{VERSION 5 0 "IBM INTEL LINUX" "5.0" }
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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 53 "Investigate Langevin nu_eff, inc
luding Krommes trick." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "restart; kernelopts(version); interface(v
ersion);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%XMaple~7.00,~IBM~INTEL~LI
NUX,~May~28~2001~Build~ID~96223G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]
pMaple~Worksheet~Interface,~Maple~7.00,~IBM~INTEL~LINUX,~May~28~2001~B
uild~ID~96223G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 305 "# We are
 working with the Langevin equation of the following form.\n# Note tha
t the conjugate(f) convention is being used, to make it similar to \n#
 the practice in the DIA/EDQNM/RMC.\n#\n# Note that the damping rate i
s denoted by eta in the paper, but by nu here.\n\ndiff(psi(t),t) = - n
u * psi + conjugate(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$
-%$psiG6#%\"tGF*,&*&%#nuG\"\"\"F(F.!\"\"-%*conjugateG6#%\"fGF." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "# Here is C0/(nu_eff+conjug
ate(nu_eff)) :\n\ne1 := (nu_f +conjugate(nu_f))/(conjugate(nu_f)-nu)*(
1/(nu+conjugate(nu_eff))/(nu+conjugate(nu))/(nu+nu_f)\n          -1/(c
onjugate(nu_f)+conjugate(nu_eff))/(nu_f+conjugate(nu_f))/(conjugate(nu
_f)+conjugate(nu)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G*(,&%%n
u_fG\"\"\"-%*conjugateG6#F'F(F(,&F)F(%#nuG!\"\"F.,&*&F(F(*(,&F-F(-F*6#
%'nu_effGF(F(,&F-F(-F*6#F-F(F(,&F-F(F'F(F(F.F(*&F(F(*(,&F)F(F3F(F(F&F(
,&F)F(F7F(F(F.F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# sim
plify(%); # too complex" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 
"# Here is C0:\n\nc0 := (nu_f +conjugate(nu_f))/(conjugate(nu_f)-nu)*(
1/(nu+conjugate(nu))/(nu+nu_f)\n                                      \+
 -1/(nu_f+conjugate(nu_f))/(conjugate(nu_f)+conjugate(nu)) );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c0G*(,&%%nu_fG\"\"\"-%*conjugateG6#
F'F(F(,&F)F(%#nuG!\"\"F.,&*&F(F(*&,&F-F(-F*6#F-F(F(,&F-F(F'F(F(F.F(*&F
(F(*&F&F(,&F)F(F3F(F(F.F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "# simplify(e1/c0);  # too long" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 437 "# My own calculation (which apparently has an error \+
in it)\n# gives 1/(nu_eff+conjugate(nu_eff)) as:\n\ne10 := ( ( nu_f + \+
conjugate(nu) ) * ( nu_f + conjugate(nu_f) + conjugate(nu_eff) )\n    \+
    +( nu + conjugate(nu_f) ) * ( nu   + conjugate(nu)   + conjugate(n
u_eff) ) \n        + nu * nu_f - conjugate(nu) * conjugate(nu_f) ) \n \+
     / (nu + conjugate(nu) + nu_f + conjugate(nu_f) ) / (nu + conjugat
e(nu_eff)) / (nu_f + conjugate(nu_eff))\n;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$e10G**,**&,&%%nu_fG\"\"\"-%*conjugateG6#%#nuGF*F*,(F
)F*-F,6#F)F*-F,6#%'nu_effGF*F*F**&,&F.F*F0F*F*,(F.F*F+F*F2F*F*F**&F.F*
F)F*F**&F+F*F0F*!\"\"F*,*F.F*F+F*F)F*F0F*F:,&F.F*F2F*F:,&F)F*F2F*F:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "e10 - e1/c0;  # If e10 is \+
right, then this should be zero." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
**,**&,&%%nu_fG\"\"\"-%*conjugateG6#%#nuGF)F),(F(F)-F+6#F(F)-F+6#%'nu_
effGF)F)F)*&,&F-F)F/F)F),(F-F)F*F)F1F)F)F)*&F-F)F(F)F)*&F*F)F/F)!\"\"F
),*F-F)F*F)F(F)F/F)F9,&F-F)F1F)F9,&F(F)F1F)F9F)*&,&*&F)F)*(F;F),&F-F)F
*F)F),&F-F)F(F)F)F9F)*&F)F)*(,&F/F)F1F)F),&F(F)F/F)F),&F/F)F*F)F)F9F9F
),&*&F)F)*&FAF)FBF)F9F)*&F)F)*&FFF)FGF)F9F9F9F9" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "# simplify(%);  # This calculation takes enorm
ously long..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "e10*c0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*0,**&,&%%nu_fG\"\"\"-%*conjugateG6#%#
nuGF(F(,(F'F(-F*6#F'F(-F*6#%'nu_effGF(F(F(*&,&F,F(F.F(F(,(F,F(F)F(F0F(
F(F(*&F,F(F'F(F(*&F)F(F.F(!\"\"F(,*F,F(F)F(F'F(F.F(F8,&F,F(F0F(F8,&F'F
(F0F(F8,&F'F(F.F(F(,&F.F(F,F8F8,&*&F(F(*&,&F,F(F)F(F(,&F,F(F'F(F(F8F(*
&F(F(*&F<F(,&F.F(F)F(F(F8F8F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 409 "# After trying lots of things, I eventually decided to try to g
uide Maple in repeating the\n# calculation I did by hand, by having it
 do polynomial division (there is a \"divide\" command).\n# However, i
n the process of putting it in a \"normal\" form (polynomial numerator
 / polynomial denominator)\n# I discovered that this automatically sim
plifies it, eliminating the apparent singularity!\n\ne12 := normal(e1/
c0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e12G**,:*$)%#nuG\"\"#\"\"
\"F+*&-%*conjugateG6#%'nu_effGF+F)F+F+*&F)F+%%nu_fGF+F+*&F)F+-F.6#F)F+
F+*&F)F+-F.6#F2F+F+*&F-F+F2F+F+*&F-F+F4F+F+*&F2F+F4F+F+*&F-F+F7F+F+*&F
7F+F2F+F+*&F4F+F7F+F+*$)F7F*F+F+F+,*F)F+F4F+F2F+F7F+!\"\",&F7F+F-F+FB,
&F)F+F-F+FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# Check the \+
real limit:\ne14 := subs(nu=g, nu_f=g_f, nu_eff=g_eff, e12);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$e14G**,:*$)%\"gG\"\"#\"\"\"F+*&-%*conjuga
teG6#%&g_effGF+F)F+F+*&F)F+%$g_fGF+F+*&F)F+-F.6#F)F+F+*&F)F+-F.6#F2F+F
+*&F-F+F2F+F+*&F-F+F4F+F+*&F2F+F4F+F+*&F-F+F7F+F+*&F7F+F2F+F+*&F4F+F7F
+F+*$)F7F*F+F+F+,*F)F+F4F+F2F+F7F+!\"\",&F7F+F-F+FB,&F)F+F-F+FB" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "e16 := simplify(evalc(%));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e16G*(,(%\"gG\"\"\"%$g_fGF(%&g_ef
fGF(F(,&F'F(F*F(!\"\",&F)F(F*F(F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "e17 := numer(%)*2*g_eff-denom(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$e17G,&*&,(%\"gG\"\"\"%$g_fGF)%&g_effGF)F)F+F)\"\"#
*&,&F(F)F+F)F),&F*F)F+F)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "e18 := factor(e17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e18G
,**&%&g_effG\"\"\"%\"gGF(F(*&F'F(%$g_fGF(F(*$)F'\"\"#F(F(*&F)F(F+F(!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# This is indeed equi
valent to  g_eff = g*g_f/(g+g_f_g_eff) \n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "e19 := solve(e18=0, g_eff);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$e19G6$,(%\"gG#!\"\"\"\"#*&#\"\"\"F*F-%$g_fGF-F)*&#F-
F*F--%%sqrtG6#,(*$)F'F*F-F-*(\"\"'F-F'F-F.F-F-*$)F.F*F-F-F-F-,(F'F(*&#
F-F*F-F.F-F)*&#F-F*F-*$F1F-F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 117 "# Now return to the general case with complex coefficients:\n
\ne20 := numer(e12)*(nu_eff+conjugate(nu_eff))-denom(e12);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%$e20G,&*&,:*$)%#nuG\"\"#\"\"\"F,*&-%*conju
gateG6#%'nu_effGF,F*F,F,*&F*F,%%nu_fGF,F,*&F*F,-F/6#F*F,F,*&F*F,-F/6#F
3F,F,*&F.F,F3F,F,*&F.F,F5F,F,*&F3F,F5F,F,*&F.F,F8F,F,*&F8F,F3F,F,*&F5F
,F8F,F,*$)F8F+F,F,F,,&F1F,F.F,F,F,*(,*F*F,F5F,F3F,F8F,F,,&F8F,F.F,F,,&
F*F,F.F,F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify
(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,F*(%%nu_fG\"\"\"-%*conjugateG
6#%#nuGF&%'nu_effGF&F&*(F*F&-F(6#F%F&F+F&F&*&)F*\"\"#F&F+F&F&*&)F-F1F&
F+F&F&*&F0F&F-F&!\"\"*&F3F&F*F&F5*(-F(6#F+F&F-F&F+F&F&*(F-F&F%F&F*F&F5
*(F8F&F'F&F+F&F&*(F'F&F-F&F+F&F&*(F-F&F%F&F+F&F&*(F*F&F-F&F8F&F5*(F8F&
F%F&F+F&F&*(F*F&F'F&F+F&F&*(F*F&F%F&F+F&F&*(F8F&F*F&F+F&F&*(F%F&F'F&F8
F&F&*(F'F&F-F&F*F&F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "exp
and(e20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,F*(%%nu_fG\"\"\"-%*conju
gateG6#%#nuGF&%'nu_effGF&F&*(F*F&-F(6#F%F&F+F&F&*&)F*\"\"#F&F+F&F&*&)F
-F1F&F+F&F&*&F0F&F-F&!\"\"*&F3F&F*F&F5*(-F(6#F+F&F-F&F+F&F&*(F-F&F%F&F
*F&F5*(F8F&F'F&F+F&F&*(F'F&F-F&F+F&F&*(F-F&F%F&F+F&F&*(F*F&F-F&F8F&F5*
(F8F&F%F&F+F&F&*(F*F&F'F&F+F&F&*(F*F&F%F&F+F&F&*(F8F&F*F&F+F&F&*(F%F&F
'F&F8F&F&*(F'F&F-F&F*F&F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "factor(e20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,F*(%%nu_fG\"\"\"-
%*conjugateG6#%#nuGF&%'nu_effGF&F&*(F*F&-F(6#F%F&F+F&F&*&)F*\"\"#F&F+F
&F&*&)F-F1F&F+F&F&*&F0F&F-F&!\"\"*&F3F&F*F&F5*(-F(6#F+F&F-F&F+F&F&*(F-
F&F%F&F*F&F5*(F8F&F'F&F+F&F&*(F'F&F-F&F+F&F&*(F-F&F%F&F+F&F&*(F*F&F-F&
F8F&F5*(F8F&F%F&F+F&F&*(F*F&F'F&F+F&F&*(F*F&F%F&F+F&F&*(F8F&F*F&F+F&F&
*(F%F&F'F&F8F&F&*(F'F&F-F&F*F&F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "factor(numer(e12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,:*$)%#nuG\"\"#\"\"\"F(*&-%*conjugateG6#%'nu_effGF(F&F(F(*&F&F(%%nu_
fGF(F(*&F&F(-F+6#F&F(F(*&F&F(-F+6#F/F(F(*&F*F(F/F(F(*&F*F(F1F(F(*&F/F(
F1F(F(*&F*F(F4F(F(*&F4F(F/F(F(*&F1F(F4F(F(*$)F4F'F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,:*$)%#nuG\"\"#\"\"\"F(*&-%*conjugateG6#%'nu_effGF(F&F(
F(*&F&F(%%nu_fGF(F(*&F&F(-F+6#F&F(F(*&F&F(-F+6#F/F(F(*&F*F(F/F(F(*&F*F
(F1F(F(*&F/F(F1F(F(*&F*F(F4F(F(*&F4F(F/F(F(*&F1F(F4F(F(*$)F4F'F(F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "# Because both nu_eff and \+
conjugate(nu_eff) appear in the equations, it isn't straightforward\n#
 to solve for nu_eff.  Denote conjugate(nu_eff) as a special symbol, a
nd then solve for\n# nu_eff:\n\ne30 := subs(conjugate(nu_eff)=nu_effcc
, e20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$e30G,&*&,:*$)%#nuG\"\"#
\"\"\"F,*&%)nu_effccGF,F*F,F,*&F*F,%%nu_fGF,F,*&F*F,-%*conjugateG6#F*F
,F,*&F*F,-F36#F0F,F,*&F.F,F0F,F,*&F.F,F2F,F,*&F0F,F2F,F,*&F.F,F6F,F,*&
F6F,F0F,F,*&F2F,F6F,F,*$)F6F+F,F,F,,&%'nu_effGF,F.F,F,F,*(,*F*F,F2F,F0
F,F6F,F,,&F6F,F.F,F,,&F*F,F.F,F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "e31 := solve(e30=0, nu_eff);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$e31G,$*&,.*&)-%*conjugateG6#%%nu_fG\"\"#\"\"\"%#nuGF
/!\"\"*(F-F/-F+6#F0F/%)nu_effccGF/F/*(F0F/F*F/F5F/F1*(F3F/F*F/F0F/F1*&
)F0F.F/F*F/F1*(F*F/F-F/F0F/F1F/,:*$F9F/F/*&F5F/F0F/F/*&F0F/F-F/F/*&F0F
/F3F/F/*&F0F/F*F/F/*&F5F/F-F/F/*&F5F/F3F/F/*&F-F/F3F/F/*&F5F/F*F/F/*&F
*F/F-F/F/*&F3F/F*F/F/*$F)F/F/F1F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 320 "####################################################
##########\n# e32 is our main answer for nu_eff as a function of nu an
d nu_eff\n# (it is a recursive definition, since conjugate(nu_eff) app
ears on the RHS,\n# but it can probably be solved by just iterating a \+
few times...)\n\ne32 := subs(nu_effcc=conjugate(nu_eff), e31);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e32G,$*&,.*&)-%*conjugateG6#%%nu_fG
\"\"#\"\"\"%#nuGF/!\"\"*(F-F/-F+6#F0F/-F+6#%'nu_effGF/F/*(F0F/F*F/F5F/
F1*(F3F/F*F/F0F/F1*&)F0F.F/F*F/F1*(F*F/F-F/F0F/F1F/,:*$F;F/F/*&F5F/F0F
/F/*&F0F/F-F/F/*&F0F/F3F/F/*&F0F/F*F/F/*&F5F/F-F/F/*&F5F/F3F/F/*&F-F/F
3F/F/*&F5F/F*F/F/*&F*F/F-F/F/*&F3F/F*F/F/*$F)F/F/F1F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "\nsimplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&,.*&)-%*conjugateG6#%%nu_fG\"\"#\"\"\"%#nuGF-!\"\"*
(F+F--F)6#F.F--F)6#%'nu_effGF-F-*(F.F-F(F-F3F-F/*(F1F-F(F-F.F-F/*&)F.F
,F-F(F-F/*(F(F-F+F-F.F-F/F-,:*$F9F-F-*&F3F-F.F-F-*&F.F-F+F-F-*&F.F-F1F
-F-*&F.F-F(F-F-*&F3F-F+F-F-*&F3F-F1F-F-*&F+F-F1F-F-*&F3F-F(F-F-*&F(F-F
+F-F-*&F1F-F(F-F-*$F'F-F-F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "factor(denom(e32));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*$)%#nu
G\"\"#\"\"\"F(*&-%*conjugateG6#%'nu_effGF(F&F(F(*&F&F(%%nu_fGF(F(*&F&F
(-F+6#F&F(F(*&F&F(-F+6#F/F(F(*&F*F(F/F(F(*&F*F(F1F(F(*&F/F(F1F(F(*&F*F
(F4F(F(*&F4F(F/F(F(*&F1F(F4F(F(*$)F4F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "# check e32 in the real limit:\nsimplify(evalc(e32)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%#nuG\"\"\"%%nu_fGF%,(F$F%%'nu
_effGF%F&F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "factor(
numer(e32));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&)-%*conjugateG6#%%
nu_fG\"\"#\"\"\"%#nuGF+F+*(F)F+-F'6#F,F+-F'6#%'nu_effGF+!\"\"*(F,F+F&F
+F0F+F+*(F.F+F&F+F,F+F+*&)F,F*F+F&F+F+*(F&F+F)F+F,F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,.*&)-%*conjugateG6#%%nu_fG\"\"#\"\"\"%#nuGF+F+*(F)F+-F
'6#F,F+-F'6#%'nu_effGF+!\"\"*(F,F+F&F+F0F+F+*(F.F+F&F+F,F+F+*&)F,F*F+F
&F+F+*(F&F+F)F+F,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "de
nom(e32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*$)%#nuG\"\"#\"\"\"F(*&
-%*conjugateG6#%'nu_effGF(F&F(F(*&F&F(%%nu_fGF(F(*&F&F(-F+6#F&F(F(*&F&
F(-F+6#F/F(F(*&F*F(F/F(F(*&F*F(F1F(F(*&F/F(F1F(F(*&F*F(F4F(F(*&F4F(F/F
(F(*&F1F(F4F(F(*$)F4F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*$)%#nuG\"\"#\"
\"\"F(*&-%*conjugateG6#%'nu_effGF(F&F(F(*&F&F(%%nu_fGF(F(*&F&F(-F+6#F&
F(F(*&F&F(-F+6#F/F(F(*&F*F(F/F(F(*&F*F(F1F(F(*&F/F(F1F(F(*&F*F(F4F(F(*
&F4F(F/F(F(*&F1F(F4F(F(*$)F4F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:*$)%#
nuG\"\"#\"\"\"F(*&-%*conjugateG6#%'nu_effGF(F&F(F(*&F&F(%%nu_fGF(F(*&F
&F(-F+6#F&F(F(*&F&F(-F+6#F/F(F(*&F*F(F/F(F(*&F*F(F1F(F(*&F/F(F1F(F(*&F
*F(F4F(F(*&F4F(F/F(F(*&F1F(F4F(F(*$)F4F'F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "coeff(%,conjugate(nu_eff));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*%#nuG\"\"\"-%*conjugateG6#F$F%%%nu_fGF%-F'6#F)F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "e40 := subs(nu=g+I*w, nu_f= \+
g_f+I*w_f, nu_eff=g_eff+I*w+eff, e32);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$e40G,$*&,.*&)-%*conjugateG6#,&%$g_fG\"\"\"*&^#F/F/%$w_fGF/F/
\"\"#F/,&%\"gGF/*&F1F/%\"wGF/F/F/!\"\"*(F-F/-F+6#F4F/-F+6#,(%&g_effGF/
F6F/%$effGF/F/F/*(F4F/F*F/F<F/F8*(F:F/F*F/F4F/F8*&)F4F3F/F*F/F8*(F*F/F
-F/F4F/F8F/,:*$FDF/F/*&F<F/F4F/F/*&F4F/F-F/F/*&F4F/F:F/F/*&F4F/F*F/F/*
&F<F/F-F/F/*&F<F/F:F/F/*&F-F/F:F/F/*&F<F/F*F/F/*&F*F/F-F/F/*&F:F/F*F/F
/*$F)F/F/F8F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*(,6*&,&*$)%$g_fG\"\"#\"\"\"!\"\"*
$)%$w_fGF+F,F,F,%\"gGF,F-**F+F,F*F,F0F,%\"wGF,F,*&,&*&F1F,F*F,F,*&F0F,
F3F,F,F,,&%&g_effGF,%$effGF,F,F-*(F+F,,&*&F0F,F1F,F,*&F*F,F3F,F-F,F3F,
F-*&,&F6F-F7F-F,F8F,F-*&,&F6F-F7F,F,F1F,F-*&,&F>F,F=F,F,F3F,F,*&,&*$)F
1F+F,F-*$)F3F+F,F,F,F*F,F-**F+F,F1F,F3F,F0F,F,*&,&F(F-F.F-F,F1F,F-F,,,
FGF+*(F+F,F8F,F1F,F,*(\"\"%F,F1F,F*F,F,*(F+F,F8F,F*F,F,*&F+F,F)F,F,F,,
&*$)FNF+F,F,*$),&F>!\"#*(F+F,F*F,F0F,F-F+F,F,F-F,*(,6*(F*F,F0F,F1F,FZ*
&F'F,F3F,F-*(F+F,F<F,F8F,F-*&F5F,F3F,F,*&F@F,F3F,F,*&FDF,F1F,F-*&FBF,F
3F,F-**F+F,F1F,F3F,F*F,F,*&FFF,F0F,F,*&FMF,F3F,F-F,FYF,FTF-F,*&^#F,F,,
&*(FgnF,FNF,FTF-F,*(F%F,FYF,FTF-F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,
\\r*,%$g_fG\"\"\"%$w_fGF(%\"wGF(%&g_effGF(%\"gGF(!\"#*.\"\"#F(F'F(F)F(
F*F(F,F(%$effGF(!\"\"**F'F()F)F/F(F,F(F+F(F1*()F'F/F(F3F(F,F(F1**F/F()
F,F/F(F+F(F5F(F1*(\"\"$F()F,F9F(F5F(F1*(F9F(F7F()F'F9F(F1*&F'F()F,\"\"
%F(F1*&F,F()F'F?F(F1*,F/F(F)F(F*F(F,F(F5F(F1**F'F()F*F/F(F,F(F0F(F1**F
'F(FDF(F+F(F,F(F1*,F/F(F'F(F)F(F*F(F7F(F1**F'F(F3F(F,F(F0F(F1**^#F(F(F
)F(F7F()F0F/F(F(**FIF(F)F(F7F()F+F/F(F(*,^#F-F(F'F(F*F(F0F(F7F(F(*,^#F
1F(F'F(F*F(FJF(F,F(F(*.FNF(F'F(F*F(F+F(F,F(F0F(F(*,^#F?F(F'F(F)F(F7F(F
+F(F(*,FSF(F'F(F)F(F7F(F0F(F(*,^#F/F(F5F(F)F(F,F(F+F(F(*,FVF(F5F(F)F(F
,F(F0F(F(*.FVF(F)F(F,F(F+F(F'F(F0F(F(*,^#!\"%F(F*F(F5F(F+F(F,F(F(**FNF
(F*F(F<F(F0F(F(**FZF(F*F(F<F(F,F(F(**FZF(F*F(F5F(F7F(F(**FVF(F5F(F)F(F
7F(F(**^#F9F(F'F(F)F(F:F(F(**FNF(F*F(F<F(F+F(F(*,FZF(F*F(F5F(F,F(F0F(F
(*,FVF(F)F(F7F(F+F(F0F(F(*,FNF(F'F(F*F(F+F(F7F(F(*,FPF(F'F(F*F(FLF(F,F
(F(**FPF(F5F(F*F(F3F(F(**FNF(F5F(F)F(FDF(F(**FPF(F:F(F*F(F'F(F(**FPF(F
5F(F*F(FJF(F(**FPF(F5F(F*F(FLF(F(**FVF(F)F(F:F(F0F(F(**FVF(F)F(F:F(F+F
(F(*,FIF(F)F(F,F(FLF(F'F(F(*,FIF(F)F(F,F(FJF(F'F(F(*(FIF(F)F(F>F(F(*,F
NF(F5F(F*F(F+F(F0F(F(*(FPF(F*F(FAF(F(*(F'F(F3F(F7F(F1*(F'F(FDF(F7F(F1*
(F'F(F:F(F0F(F1*(F'F(F:F(F+F(F1*(FDF(F,F(F5F(F1*(FPF(F5F()F*F9F(F(**F/
F(F7F(F5F(F0F(F1*(F,F(F0F(F<F(F1*(F+F(F,F(F<F(F1F(,T**F+F(F,F(F'F(F0F(
F?*(F/F(F:F(F+F(F(*(F/F(F:F(F0F(F(*(F?F(F:F(F'F(F(*(\"\"'F(F7F(F5F(F(*
&FLF(F7F(F(*&F7F(FJF(F(**F]qF(F7F(F+F(F'F(F(*$F>F(F(*$FAF(F(*&F5F(FDF(
F(*&F5F(F3F(F(*(F?F(F,F(F<F(F(*&FLF(F5F(F(*(F/F(F+F(F<F(F(*&F5F(FJF(F(
*(F/F(F<F(F0F(F(**F/F(F5F(F*F(F)F(F(**F/F(F,F(FJF(F'F(F(**F]qF(F+F(F,F
(F5F(F(**F/F(FLF(F,F(F'F(F(**F/F(F+F(F7F(F0F(F(**F/F(F+F(F5F(F0F(F(**F
]qF(F,F(F0F(F5F(F(**F]qF(F7F(F'F(F0F(F(F1F1" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "numer(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,\\r*(
^#\"\"\"F&%\"wGF&)%$g_fG\"\"%F&F&*(^#!\"\"F&%$w_fGF&)%\"gGF*F&F&*.\"\"
#F&F)F&F.F&F'F&%&g_effGF&F0F&F&*.F2F&F)F&F.F&F'F&F0F&%$effGF&F&**F)F&)
F.F2F&F0F&F3F&F&*()F)F2F&F7F&F0F&F&**F2F&)F0F2F&F3F&F9F&F&*(\"\"$F&)F0
F=F&F9F&F&*(F=F&F;F&)F)F=F&F&*&F)F&F/F&F&*&F0F&F(F&F&*,F2F&F.F&F'F&F0F
&F9F&F&**F)F&)F'F2F&F0F&F5F&F&**F)F&FEF&F3F&F0F&F&*,F2F&F)F&F.F&F'F&F;
F&F&**F)F&F7F&F0F&F5F&F&**F%F&F9F&F'F&F7F&F&**F%F&F>F&F'F&F)F&F&**F%F&
F9F&F'F&)F5F2F&F&**F%F&F9F&F'F&)F3F2F&F&*,F%F&F)F&F'F&FLF&F0F&F&*,F%F&
F)F&F'F&FNF&F0F&F&*(F%F&F9F&)F'F=F&F&**^#F2F&F9F&F.F&FEF&F&**F,F&F.F&F
;F&FLF&F&**^#!\"#F&F.F&F>F&F5F&F&**F,F&F.F&F;F&FNF&F&**FWF&F.F&F>F&F3F
&F&**FTF&F'F&F@F&F5F&F&**FTF&F'F&F@F&F3F&F&**^#F*F&F'F&F@F&F0F&F&**Fhn
F&F'F&F9F&F;F&F&**FWF&F9F&F.F&F;F&F&**^#!\"$F&F)F&F.F&F>F&F&*,FhnF&F'F
&F9F&F0F&F5F&F&*,FWF&F.F&F;F&F3F&F5F&F&*,F,F&F.F&F0F&FNF&F)F&F&*,F,F&F
.F&F0F&FLF&F)F&F&*,FTF&F)F&F'F&F3F&F;F&F&*,FTF&F9F&F'F&F3F&F5F&F&*,FTF
&F)F&F'F&F5F&F;F&F&*.FTF&F)F&F'F&F3F&F0F&F5F&F&*,^#!\"%F&F)F&F.F&F;F&F
3F&F&*,FgoF&F)F&F.F&F;F&F5F&F&*,FWF&F9F&F.F&F0F&F3F&F&*,FWF&F9F&F.F&F0
F&F5F&F&*.FWF&F.F&F0F&F3F&F)F&F5F&F&*,FhnF&F'F&F9F&F3F&F0F&F&*(F)F&F7F
&F;F&F&*(F)F&FEF&F;F&F&*(F)F&F>F&F5F&F&*(F)F&F>F&F3F&F&*(FEF&F0F&F9F&F
&**F2F&F;F&F9F&F5F&F&*(F0F&F5F&F@F&F&*(F3F&F0F&F@F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "denom(%%);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,T**%&g_effG\"\"\"%\"gGF&%$g_fGF&%$effGF&\"\"%*(\"\"#F&
)F'\"\"$F&F%F&F&*(F,F&F-F&F)F&F&*(F*F&F-F&F(F&F&*(\"\"'F&)F'F,F&)F(F,F
&F&*&)F%F,F&F3F&F&*&F3F&)F)F,F&F&**F2F&F3F&F%F&F(F&F&*$)F'F*F&F&*$)F(F
*F&F&*&F4F&)%\"wGF,F&F&*&F4F&)%$w_fGF,F&F&*(F*F&F'F&)F(F.F&F&*&F6F&F4F
&F&*(F,F&F%F&FEF&F&*&F4F&F8F&F&*(F,F&FEF&F)F&F&**F,F&F4F&F@F&FCF&F&**F
,F&F'F&F8F&F(F&F&**F2F&F%F&F'F&F4F&F&**F,F&F6F&F'F&F(F&F&**F,F&F%F&F3F
&F)F&F&**F,F&F%F&F4F&F)F&F&**F2F&F'F&F)F&F4F&F&**F2F&F3F&F(F&F)F&F&" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#,T**%&g_effG\"\"\"%\"gGF&%$g_fGF&%$effGF&\"\"%*(\"
\"#F&)F'\"\"$F&F%F&F&*(F,F&F-F&F)F&F&*(F*F&F-F&F(F&F&*(\"\"'F&)F'F,F&)
F(F,F&F&*&)F%F,F&F3F&F&*&F3F&)F)F,F&F&**F2F&F3F&F%F&F(F&F&*$)F'F*F&F&*
$)F(F*F&F&*&F4F&)%\"wGF,F&F&*&F4F&)%$w_fGF,F&F&*(F*F&F'F&)F(F.F&F&*&F6
F&F4F&F&*(F,F&F%F&FEF&F&*&F4F&F8F&F&*(F,F&FEF&F)F&F&**F,F&F4F&F@F&FCF&
F&**F,F&F'F&F8F&F(F&F&**F2F&F%F&F'F&F4F&F&**F,F&F6F&F'F&F(F&F&**F,F&F%
F&F3F&F)F&F&**F,F&F%F&F4F&F)F&F&**F2F&F'F&F)F&F4F&F&**F2F&F3F&F(F&F)F&
F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 392 "\n# I couldn't simpl
ify e32 by computer, but I could by hand.  My claim is that nu_eff is \+
given by:\n\ne40 := ( nu*conjugate(nu_f)*(nu+nu_f + conjugate(nu)+conj
ugate(nu_f)) \n       + conjugate(nu_eff) * (nu * conjugate(nu_f) - nu
_f*conjugate(nu)) )\n     / ( (nu +conjugate(nu_eff)) * (nu + nu_f + c
onjugate(nu) + conjugate(nu_f)) \n    + (conjugate(nu_f)+conjugate(nu)
)*(nu_f+conjugate(nu_f)) ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e40
G*&,&*(%#nuG\"\"\"-%*conjugateG6#%%nu_fGF),*F(F)-F+6#F(F)F-F)F*F)F)F)*
&-F+6#%'nu_effGF),&*&F(F)F*F)F)*&F-F)F/F)!\"\"F)F)F),&*&,&F(F)F2F)F)F.
F)F)*&,&F*F)F/F)F),&F-F)F*F)F)F)F8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "# Verify that this is true:\n\nnormal(e40-e32);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 244 "# From my Red RMC treatment of the Langevin equation
, I found that nu_eff should be given by:\ne50 := nu - conjugate(theta
)*(nu_eff +conjugate(nu_eff))*(nu+conjugate(nu))\n       / (theta + co
njugate(theta))/(conjugate(nu_eff) + conjugate(nu_f));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$e50G,&%#nuG\"\"\"*,-%*conjugateG6#%&thetaGF',&%
'nu_effGF'-F*6#F.F'F',&F&F'-F*6#F&F'F',&F,F'F)F'!\"\",&-F*6#%%nu_fGF'F
/F'F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "e51 := subs(conj
ugate(nu_eff)=nu_effcc, e50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e5
1G,&%#nuG\"\"\"*,-%*conjugateG6#%&thetaGF',&%'nu_effGF'%)nu_effccGF'F'
,&F&F'-F*6#F&F'F',&F,F'F)F'!\"\",&-F*6#%%nu_fGF'F/F'F4F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "e52 := nu_eff -e51;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$e52G,(%'nu_effG\"\"\"%#nuG!\"\"*,-%*conjugateG6
#%&thetaGF',&F&F'%)nu_effccGF'F',&F(F'-F,6#F(F'F',&F.F'F+F'F),&-F,6#%%
nu_fGF'F0F'F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "e53 := s
olve(e52=0, nu_eff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e53G,$*&,**
(%#nuG\"\"\"%&thetaGF*-%*conjugateG6#%%nu_fGF*!\"\"*(F)F*F+F*%)nu_effc
cGF*F0*(F)F*-F-6#F+F*F,F*F0*(F4F*F2F*-F-6#F)F*F*F*,.*&F+F*F,F*F**&F+F*
F2F*F**&F4F*F,F*F**&F4F*F2F*F**&F4F*F)F*F**&F4F*F7F*F*F0F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "e54 := subs(theta=1/(nu+nu_f), nu_e
ffcc=conjugate(nu_eff), e53);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e5
4G,$*&,**(%#nuG\"\"\",&F)F*%%nu_fGF*!\"\"-%*conjugateG6#F,F*F-*(F)F*F+
F--F/6#%'nu_effGF*F-*(F)F*-F/6#*&F*F*F+F-F*F.F*F-*(F6F*F2F*-F/6#F)F*F*
F*,.*&F+F-F.F*F**&F+F-F2F*F**&F6F*F.F*F**&F6F*F2F*F**&F6F*F)F*F**&F6F*
F:F*F*F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "e55 := normal
(e54);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e55G,$*&,.*(%#nuG\"\"\"-%
*conjugateG6#%%nu_fGF*-F,6#,&F)F*F.F*F*!\"\"*(-F,6#%'nu_effGF*F)F*F/F*
F2*&)F)\"\"#F*F+F*F2*(F+F*F.F*F)F*F2*(F4F*-F,6#F)F*F)F*F**(F.F*F<F*F4F
*F*F*,6*&F+F*F/F*F**&F4F*F/F*F**&F)F*F+F*F**&F+F*F.F*F**&F4F*F)F*F**&F
4F*F.F*F**$F8F*F**&F)F*F.F*F**&F)F*F<F*F**&F.F*F<F*F*F2F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "numer(e55)-numer(e32);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,.*(%#nuG\"\"\"-%*conjugateG6#%%nu_fGF&-F(6#,&F
%F&F*F&F&F&*(-F(6#%'nu_effGF&F%F&F+F&F&*(F/F&-F(6#F%F&F%F&!\"\"*&)F'\"
\"#F&F%F&F5*(F%F&F'F&F/F&F5*(F3F&F'F&F%F&F5" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 115 "# Maple doesn't realize conjugate(nu+nu_f)=conjug
ate(nu)+conjugate(nu_f) until you force\n# it to expand:\nexpand(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "expand(denom(e55)-denom(e32));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "##
###############################################################\n# We \+
can use e32 recursively to deterine nu_eff in most cases.  But in case
s where we want\n# a direct solution, we will need to solve some simul
taneous equations:\n\ne60 := subs(nu_eff=g_eff+I*w_eff,nu=g+I*w,nu_f=g
_f+I*w_f,e20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$e60G,&*&,:*$),&%
\"gG\"\"\"*&^#F,F,%\"wGF,F,\"\"#F,F,*&-%*conjugateG6#,&%&g_effGF,*&F.F
,%&w_effGF,F,F,F*F,F,*&F*F,,&%$g_fGF,*&F.F,%$w_fGF,F,F,F,*&F*F,-F36#F*
F,F,*&F*F,-F36#F:F,F,*&F2F,F:F,F,*&F2F,F?F,F,*&F:F,F?F,F,*&F2F,FBF,F,*
&FBF,F:F,F,*&F?F,FBF,F,*$)FBF0F,F,F,,(F6F,F7F,F2F,F,F,*(,.F+F,F-F,F?F,
F;F,F<F,FBF,F,,&FBF,F2F,F,,(F+F,F-F,F2F,F,!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "e61 := evalc(%);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$e61G,**&,,*$)%\"gG\"\"#\"\"\"F+*(F+F,%&g_effGF,F*F,F,*(\"\"%F
,F*F,%$g_fGF,F,*(F+F,F.F,F1F,F,*&F+F,)F1F+F,F,F,F.F,F+*(,&F*!\"#*&F+F,
F1F,!\"\"F,,&F1F,F.F,F,,&F*F,F.F,F,F,*(F6F,,&%$w_fGF9%&w_effGF9F,,&%\"
wGF,F?F9F,F9*&^#F,F,,(*&,**&F*F,FAF,F+*(F+F,F?F,F*F,F9*(F+F,F?F,F1F,F9
*(F+F,F1F,F>F,F9F,F.F,F+*(F6F,F=F,F;F,F,*(F6F,F:F,F@F,F,F,F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "e62 := evalc(Re(e61));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$e62G,@*(%&g_effG\"\"\"%\"gGF(%$g_fG
F(\"\"%**\"\"#F(%&w_effGF(F*F(%\"wGF(!\"\"**F-F(F*F(%$w_fGF(F.F(F(**F-
F(F*F(F2F(F/F(F0**F-F(F.F(F)F(F/F(F0**F-F(F)F(F2F(F.F(F(**F-F(F)F(F2F(
F/F(F0*(F-F()F.F-F(F*F(F(*(F-F(F8F(F)F(F(*(F-F()F*F-F(F)F(F0*(F-F(F'F(
)F)F-F(F(*(F-F()F'F-F(F)F(F(*(F-F(F?F(F*F(F(*(F-F(F'F(F;F(F(*(F-F(F=F(
F*F(F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "e63 := evalc(Im(e
61));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e63G,8*(%&g_effG\"\"\"%\"g
GF(%\"wGF(\"\"#**F+F(F'F(%$g_fGF(%$w_fGF(!\"\"*(F+F()F)F+F(F.F(F(**F+F
(F)F(F.F(F'F(F(*(F+F(%&w_effGF(F1F(F(**F+F(F-F(F.F(F)F(F(**\"\"%F(F4F(
F-F(F)F(F(**F+F(F)F(F-F(F*F(F/*(F+F()F-F+F(F*F(F/*(F+F(F:F(F4F(F(**F+F
(F'F(F-F(F*F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "# Brute
 strength solution doesn't work (it's a 4th order polynomial):\ne65 :=
 solve(\{e62=0, e63=0\}, \{g_eff,w_eff\});" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$e65G<$/%&g_effG*&-%'RootOfG6$,2*&,>*&)%\"gG\"\"$\"\"
\"%$g_fGF3\"\"%*$)F4F5F3F3*$)F1F5F3F3*&)F1\"\"#F3)%$w_fGF<F3F3*&)F4F<F
3)%\"wGF<F3F3*(F5F3)F4F2F3F1F3F3**F<F3F;F3FBF3F>F3F3**F<F3F4F3F=F3F1F3
!\"\"*,F5F3F1F3FBF3F4F3F>F3FG**F<F3F1F3FAF3F4F3FG**F<F3F@F3F>F3FBF3F3*
(\"\"'F3F;F3F@F3F3*&F@F3F=F3F3*&F;F3FAF3F3F3)%#_ZGF<F3F3F/FG*(F<F3F;F3
F@F3FG*(F4F3F=F3F1F3FG*(F1F3FAF3F4F3FG*&FDF3F1F3FG*,F<F3F1F3FBF3F4F3F>
F3FG*&F.F3FPF3F3/%&labelG%$_L1GF3,&F1F3F4F3F3/%&w_effG,$*&,.*(F)F3F1F3
FBF3F3*&F>F3F1F3F3*(F1F3F>F3F)F3F3*(F)F3F4F3F>F3FG*&F4F3FBF3FG*(F)F3F4
F3FBF3FGF3FZFGFG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "e66 := \+
solve(e62=0,g_eff);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$e66G6$,(%\"g
G#!\"\"\"\"#*&#\"\"\"F*F-%$g_fGF-F)*&#F-F*F--%%sqrtG6#,0*&F'F-F.F-\"\"
'*$)F.F*F-F-*$)F'F*F-F-*(\"\"%F-%&w_effGF-%\"wGF-F-*(F<F-%$w_fGF-F=F-F
)*(F<F-F>F-F@F-F-*&F<F-)F=F*F-F)F-F-,(F'F(*&#F-F*F-F.F-F)*&#F-F*F-*$F1
F-F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "e67 := solve(e63=
0,w_eff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e67G,$*&,2*(%&g_effG\"
\"\"%\"gGF*%\"wGF*F**(F)F*%$g_fGF*%$w_fGF*!\"\"*&)F+\"\"#F*F/F*F**(F+F
*F/F*F)F*F**&)F.F3F*F,F*F0*(F.F*F/F*F+F*F**(F)F*F.F*F,F*F0*(F+F*F.F*F,
F*F0F*,(*&F+F*F.F*F3*$F6F*F**$F2F*F*F0F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "e68 := solve(e62=0,w_eff);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$e68G6$,(%\"wG#\"\"\"\"\"#*&#F)F*F)%$w_fGF)!\"\"*&F(F
)-%%sqrtG6#,0*$)F'F*F)F)*(F*F)F'F)F-F)F)*$)F-F*F)F)*(\"\"%F)%&g_effGF)
%\"gGF)F.*(F:F)F<F)%$g_fGF)F)*&F:F))F;F*F)F.*(F:F)F;F)F>F)F.F)F),(F'F(
*&#F)F*F)F-F)F.*&#F)F*F)*$F0F)F)F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "e69 := solve(e63=0,g_eff);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$e69G,$*&,0*&%&w_effG\"\"\")%\"gG\"\"#F*F**(%$g_fGF*%
$w_fGF*F,F*F**&F+F*F0F*F**(F,F*F/F*%\"wGF*!\"\"*&)F/F-F*F3F*F4*&F6F*F)
F*F***F-F*F)F*F/F*F,F*F*F*,**&F,F*F3F*F**&F/F*F0F*F4*&F0F*F,F*F**&F/F*
F3F*F4F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 397 "# Surprising
ly, The imaginary part of the equation e61 (in e63) is linear in w_eff
 or g_eff, and\n# so can be solved more easily.  It seems more natural
 to let this determine w_eff, since in the\n# real limit w=w_f=0, w_ef
f should be zero also, while g_eff should still involve a quadratic.\n
# Thus use e67 to determine w_eff, and substitute into e66 to determin
e g_eff:\n\ne70 := subs(w_eff=e67, e62);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$e70G,@*(%&g_effG\"\"\"%\"gGF(%$g_fGF(\"\"%*,\"\"#F(,2*(F'F(F)
F(%\"wGF(F(*(F'F(F*F(%$w_fGF(!\"\"*&)F)F-F(F2F(F(*(F)F(F2F(F'F(F(*&)F*
F-F(F0F(F3*(F*F(F2F(F)F(F(*(F'F(F*F(F0F(F3*(F)F(F*F(F0F(F3F(,(*&F)F(F*
F(F-*$F8F(F(*$F5F(F(F3F*F(F0F(F(*,F-F(F*F(F2F(F.F(F<F3F3**F-F(F*F(F2F(
F0F(F3*,F-F(F.F(F<F3F)F(F0F(F(*,F-F(F)F(F2F(F.F(F<F3F3**F-F(F)F(F2F(F0
F(F3**F-F(F.F-F<!\"#F*F(F(**F-F(F.F-F<FFF)F(F(*(F-F(F8F(F)F(F3*(F-F(F'
F(F5F(F(*(F-F()F'F-F(F)F(F(*(F-F(FKF(F*F(F(*(F-F(F'F(F8F(F(*(F-F(F5F(F
*F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*(,hp*(%$g_fG\"\"\")%$w_fG\"\"#F()%
\"gG\"\"$F(!\"\"**F+F()F'F+F(F)F()F-F+F(F/**F+F(F2F(F1F()%\"wGF+F(F/*(
F-F()F'F.F(F4F(F/*(F)F(F7F(F-F(F/*(F4F(F,F(F'F(F/*&)F-\"\"&F(F'F(F/*(
\"\"'F(F,F(F7F(F/*(\"\"%F()F-F@F(F1F(F/*&F-F()F'F<F(F/*(F@F(F2F()F'F@F
(F/*,F@F(F1F(F*F(F2F(F5F(F/*,F+F(F7F(F*F(F-F(F5F(F/*,F+F(F,F(F'F(F5F(F
*F(F/*(F,F(F)F(%&g_effGF(F(*(F2F(F)F()FJF+F(F(*(F7F(F4F(FJF(F(*(FLF(F1
F(F4F(F(*.F+F(FJF(F1F(F*F(F-F(F5F(F/*,F+F(FLF(F'F(F)F(F-F(F/*,F+F(FJF(
F7F(F*F(F5F(F(**FJF(F1F(F)F(F-F(F/*,F+F(FLF(F1F(F*F(F5F(F(*.F+F(FJF(F2
F(F5F(F'F(F*F(F/*,F+F(FLF(F-F(F4F(F'F(F/**FJF(F2F(F4F(F'F(F/**FJF(F'F(
F)F(F2F(F/*.F@F(FLF(F-F(F5F(F'F(F*F(F/*,F+F(FJF(F,F(F5F(F*F(F(*,F+F(FL
F(F2F(F5F(F*F(F(**FJF(F-F(F4F(F1F(F/*(FLF(F2F(F4F(F(*(FLF(F1F(F)F(F(*&
FLF(FEF(F(*(F7F(F)F(FJF(F(*&FJF(FCF(F(*&F;F(FJF(F(**F@F(F-F(FLF(F7F(F(
*&FAF(FLF(F(**F<F(FAF(FJF(F'F(F(*(F,F(F4F(FJF(F(**F@F(F,F(FLF(F'F(F(**
\"#5F(F,F(FJF(F1F(F(**FboF(F2F(FJF(F7F(F(**F>F(F2F(FLF(F1F(F(**F<F(F-F
(FEF(FJF(F(F(,&F-F(F'F(F/,(*&F-F(F'F(F+*$F1F(F(*$F2F(F(F/F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "e71 := solve(%=0, g_eff);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$e71G6$,$*(,>*&)%\"gG\"\"$\"\"\"%$g
_fGF-\"\"%*$)F.F/F-F-*$)F+F/F-F-*&)F+\"\"#F-)%$w_fGF6F-F-*&)F.F6F-)%\"
wGF6F-F-*(F/F-)F.F,F-F+F-F-**F6F-F5F-F<F-F8F-F-**F6F-F.F-F7F-F+F-!\"\"
*,F/F-F+F-F<F-F.F-F8F-FA**F6F-F+F-F;F-F.F-FA**F6F-F:F-F8F-F<F-F-*(\"\"
'F-F5F-F:F-F-*&F:F-F7F-F-*&F5F-F;F-F-FA,@F)!\"%F0FAF2FAF4FAF9FA*(F/F-F
>F-F+F-FA**F6F-F5F-F<F-F8F-FA**F6F-F.F-F7F-F+F-F-*,F/F-F+F-F<F-F.F-F8F
-F-**F6F-F+F-F;F-F.F-F-**F6F-F:F-F8F-F<F-FA*(FFF-F5F-F:F-FAFGFAFHFA*$-
%%sqrtG6#,fp*&F1F-)F8F/F-F-*&FXF-F3F-F-**F6F-FXF-F5F-F:F-FA*(\"$;\"F-)
F.\"\"&F-F*F-F-*(\"#7F-)F.\"\"(F-F+F-F-*(\"#_F-)F+FFF-F:F-F-*(FjnF-)F+
F\\oF-F.F-F-*(FfnF-)F+FhnF-F>F-F-*(\"$]\"F-F3F-F1F-F-*(F6F-)F.FFF-F;F-
F-*(F6F-FgoF-F7F-F-*(F6F-F_oF-F7F-F-**FjnF-FgnF-F7F-F+F-F-*,\"#GF-F3F-
F<F-F8F-F:F-F-*,\"#CF-FcoF-F<F-F8F-F.F-F-**FjnF-FcoF-F;F-F.F-F-**\"#9F
-F:F-F7F-F3F-F-**FapF-F1F-F7F-F5F-F-**F6F-F5F-)F<F/F-F:F-FA**\"\")F-F*
F-F;F-F>F-F-**FapF-F3F-F;F-F:F-F-**FapF-F5F-F;F-F1F-F-**FjnF-FgnF-F;F-
F+F-F-**FjnF-FcoF-F7F-F.F-F-*,FfpF-F:F-)F8F,F-F5F-F<F-FA*,FfpF-F5F-)F<
F,F-F:F-F8F-FA*,\"#;F-F>F-F8F-F<F-F*F-F-*,F^pF-FgnF-F8F-F<F-F+F-F-*,Fj
nF-F:F-F7F-F5F-F;F-FA*,F\\pF-F1F-F8F-F<F-F5F-F-**F/F-F1F-F\\qF-F<F-F-*
*F/F-F3F-F\\qF-F<F-F-**FFF-F3F-F7F-F;F-F-**F/F-F1F-F^qF-F8F-F-**FFF-F1
F-F;F-F7F-F-**F/F-F3F-F^qF-F8F-F-*(F6F-F_oF-F;F-F-*&F1F-FdpF-F-*&F3F-F
dpF-F-**F/F-F_oF-F<F-F8F-F-**F/F-FgoF-F8F-F<F-F-*$)F.FfpF-F-*$)F+FfpF-
F-*(F^oF-FgoF-F5F-F-**FfpF-F>F-F7F-F*F-F-F-F-F-,&F+F-F.F-F-#F-F6,$*(F(
FA,@F)FJF0FAF2FAF4FAF9FA*(F/F-F>F-F+F-FA**F6F-F5F-F<F-F8F-FA**F6F-F.F-
F7F-F+F-F-*,F/F-F+F-F<F-F.F-F8F-F-**F6F-F+F-F;F-F.F-F-**F6F-F:F-F8F-F<
F-FA*(FFF-F5F-F:F-FAFGFAFHFAFRFAF-FerF-Ffr" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "e72 := simplify(%[1]);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$e72G,$*(,@*&)%\"gG\"\"$\"\"\"%$g_fGF,\"\"%*$)F-F.F,F
,*$)F*F.F,F,*&)F*\"\"#F,)%$w_fGF5F,F,*&)F-F5F,)%\"wGF5F,F,*(F.F,)F-F+F
,F*F,F,**F5F,F4F,F;F,F7F,F,**F5F,F-F,F6F,F*F,!\"\"*,F.F,F*F,F;F,F-F,F7
F,F@**F5F,F*F,F:F,F-F,F@**F5F,F9F,F7F,F;F,F,*(\"\"'F,F4F,F9F,F,*&F9F,F
6F,F,*&F4F,F:F,F,*$-%%sqrtG6#*&),&F*F,F-F,F5F,,fo*&F9F,)F7F.F,F,*&FQF,
F4F,F,**F5F,F-F,FQF,F*F,F@**\"\")F,F-F,F6F,F)F,F,**F.F,F9F,F6F,F4F,F@*
*F.F,F4F,F9F,F:F,F@**FUF,F*F,F=F,F:F,F,**FUF,F6F,F=F,F*F,F,**FUF,F:F,F
)F,F-F,F,*(\"#5F,)F*\"\"&F,F-F,F,*(\"#WF,F)F,F=F,F,*(\"#JF,F2F,F9F,F,*
(FfnF,F*F,)F-FhnF,F,*(F\\oF,F4F,F0F,F,*,FUF,F9F,F7F,F4F,F;F,F@*,\"#;F,
F=F,F7F,F*F,F;F,F,*,FboF,F)F,F-F,F;F,F7F,F,*(F5F,F2F,F:F,F,*(F5F,F0F,F
:F,F,*$)F-FEF,F,*$)F*FEF,F,**F.F,F2F,F;F,F7F,F,**FEF,F4F,F:F,F6F,F,**F
.F,F4F,F;F,)F7F+F,F,**F.F,F4F,F7F,)F;F+F,F,**F5F,F*F,F-F,)F;F.F,F@*(F5
F,F2F,F6F,F,*,\"#7F,F*F,F-F,F:F,F6F,F@*,FUF,F*F,F-F,F;F,F]pF,F@*,FUF,F
*F,F-F,F7F,F_pF,F@**F.F,F9F,F]pF,F;F,F,**F.F,F_pF,F9F,F7F,F,**FEF,F9F,
F6F,F:F,F,**F.F,F0F,F7F,F;F,F,*(F5F,F0F,F6F,F,*&FapF,F9F,F,*&F4F,FapF,
F,F,F,F@F,FNF,,>F(F.F/F,F1F,F3F,F8F,*(F.F,F=F,F*F,F,**F5F,F4F,F;F,F7F,
F,**F5F,F-F,F6F,F*F,F@*,F.F,F*F,F;F,F-F,F7F,F@**F5F,F*F,F:F,F-F,F@**F5
F,F9F,F7F,F;F,F,*(FEF,F4F,F9F,F,FFF,FGF,F@#F@F5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1181 "\n# Define a procedure to evaluate nueff:\n
\nfunc_nueff := proc(nu,nu_f) \n      local g, w, g_f, w_f , g_eff, w_
eff;\n      g   := Re(nu)   ; w   := Im(nu);\n      g_f := Re(nu_f) ; \+
w_f := Im(nu_f);\ng_eff := -1/2*(g^4+g_f^2*w^2+w^2*g^2+6*g^2*g_f^2+4*g
^3*g_f+4*g*g_f^3-2*g*g_f*w^2+g_f^4-4*g*w*w_f*g_f+w_f^2*g_f^2+g^2*w_f^2
+2*g^2*w*w_f-2*w_f^2*g_f*g+2*g_f^2*w*w_f-sqrt((g+g_f)^2*(g^6-2*w_f^4*g
_f*g+w^4*g^2+w^4*g_f^2+2*w_f^2*g_f^4+4*g_f^4*w*w_f+4*g_f^2*w^3*w_f+6*w
^2*w_f^2*g_f^2+4*w*w_f^3*g_f^2-12*g*g_f*w_f^2*w^2-8*g*g_f*w_f*w^3-8*g*
g_f*w*w_f^3+2*g^4*w_f^2+4*g^4*w*w_f+6*g^2*w_f^2*w^2+4*g^2*w_f*w^3+4*g^
2*w*w_f^3+2*g^4*w^2+2*g_f^4*w^2+g^2*w_f^4+w_f^4*g_f^2+g_f^6-2*w^4*g*g_
f-4*g_f^2*g^2*w^2+8*g*g_f^3*w^2+8*g^3*g_f*w^2+31*g^2*g_f^4+44*g^3*g_f^
3+8*w_f^2*g_f*g^3-4*w_f^2*g_f^2*g^2+8*w_f^2*g_f^3*g+31*g^4*g_f^2-8*w_f
*g_f^2*g^2*w+16*w_f*g_f^3*g*w+16*g^3*g_f*w*w_f+10*g^5*g_f+10*g*g_f^5))
)*(g+g_f)/(g^4+g_f^2*w^2+w^2*g^2+6*g^2*g_f^2+4*g^3*g_f+4*g*g_f^3-2*g*g
_f*w^2+g_f^4-4*g*w*w_f*g_f+w_f^2*g_f^2+g^2*w_f^2+2*g^2*w*w_f-2*w_f^2*g
_f*g+2*g_f^2*w*w_f)           ;\nw_eff := -(g*w*g_eff-g_eff*w_f*g_f+g^
2*w_f+g*w_f*g_eff+w_f*g_f*g-g*g_f*w-g_f^2*w-g_f*w*g_eff)/(g^2+g_f^2+2*
g*g_f) ;\ng+I*w_eff;\nend ;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+fu
nc_nueffGf*6$%#nuG%%nu_fG6(%\"gG%\"wG%$g_fG%$w_fG%&g_effG%&w_effG6\"F0
C)>8$-%#ReG6#9$>8%-%#ImGF6>8&-F56#9%>8'-F;F?>8(,$*(,@*$)F3\"\"%\"\"\"F
L*&)F=\"\"#FL)F9FOFLFL*&FPFL)F3FOFLFL*(\"\"'FLFRFLFNFLFL*(FKFL)F3\"\"$
FLF=FLFL*(FKFLF3FL)F=FWFLFL**FOFLF3FLF=FLFPFL!\"\"*$)F=FKFLFL*,FKFLF3F
LF9FLFBFLF=FLFen*&)FBFOFLFNFLFL*&FRFLFjnFLFL**FOFLFRFLF9FLFBFLFL**FOFL
FjnFLF=FLF3FLFen**FOFLFNFLF9FLFBFLFL-%%sqrtG6#*&),&F3FLF=FLFOFL,fo**F3
FLF=FLFBFL)F9FWFL!\")*,\"\")FLF3FLF=FLF9FL)FBFWFLFen*,FjoFLFBFLFNFLFRF
LF9FLFen*,\"#;FLFBFLFYFLF3FLF9FLFL*,F^pFLFVFLF=FLF9FLFBFLFL*(FOFLFJFLF
jnFLFL*,\"#7FLF3FLF=FLFjnFLFPFLFen*$)F3FTFLFL*$)F=FTFLFL*&)FBFKFLFNFLF
L*&FRFLFhpFLFL*&)F9FKFLFNFLFL*&F[qFLFRFLFL*(\"#JFLFRFLFgnFLFL*(FOFLFgn
FLFPFLFL*(FOFLFJFLFPFLFL*(FOFLFjnFLFgnFLFL*(\"#5FLF3FL)F=\"\"&FLFL*(Fc
qFL)F3FeqFLF=FLFL*(F^qFLFJFLFNFLFL*(\"#WFLFVFLFYFLFL**FOFLF[qFLF3FLF=F
LFen**FKFLFRFLF9FLF[pFLFL**FKFLFRFLFBFLFgoFLFL**FTFLFRFLFjnFLFPFLFL**F
KFLFJFLF9FLFBFLFL**FKFLF9FLF[pFLFNFLFL**FTFLFPFLFjnFLFNFLFL**FKFLFNFLF
goFLFBFLFL**FKFLFgnFLF9FLFBFLFL**FOFLFhpFLF=FLF3FLFen**FjoFLFjnFLFYFLF
3FLFL**FKFLFjnFLFNFLFRFLFen**FjoFLFjnFLF=FLFVFLFL**FjoFLFVFLF=FLFPFLFL
**FjoFLF3FLFYFLFPFLFL**FKFLFNFLFRFLFPFLFenFLFenFLFdoFL,>FIFLFMFLFQFL*(
FTFLFRFLFNFLFL*(FKFLFVFLF=FLFL*(FKFLF3FLFYFLFL**FOFLF3FLF=FLFPFLFenFfn
FL*,FKFLF3FLF9FLFBFLF=FLFenFinFLF[oFL**FOFLFRFLF9FLFBFLFL**FOFLFjnFLF=
FLF3FLFen**FOFLFNFLF9FLFBFLFLFen#FenFO>8),$*&,2*(F3FLF9FLFEFLFL*(FEFLF
BFLF=FLFen*&FRFLFBFLFL*(F3FLFBFLFEFLFL*(FBFLF=FLF3FLFL*(F3FLF=FLF9FLFe
n*&FNFLF9FLFen*(F=FLF9FLFEFLFenFL,(*$FRFLFL*$FNFLFL*(FOFLF3FLF=FLFLFen
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