), (5-160)

 h d =- (- ()), (5-161)

and by combination

 hSe + h^^S e + hS = ({e()) (e(')) (g()), (5-162)

where H (4)) is the f-th component of the expansions of h "ace (6, 4) in P,(X) as

in Eq. (5-157), and h((I) and (4()) are the L-th components of the expansions of

F(,, 4) cos(24)/(1- X2) and G(H, 4) sin(24)/( X2) in d2P(X)/dX2 as in
Eqs. (5-152) and (5-153), respectively, and hST ace (, )), F(0, 4) and G(0, 4) are

obtained from Eqs. (5-156), (5-147) and (5-148) together with the tensor source
functions, {hS, (t, r, 0, h() (t, r, 0,) hs, (t, r, 0, ))}, via the coordinate rotation.

 hS0Trace has the same structure as scalars as in Eq. (5-72), thus the same
structure analyses for c -term and oe-term apply to find the non-vanishing

regularization parameters via Eq. (5-159). hS ev/od has the structure to the first two

highest orders,