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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 42 "\nA supplementing worksheet to
 the article\n" }}{PARA 257 "" 0 "" {TEXT 257 73 "\"Symbolic evaluatio
n of coefficients in\nAiry-type asymtotic Expansions\"\n\n" }{TEXT 
258 3 "by " }{TEXT 260 9 "R.Vidunas" }{TEXT 259 5 " and " }{TEXT 261 
7 "N.Temme" }{TEXT 262 18 " (CWI, Amsterdam)\n" }{TEXT 264 12 "Uses pa
ckage" }{TEXT 263 14 "  airypcf.mpl\n" }}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 42 "First read package, then define a function" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "read `airypcf.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%_oThis~package~computes~Airy-type~asymptotic~expansion~of~the~i
ntegralG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*&\"\"\"F%*&%#piGF%%\"iG
F%!\"\"#F%\"\"#-%$intG6$*&-%$expG6#*&%\"zGF%,&*$)%\"wG\"\"$F%#F%F9*&%$
etaGF%F8F%F)F%F%-%\"fG6#F8F%F8%\".G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%RFor~the~input~one~has~to~redefine~Maple~functionsG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%-%'AiryPwG6#%\"kG%&~and~G-%'AiryPmGF%" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6)%Gwhich~specify~the~k-th~coefficient~of~G-%\"fG
6#%\"wG%&~and~G-F%6#,$F'!\"\"%5~at~the~saddle~pointG/F'%&AiryBG%\".G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6%%&Here~G/%&AiryBG-%%sqrtG6#%$etaG%7,
~is~a~global~variableG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%:For~exampl
e,~for~general~G-%\"fG6#%\"wG%0one~may~define:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%HAiryPw:=~k->p[k];~~~~~AiryPm:=~k->q[k];G" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6%%)And~for~G/-%\"fG6#%\"wG*&\"\"\"F*,&F*F*F(!\"
\"F,%*~we~have:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%'AiryPwG6#%\"kG
*&)!\"\"F'\"\"\"),&F+F+%&AiryBGF+F'F*%\"~G/-%'AiryPmGF&*&F+F+),&F+F+F.
F*F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@For~the~output,~call~funct
ions~G3-%*AiryAlphaG6#%\"kG-%)AiryBetaGF'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%Uto~get~the~coefficients~in~the~assymptotic~expansionG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*()%\"zG%%-1/3G\"\"\"-%#AiG6#*&%
$etaGF()F&%$2/3GF(F(-%$sumG6$*&*&)!\"\"%\"kGF(&%&alphaG6#F7F(F()F&F7F6
/F7;\"\"!%)infinityGF(F(*()F&%%-2/3GF(--%\"DG6#F*F+F(-F16$*&*&F5F(&%%b
etaGF:F(F(F;F6F<F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&%=These~coef
fcients~depend~on~G\"\"\"%$etaGF%%<(or~equivalently,~on~AiryB)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%OIt~is~convenient~to~rename~AiryB,~fo
r~example~G-%&aliasG6#/%\"bG%'AiryB~G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%eo(However,~either~define~AipyPw,~AiryPm~using~AiryB,~or~first~def
ine~alias)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%UTo~work~with~Weber~
parabolic~cylinder~function,~callG\"\"\"-%*ParCyInfoG6\"F%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%8Please~define~functionsG3-%'AiryPwG6#%\"kG
-%'AiryPmGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "AiryPw:= k-
>(-1)^k/(1+AiryB)^k;\nAiryPm:= k->1/(1-AiryB)^k;\nalias(b=AiryB);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AiryPwGR6#%\"kG6\"6$%)operatorG%&ar
rowGF(*&)!\"\"9$\"\"\"),&F0F0%&AiryBGF0F/F.F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'AiryPmGR6#%\"kG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-)
,&F-F-%&AiryBG!\"\"9$F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%\"IG
%(ParCyUaG%\"bG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "AiryAlph
a(5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,6*$)%\"bG\"\"#\"\"\"\"%h
Q!$H%F**$)F(\"#;F*\"(VaF#*$)F(\"#=F*\"'<PZ*$)F(\"\")F*!&Mn%*$)F(\"#5F*
\"&Is%*$)F(\"\"%F*!&K^\"*$)F(\"\"'F*\"&'fL*$)F(\"#7F*!&+V&*$)F(\"#9F*
\"'#p@)F**()F(FKF*),&F*F*F(F*F;F*),&!\"\"F*F(F*F;F*FS#!#N\"%#>)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "AiryBeta(4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&,2*$)%\"bG\"\"#\"\"\"\"%<;*$)F(\"\")F*!%*e&*$)
F(\"#5F*\"&r:\"*$)F(\"\"%F*!%FY*$)F(\"\"'F*\"%4n!$J#F**$)F(\"#7F*\"'()
)R#*$)F(\"#9F*\"&FZ*F**()F(F?F*),&F*F*F(F*F.F*),&!\"\"F*F(F*F.F*FK#!\"
&\"%[?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 42 "First define a function, then read packag
e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "alias(b=AiryB);\nAiryPw:= k->p[k];
\nAiryPm:= k->q[k];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"IG%\"bG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AiryPwGR6#%\"kG6\"6$%)operatorG%&ar
rowGF(&%\"pG6#9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AiryPmGR6
#%\"kG6\"6$%)operatorG%&arrowGF(&%\"qG6#9$F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "read `airypcf.mpl`;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%_oThis~package~computes~Airy-type~asymptotic~expansion
~of~the~integralG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*&\"\"\"F%*&%#p
iGF%%\"iGF%!\"\"#F%\"\"#-%$intG6$*&-%$expG6#*&%\"zGF%,&*$)%\"wG\"\"$F%
#F%F9*&%$etaGF%F8F%F)F%F%-%\"fG6#F8F%F8%\".G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%RFor~the~input~one~has~to~redefine~Maple~functionsG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%'AiryPwG6#%\"kG%&~and~G-%'AiryPmGF%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%Gwhich~specify~the~k-th~coefficie
nt~of~G-%\"fG6#%\"wG%&~and~G-F%6#,$F'!\"\"%5~at~the~saddle~pointG/F'%
\"bG%\".G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%&Here~G/%\"bG-%%sqrtG6#%
$etaG%7,~is~a~global~variableG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%:Fo
r~example,~for~general~G-%\"fG6#%\"wG%0one~may~define:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%HAiryPw:=~k->p[k];~~~~~AiryPm:=~k->q[k];G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%@For~the~output,~call~functions~G3-%*
AiryAlphaG6#%\"kG-%)AiryBetaGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Ut
o~get~the~coefficients~in~the~assymptotic~expansionG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,&*()%\"zG%%-1/3G\"\"\"-%#AiG6#*&%$etaGF()F&%$2/3GF(
F(-%$sumG6$*&*&)!\"\"%\"kGF(&%&alphaG6#F7F(F()F&F7F6/F7;\"\"!%)infinit
yGF(F(*()F&%%-2/3GF(--%\"DG6#F*F+F(-F16$*&*&F5F(&%%betaGF:F(F(F;F6F<F(
F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&%=These~coeffcients~depend~on~
G\"\"\"%$etaGF%%<(or~equivalently,~on~AiryB)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%OIt~is~convenient~to~rename~AiryB,~for~example~G-%&ali
asG6#/%\"bG%'AiryB~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%eo(However,~e
ither~define~AipyPw,~AiryPm~using~AiryB,~or~first~define~alias)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%UTo~work~with~Weber~parabolic~cylin
der~function,~callG\"\"\"-%*ParCyInfoG6\"F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%@At~the~moment~you~have~defined:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%'AiryPwGR6#%\"kG6\"6$%)operatorG%&arrowGF(&%\"pG6#9$F
(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'AiryPmGR6#%\"kG6\"6$%)oper
atorG%&arrowGF(&%\"qG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 52 "for i from 0 to 3 do  AiryAlpha(i), AiryBeta(i)  od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$,&&%\"pG6#\"\"!#\"\"\"\"\"#&%\"qGF&F(,$*&,&F
$F)F+!\"\"F)%\"bGF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&,.*&&%\"p
G6#\"\"\"F*%\"bGF*!\"\"*&&%\"qGF)F*F+F*F*&F(6#\"\"!F*&F/F1F,*&&F(6#\"
\"#F*)F+F7F*F7*&&F/F6F*F8F*!\"#F**$)F+\"\"$F*F,#F*\"\"),$*&,*F'F,F.F,*
&F+F*F5F*F7*&F+F*F:F*F7F**$)F+F>F*F,F?" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6$,$*&,2&%\"pG6#\"\"\"!\"(&%\"qGF(F**&%\"bGF)&F'6#\"\"#F)\"#9*&F.F
)&F,F0F)F2*&)F.F1F)&F'6#\"\"$F)!#?*&F6F)&F,F8F)F:*&)F.F9F)&F'6#\"\"%F)
\"#C*&F>F)&F,F@F)FBF)*$)F.\"\"&F)!\"\"#F)\"#k,$*&,6*&F&F)F.F)!\"&*&F+F
)F.F)FG&F'6#\"\"!FG&F,FRFO*&F/F)F6F)\"#5*&F4F)F6F)!#5*&F7F)F>F)F:*&F<F
)F>F)\"#?*&F?F))F.FAF)FB*&FDF)FgnF)!#CF)*$)F.\"\"(F)FHFI" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6$,$*&,>*&&%\"pG6#\"\"%\"\"\")%\"bGF*F+\"$C#*&&F(6
#\"\"#F+)F-F2F+\"#q*&&%\"qGF1F+F3F+!#q*&&F(6#F+F+F-F+!#N*&&F7F;F+F-F+
\"#N*&&F(6#\"\"$F+)F-FCF+!$S\"*&&F7FBF+FDF+\"$S\"&F(6#\"\"!F?&F7FJF<*&
&F7F)F+F,F+!$C#*&&F(6#\"\"&F+)F-FSF+!$!G*&&F7FRF+FTF+\"$!G*&&F(6#\"\"'
F+)F-FfnF+\"$S#*&&F7FenF+FgnF+!$S#F+*$)F-\"\"*F+!\"\"#F+\"$c#,$*&,:F:!
\"(F>Feo*&F-F+F0F+\"#9*&F-F+F6F+Fgo*&F3F+FAF+!#C*&F3F+FGF+Fjo*&FDF+F'F
+\"#S*&FDF+FNF+F]p*&F,F+FQF+!#c*&F,F+FWF+F`p*&FTF+FZF+\"#[*&FTF+FjnF+F
cpF+*$)F-F^oF+F_o#FSFao" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
ParCyInfo();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oTo~work~with~Weber~
parabolic~cylinder~function,~you~have~to~assign:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%DAiryPw:=ParCyPw;~~~AiryPm:=ParCyPm;G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%^oParabolic~cylinder~functions~are~solutions~of~d
ifferential~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%
\"yG6#%\"xG-%\"$G6$F*\"\"#*&,&*$)F*F.\"\"\"#F3\"\"%%\"aGF3F3F'F3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oFollowing~paper~[VT],~we~take~the~e
xponentially~decreasing~solutionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-
%\"UG6$%\"aG%\"xG%$~=~G*&-%$expG6#,$*$)F'\"\"#\"\"\"#F1\"\"%F1*&%\"iGF
1-%%sqrtG6#,$%#piGF0F1!\"\"-%$intG6$*&-F+6#,&*&F'F1%\"sGF1F;*$)FDF0F1#
F1F0F1)FD,&F&F;#F;F0F1F1FD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnThis~
integral~representation~can~be~analytically~transformed~toG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%eoa~standard~Airy-type~integral.~The~obtain
ed~Airy-type~asymptotic~expansionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
%Cis~in~the~(non-positive)~powers~ofG/%\"zG-%%sqrtG6#,$%\"aG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%VThe~coefficients~depend~on~mutally~r
elated~parametersG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/,$*$)%\"bG\"\"$
\"\"\"#\"\"#F(,&*&%\"tGF)-%%sqrtG6#,&*$)F.F+F)F)!\"\"F)F)F)-%$logG6#,&
F.F)F/F)F5%(~where~G/F.,$*&%\"xGF)-F06#,$%\"aGF5F5#F)F+" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$/%'ParCyUG*&-%%sqrtG6#,$%\"bG\"\"#\"\"\")*&,&%\"
tGF,!\"\"F,F,,&F0F,F,F,F1%$1/4GF,/%(ParCyXiG*&*&F0F,F*F,F,-F'6#,&*$)F0
F+F,F,F1F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%1Coefficients~in~G3%
\"bG%'ParCyUG%2~are~returned~by~G-%*AiryAlphaG6#%\"kG-%)AiryBetaGF*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6'%1Coefficients~in~G3%\"bG%(ParCyXiG%2
~are~returned~by~G-%+ParCyAlphaG6#%\"kG-%*ParCyBetaGF*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%ZIt~is~convenient~to~rename~AiryB,~ParCyU~and~Par
CyXi,~sayG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&aliasG6%/%\"bG%'AiryB
~G/%\"uG%(ParCyU~G/%#xiG%)ParCyXi~G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "Working wit
h parabolic cylinder function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "AiryPw:= P
arCyPw;\nAiryPm:= ParCyPm;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AiryP
wG%(ParCyPwG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AiryPmG%(ParCyPmG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "read `airypcf.mpl`;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oThis~package~computes~Airy-type~asy
mptotic~expansion~of~the~integralG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%
,$*&\"\"\"F%*&%#piGF%%\"iGF%!\"\"#F%\"\"#-%$intG6$*&-%$expG6#*&%\"zGF%
,&*$)%\"wG\"\"$F%#F%F9*&%$etaGF%F8F%F)F%F%-%\"fG6#F8F%F8%\".G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%RFor~the~input~one~has~to~redefine~Ma
ple~functionsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%'AiryPwG6#%\"kG%&~
and~G-%'AiryPmGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%Gwhich~specify~t
he~k-th~coefficient~of~G-%\"fG6#%\"wG%&~and~G-F%6#,$F'!\"\"%5~at~the~s
addle~pointG/F'%&AiryBG%\".G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%&Here
~G/%&AiryBG-%%sqrtG6#%$etaG%7,~is~a~global~variableG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%%:For~example,~for~general~G-%\"fG6#%\"wG%0one~may~d
efine:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%HAiryPw:=~k->p[k];~~~~~Air
yPm:=~k->q[k];G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@For~the~output,~c
all~functions~G3-%*AiryAlphaG6#%\"kG-%)AiryBetaGF'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%Uto~get~the~coefficients~in~the~assymptotic~expansio
nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*()%\"zG%%-1/3G\"\"\"-%#AiG6#*
&%$etaGF()F&%$2/3GF(F(-%$sumG6$*&*&)!\"\"%\"kGF(&%&alphaG6#F7F(F()F&F7
F6/F7;\"\"!%)infinityGF(F(*()F&%%-2/3GF(--%\"DG6#F*F+F(-F16$*&*&F5F(&%
%betaGF:F(F(F;F6F<F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&%=These~co
effcients~depend~on~G\"\"\"%$etaGF%%<(or~equivalently,~on~AiryB)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%OIt~is~convenient~to~rename~AiryB,~fo
r~example~G-%&aliasG6#/%\"bG%'AiryB~G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%eo(However,~either~define~AipyPw,~AiryPm~using~AiryB,~or~first~def
ine~alias)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%aoYou~already~have~cho
sen~to~work~with~Weber~parabolic~cylinder~functionG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%'AiryPwG%(ParCyPwG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%'AiryPmG%(ParCyPmG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%BFor~mor
e~information~please~call~G\"\"\"-%*ParCyInfoG6\"F%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "ParCyInfo();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%aoYou~already~have~chosen~to~work~with~Weber~parabolic
~cylinder~functionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'AiryPwG%(Par
CyPwG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'AiryPmG%(ParCyPmG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oParabolic~cylinder~functions~are~so
lutions~of~differential~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#*&,&*$)F*F.\"\"\"#F3\"\"%%\"aGF3
F3F'F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oFollowing~paper~[VT],~we~
take~the~exponentially~decreasing~solutionG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&-%\"UG6$%\"aG%\"xG%$~=~G*&-%$expG6#,$*$)F'\"\"#\"\"\"#F
1\"\"%F1*&%\"iGF1-%%sqrtG6#,$%#piGF0F1!\"\"-%$intG6$*&-F+6#,&*&F'F1%\"
sGF1F;*$)FDF0F1#F1F0F1)FD,&F&F;#F;F0F1F1FD" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%jnThis~integral~representation~can~be~analytically~tra
nsformed~toG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%eoa~standard~Airy-typ
e~integral.~The~obtained~Airy-type~asymptotic~expansionG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$%Cis~in~the~(non-positive)~powers~ofG/%\"zG-%%sq
rtG6#,$%\"aG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VThe~coefficient
s~depend~on~mutally~related~parametersG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%/,$*$)%&AiryBG\"\"$\"\"\"#\"\"#F(,&*&%\"tGF)-%%sqrtG6#,&*$)F.F+F
)F)!\"\"F)F)F)-%$logG6#,&F.F)F/F)F5%(~where~G/F.,$*&%\"xGF)-F06#,$%\"a
GF5F5#F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'ParCyUG*&-%%sqrtG6#,$
%&AiryBG\"\"#\"\"\")*&,&%\"tGF,!\"\"F,F,,&F0F,F,F,F1%$1/4GF,/%(ParCyXi
G*&*&F0F,F*F,F,-F'6#,&*$)F0F+F,F,F1F,F1" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6'%1Coefficients~in~G3%&AiryBG%'ParCyUG%2~are~returned~by~G-%*Airy
AlphaG6#%\"kG-%)AiryBetaGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%1Coeff
icients~in~G3%&AiryBG%(ParCyXiG%2~are~returned~by~G-%+ParCyAlphaG6#%\"
kG-%*ParCyBetaGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZIt~is~convenien
t~to~rename~AiryB,~ParCyU~and~ParCyXi,~sayG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%&aliasG6%/%\"bG%'AiryB~G/%\"uG%(ParCyU~G/%#xiG%)ParCy
Xi~G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "alias(b=AiryB, u=Pa
rCyU, xi=ParCyXi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%\"IG%(ParCyUaG%
\"bG%\"uG%#xiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "AiryAlpha
(2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,:*$)%\"bG\"#7\"\"\"\"(gpd
\"*$)%\"uGF)F*!(!oj=*&)F(\"\"%F*)F.\"#5F*!'C!H\"*&)F(\"\"'F*)F.F8F*\"'
?nG*&)F(\"\"#F*)F.\"#9F*!&cA$*$)F.\"#=F*\"%![%*$)F.\"#CF*\"$&Q*&F<F*)F
.\"#?F*!%Wb*&)F(\"\")F*)F.FOF*\"'g`\\*&F7F*F-F*!'g86*&)F(F4F*)F.F2F*!(
k#>9*&F1F*)F.\"#;F*\"&g4$F**&)F(F8F*)F.F)F*!\"\"#F*\")oV()=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for i from 0 to 3 do  ParCyA
lpha(i), ParCyBeta(i)  od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"#[,$*&,(*$)%#xiG\"\"$F
$\"\"&*&F+F$)%\"bG\"\"#F$!\"'!\"&F$F$*$)F0\"\"%F$!\"\"F#" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6$,$*&,0!$b%\"\"\"*$)%\"bG\"\"'F'!$V\"*&)F*\"\"#F'
)%#xiG\"\"%F'!$C**&)F*F2F')F1F/F'\"$%o*$)F1F+F'\"$&Q*$)F1\"\"$F'\"#q*&
F1F'F.F'!#%)F'*$)F*F+F'!\"\"#F'\"%3Y,$*&,(F;\"\"&F?!\"'!\"&F'F'*$)F*F2
F'FC#F'\"%/B" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$,$*&,0!%Do\"\"\"*$)%\"
bG\"\"'F'!%(='*&)F*\"\"#F')%#xiG\"\"%F'!&gQ\"*&)F*F2F')F1F/F'\"&g-\"*$
)F1F+F'\"%vd*$)F1\"\"$F'\"%]5*&F1F'F.F'!%g7F'*$)F*F+F'!\"\"#F'\"(gxJ$,
$*&,:F(\"&D2\"!'DaUF'*&)F*\"\")F'F1F'\"'5\"f#F;\"&v)G*&F.F')F1\"\"(F'!
(I:`\"*&F5F')F1\"\"&F'\"(7+/#*&F)F'F<F'!(0!*=\"*$)F1\"\"*F'\"'DaUF-\"&
+$pF4!&+8&F8!&v)GF?!&]Y$F'*$)F*\"#5F'FCFD" }}}{EXCHG {PARA 0 "> " 0 "
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