Support computing Up solutions using HeunC method

Co-authored-by: Adrian Ottewill <[email protected]>
Co-authored-by: Marc Casals <[email protected]>

Kernel/TeukolskyRadial.m

@@ -293,7 +293,8 @@
   (* Solution functions for the specified boundary conditions *)
   solFuncs =
     <|"In" :> ((2^((-I a m-Sqrt[1-a^2] s+2 I \[Omega])/Sqrt[1-a^2]) (1-a^2)^((I a m)/(2 (1+Sqrt[1-a^2]))-s-I \[Omega]) E^(1/2 I (a m-2 # \[Omega])) ((-1-Sqrt[1-a^2]+#)/Sqrt[1-a^2])^((I a m)/(2 Sqrt[1-a^2])-s-I (1+1/Sqrt[1-a^2]) \[Omega]) ((-1+Sqrt[1-a^2]+#)/Sqrt[1-a^2])^((I (a m+2 (-1+Sqrt[1-a^2]) \[Omega]))/(2 Sqrt[1-a^2])) HeunC[s+s^2+\[Lambda]-(a m-2 \[Omega])^2/(-1+a^2)-4 \[Omega]^2+(I (-a m-4 (-1+s) \[Omega]+2 a^2 (-1+2 s) \[Omega]))/Sqrt[1-a^2],-4 \[Omega] (-I Sqrt[1-a^2]+a m+I Sqrt[1-a^2] s-2 \[Omega]+2 Sqrt[1-a^2] \[Omega]),1-s+(I (a m-2 \[Omega]))/Sqrt[1-a^2]-2 I \[Omega],1+s+(I (a m-2 \[Omega]))/Sqrt[1-a^2]+2 I \[Omega],4 I Sqrt[1-a^2] \[Omega],(1+Sqrt[1-a^2]-#)/(2 Sqrt[1-a^2])])&),
-      "Up" :> $Failed |>;
+      "Up" :> (norms["Up"]["Reflection"]/norms["Up"]["Transmission"](2^((-I a m-Sqrt[1-a^2] s+2 I \[Omega])/Sqrt[1-a^2]) (1-a^2)^((I a m)/(2 (1+Sqrt[1-a^2]))-s-I \[Omega]) E^(1/2 I (a m-2 # \[Omega])) ((-1-Sqrt[1-a^2]+#)/Sqrt[1-a^2])^((I a m)/(2 Sqrt[1-a^2])-s-I (1+1/Sqrt[1-a^2]) \[Omega]) ((-1+Sqrt[1-a^2]+#)/Sqrt[1-a^2])^((I (a m+2 (-1+Sqrt[1-a^2]) \[Omega]))/(2 Sqrt[1-a^2])) HeunC[s+s^2+\[Lambda]-(a m-2 \[Omega])^2/(-1+a^2)-4 \[Omega]^2+(I (-a m-4 (-1+s) \[Omega]+2 a^2 (-1+2 s) \[Omega]))/Sqrt[1-a^2],-4 \[Omega] (-I Sqrt[1-a^2]+a m+I Sqrt[1-a^2] s-2 \[Omega]+2 Sqrt[1-a^2] \[Omega]),1-s+(I (a m-2 \[Omega]))/Sqrt[1-a^2]-2 I \[Omega],1+s+(I (a m-2 \[Omega]))/Sqrt[1-a^2]+2 I \[Omega],4 I Sqrt[1-a^2] \[Omega],(1+Sqrt[1-a^2]-#)/(2 Sqrt[1-a^2])])+
+              norms["Up"]["Incidence"]/norms["Up"]["Transmission"]Conjugate[(#^2-2 #+a^2)^-s (2^((-I a m-Sqrt[1-a^2] (-s)+2 I \[Omega])/Sqrt[1-a^2]) (1-a^2)^((I a m)/(2 (1+Sqrt[1-a^2]))-(-s)-I \[Omega]) E^(1/2 I (a m-2 # \[Omega])) ((-1-Sqrt[1-a^2]+#)/Sqrt[1-a^2])^((I a m)/(2 Sqrt[1-a^2])-(-s)-I (1+1/Sqrt[1-a^2]) \[Omega]) ((-1+Sqrt[1-a^2]+#)/Sqrt[1-a^2])^((I (a m+2 (-1+Sqrt[1-a^2]) \[Omega]))/(2 Sqrt[1-a^2])) HeunC[(-s)+(-s)^2+(\[Lambda]+2s)-(a m-2 \[Omega])^2/(-1+a^2)-4 \[Omega]^2+(I (-a m-4 (-1+(-s)) \[Omega]+2 a^2 (-1+2 (-s)) \[Omega]))/Sqrt[1-a^2],-4 \[Omega] (-I Sqrt[1-a^2]+a m+I Sqrt[1-a^2] (-s)-2 \[Omega]+2 Sqrt[1-a^2] \[Omega]),1-(-s)+(I (a m-2 \[Omega]))/Sqrt[1-a^2]-2 I \[Omega],1+(-s)+(I (a m-2 \[Omega]))/Sqrt[1-a^2]+2 I \[Omega],4 I Sqrt[1-a^2] \[Omega],(1+Sqrt[1-a^2]-#)/(2 Sqrt[1-a^2])])]&) |>;
   solFuncs = Lookup[solFuncs, BCs];
 
   (* Select normalisation coefficients for the specified boundary conditions *)