My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX C. INTEGRATING THE CUSP FUNCTION
with the same substitution described in equation (C.8). This gives
I 2 = L √ [ (
c π z − a(z)
erfc + k ) (
cL c z
− erfc + k )]
cL c
. (C.13)
2
2
L c 2
L c
Finally,
I 4 =
∫ s
a(z)
( [ z ′ − z
exp − + k ] 2 )
cL c
dz ′
L c 2
= L c
∫ (s−z)/Lc+k cL c/2
[a(z)−z]/L c+k cL c/2
e −w2 dw
(C.14)
using the substitution defined in equation (C.10), which gives
I 4 = L √ [ (
c π b(z) − z
erfc + k ) (
cL c s − z
− erfc
2
2
L c
L c
+ k cL c
2
)]
. (C.15)
Collating the separate sub-integrals gives the final result:
I = n 0k c L 4 cπ √ π
2
( s − z
− + k cL c
L c 2
( z − a(z)
+
L
( c
a(z) − z
+
L c
(
e k2 c L2 c
z
{−
/4 + k cL c
L c 2
+ k cL c
2
+ k cL c
2
+ 1
k c L c
)
erfc
+ 1
k c L c
)
+ 1
k c L c
)
erfc
+ 1 ) ( z
erfc + k )
cL c
k c L c L c 2
( s − z
+ k )
cL c
L c 2
( z − a(z)
erfc + k )
cL c
L c 2
( a(z) − z
+ k )
cL c
2
L c
+ 1 √ π
e −(z/Lc+kcLc/2)2 + 1 √ π
e −[(s−z)/Lc+kcLc/2]2
(C.16)
− 1 √ π
e −([z−a(z)]/Lc+kcLc/2)2 − 1 √ π
e −([a(z)−z]/Lc+kcLc/2)2 }.
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