= κσ exp { − 1 2 α $^{2}$p2 α$_{−}$ }   [ α Φ( z ) exp { − αp$_{α}$$_{−}$z } − αp$_{α}$$_{−}$ ] b ∗ α − αp$_{α}$$_{−}$ a ∗ α − αp$_{α}$$_{−}$ − α − αp$_{α}$$_{−}$ ∫ b ∗ α − αp$_{α}$$_{−}$ a ∗ α − αp$_{α}$$_{−}$ 1 √$_{2}$$_{π}$ exp { − 1 2 ( z 2 + 2 αp$_{α}$$_{−}$z + α $^{2}$p2 α$_{−}$ − α $^{2}$p2 $_{α$_{−}$}$) } dz   = κσ exp { − 1 2 α $^{2}$p2 α$_{−}$ }  $^{exp}$ { α $^{2}$p2 α$_{−}$ } − p$_{α}$$_{−}$ [ Φ ( b ∗ α − αp$_{α}$$_{−}$ ) exp { − b $^{∗}$p$_{α}$$_{−}$ } − Φ ( a ∗ α − αp$_{α}$$_{−}$ ) exp { − a $^{∗}$p$_{α}$$_{−}$ } ] + 1 p$_{α}$$_{−}$ exp { 1 2 α $^{2}$p2 α$_{−}$ } Φ ( z + αp$_{α}$$_{−}$ ) b ∗ α − αp$_{α}$$_{−}$ a ∗ α − αp$_{α}$$_{−}$   = − κσ p$_{α}$$_{−}$ exp { 1 2 α $^{2}$p2 α$_{−}$ } [ Φ ( b ∗ α − αp$_{α}$$_{−}$ ) exp { − b $^{∗}$p$_{α}$$_{−}$ } − Φ ( a ∗ α − αp$_{α}$$_{−}$ ) exp { − a $^{∗}$p$_{α}$$_{−}$ } ] + κσ p$_{α}$$_{−}$ [ Φ ( b ∗ α ) − Φ ( a ∗ α ) ] . (A.12) C$_{2}$ = − κ Φ( − αp$_{α}$$_{−}$ ) ∫ b a exp { − v $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } dv = − κ Φ( − αp$_{α}$$_{−}$ ) exp { 1 2 α $^{2}$p2 α$_{−}$ }  $^{exp}$ { − v $^{∗}$p$_{α}$$_{−}$ } − p$_{α}$$_{−}$/σ  $^{b}$ a = κσ p$_{α}$$_{−}$ Φ( − αp$_{α}$$_{−}$ ) exp { 1 2 α $^{2}$p2 α$_{−}$ } [ exp { − b $^{∗}$p$_{α}$$_{−}$ } − exp { − a $^{∗}$p$_{α}$$_{−}$ }] . (A.13) 32 Note that Φ ( b ∗ α ) − Φ ( a ∗ α ) = − Φ ( − b ∗ α ) + Φ ( − a ∗ α ) and hence the relevant term from equa- tion (A.12) and equation (A.14) can be collected together when adding the expressions. Finally, to evaluate the third component denoted C$_{3}$ we use the substitution z = αp$_{α}$$_{+}$ − v $^{∗}$/α and integrate-by-parts as done in C$_{1}$ . We suppress the details for brevity and present the final expression. C$_{3}$ = κ ∫ b a Φ ( αp$_{α}$$_{+}$ − v ∗ α exp { − v $^{∗}$p$_{α}$$_{+}$ + 1 2 α $^{2}$p2 α + }) dv = − κσ p$_{α}$$_{+}$ exp { 1 2 α $^{2}$p2 α + } [ Φ ( αp$_{α}$$_{+}$ − b ∗ α ) exp {− b $^{∗}$p$_{α}$$_{+}$ } − Φ ( αp$_{α}$$_{+}$ − a ∗ α ) exp {− a $^{∗}$p$_{α}$$_{+}$ } ] + κσ p$_{α}$$_{+}$ [ Φ ( − b ∗ α ) − Φ ( − a ∗ α ) ] . (A.14) We next evaluate the second component denoted C$_{2}$ directly as follows,