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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(MS):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 303 "This example will demonstrate how the me
thods of multisectioning can be applied to functions with symbolic par
ameters for parameters of the exponentials of rational poly-exponentia
l functions.  Define the ``\{\\em Bernoulli polynomials\}'' to be the \+
coefficients of the exponential generating function of  " }{XPPEDIT 
18 0 "x*exp(t*x)/(exp(x)-1);" "6#*(%\"xG\"\"\"-%$expG6#*&%\"tGF%F$F%F%
,&-F'6#F$F%\"\"\"!\"\"F/" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "x;" "6#%
\"xG" }{TEXT -1 0 "" }{TEXT -1 163 ".  The denominator and numerator o
f this function have very complicated lacunary recurrence relations, e
ven when multisectioning by a small value such as 3 (at 0)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "top := x* exp(t*x):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bot := exp(x)-1:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "botlrr := `bottom/ms`(bot, f, x, 3)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'botlrrG6&/-%\"fG6#%\"xG-F(6#,&
F*\"\"\"!\"'F.F(F*7(/-F(6#\"\"!F4/-F(6#F.F4/-F(6#\"\"#F4/-F(6#\"\"$\"
\"'/-F(6#\"\"%F4/-F(6#\"\"&F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 64 "toplrr := collect([`top/ms/linalg/sym`(top,bot, f, x, 3, 0)],f);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'toplrrG7&/-%\"fG6#%\"xG,2*&,H*$
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oFhw/-F(6#Fio,,F=F\\pF[rF^vFdq\"%g7FC\"$/&F`q!%7:/-F(6#FfoFhw/-F(6#F9F
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FUFhw/-F(6#F1Fhw/-F(6#FM,4\"#IF\\pF[r!%l8Fdq\"&I+$F7!&!Q;FC\"&!4!*F[o!
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n$Fgq\"&![CF[r!%WtF7!'oFAFK\"%'*[FS\"&!o&)FV\"'G,qF[o!(SFJ\"F`q!'%Q6\"
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q\"'!\\.#F4F_^lF7!)![\"e5F;\"%!)zFC\"(?zi\"FG!&?=(F[o!(q`k#Fdo\"(_(fxF
`q!'_n(*Fjo\"(S2H&/-F(6#FfnFhw/-F(6#FcoFhw" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 9 "Now, if  " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT 
-1 78 " this will reduce to the situation of looking at the normal Ber
noulli numbers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(t=0
,[toplrr]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7#7&/-%\"fG6#%\"xG,.-F'
6#,&F)\"\"\"!#=F.!\"\"-F'6#,&F)F.!#:F.\"\"#-F'6#,&F)F.!#7F.F.-F'6#,&F)
F.!\"*F.!\"%-F'6#,&F)F.!\"'F.F.-F'6#,&F)F.!\"$F.F5F'F)7:/-F'6#\"\"!FK/
-F'6#F.FK/-F'6#F5FK/-F'6#\"\"$\"\"'/-F'6#\"\"%FK/-F'6#\"\"&FK/-F'6#FVF
K/-F'6#\"\"(FK/-F'6#\"\")FK/-F'6#\"\"*\"#=/-F'6#\"#5FK/-F'6#\"#6FK/-F'
6#\"#7FK/-F'6#\"#8FK/-F'6#\"#9FK/-F'6#\"#:\"#I/-F'6#\"#;FK/-F'6#\"#<FK
/-F'6#FhoFK/-F'6#\"#>FK/-F'6#\"#?FK/-F'6#\"#@\"#U/-F'6#\"#AFK/-F'6#\"#
BFK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 179 "This example is interesting because it demonstrat
es how large and complicated the results get when done symbolically, b
ut still shows that feasibility of doing these calculations." }}}}
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