{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "Help Head ing" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 37 " \+ expands - symbolic version of expand" }}{PARA 0 "" 0 "" {TEXT 26 18 "C alling sequences:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 14 "expands(expr); " }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 5 "expr " }{TEXT -1 32 "- the expression to b e expanded." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 155 "This procedure distributes powers over \+ sums and applies expansion formulas for many functions. The main diffe rence between this and the built-in procedure " }{MPLTEXT 0 21 6 "expa nd" }{TEXT -1 9 " is that " }{MPLTEXT 0 21 7 "expands" }{TEXT -1 154 " applies the formulas even in cases when they are not known to be vali d. Thus this is a \"symbolic\" expansion, analogous to the symbolic si mplification of " }{MPLTEXT 0 21 23 "simplify(..., symbolic)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {MPLTEXT 0 21 6 "expand" }{TEXT -1 161 " i s the procedure normally used to expand an expression. However, the ex pansion formulas for certain functions are only valid under restrictiv e assumptions, and " }{MPLTEXT 0 21 6 "expand" }{TEXT -1 87 " will not apply them unless Maple knows that these assumptions are valid. For e xample, " }{XPPEDIT 18 0 "ln(a*b) = ln(a)+ln(b)" "6#/-%#lnG6#*&%\"aG\" \"\"%\"bGF),&-F%6#F(F)-F%6#F*F)" }{TEXT -1 42 " is not always valid (e .g. it is false if " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 32 " are negative reals). There fore " }{MPLTEXT 0 21 6 "expand" }{TEXT -1 28 " will only apply it in \+ case " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 74 " is known to be a positive real. This can \+ be made known to Maple by using " }{MPLTEXT 0 21 6 "assume" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 190 "However, in many cases it is in convenient to make the necessary assumptions, or the assumptions are e xcessively restrictive (e.g. the expansion formula for ln given above \+ is also valid when " }{XPPEDIT 18 0 "abs(arg(a)+arg(b)) < Pi" "6#2-%$a bsG6#,&-%$argG6#%\"aG\"\"\"-F)6#%\"bGF,%#PiG" }{TEXT -1 6 ", but " } {MPLTEXT 0 21 6 "expand" }{TEXT -1 18 " will not use it)." }}{PARA 15 "" 0 "" {TEXT -1 45 "To go in the reverse direction, there is the " } {MPLTEXT 0 21 7 "combine" }{TEXT -1 85 " procedure. Note that some of \+ the sub-procedures of this have symbolic options, e.g. " }{MPLTEXT 0 21 29 "combine(..., power, symbolic)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {MPLTEXT 0 21 7 "expands" }{TEXT -1 45 " will also expand the arcta n function, which " }{MPLTEXT 0 21 6 "expand" }{TEXT -1 34 " will not. This uses the formulas " }{XPPEDIT 18 0 "arctan((a+b)/(1-a*b)) = arct an(a)+arctan(b)" "6#/-%'arctanG6#*&,&%\"aG\"\"\"%\"bGF*F*,&F*F**&F)F*F +F*!\"\"F.,&-F%6#F)F*-F%6#F+F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "ar ctan(2*a/(1-a^2))=2*arctan(a)" "6#/-%'arctanG6#*(\"\"#\"\"\"%\"aGF),&F )F)*$F*F(!\"\"F-*&F(F)-F%6#F*F)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 24 "There is one case where " }{MPLTEXT 0 21 6 "expand" } {TEXT -1 57 " will produce an expansion which may be undesirable, and \+ " }{MPLTEXT 0 21 7 "expands" }{TEXT -1 16 " will not: when " } {XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 24 " is assumed to be real, " } {MPLTEXT 0 21 16 "expand(exp(I*a))" }{TEXT -1 12 " results in " } {XPPEDIT 18 0 "(-1)^a" "6#),$\"\"\"!\"\"%\"aG" }{TEXT -1 8 ", while " }{MPLTEXT 0 21 17 "expands(exp(I*a))" }{TEXT -1 14 " leaves it as " } {XPPEDIT 18 0 "exp(I*a)" "6#-%$expG6#*&%\"IG\"\"\"%\"aGF(" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 29 "This function is part of the " } {TEXT 256 22 "Maple Advisor Database" }{TEXT -1 9 " library." }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 5 "Both " }{MPLTEXT 0 21 6 "expand" }{TEXT -1 5 " and " } {MPLTEXT 0 21 7 "expands" }{TEXT -1 37 " produce many of the same expa nsions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "expands((x+1)*(x+ 2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&\"\"$ F(F&F(F(F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expands(sin (x+y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%$ cosG6#%\"yGF)F)*&-F+F'F)-F&F,F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 106 "Th e expansion of the logarithm of a product. This is not always valid, a s there may be an additional term " }{XPPEDIT 18 0 "2*Pi*I*n" "6#**\" \"#\"\"\"%#PiGF%%\"IGF%%\"nGF%" }{TEXT -1 7 " where " }{XPPEDIT 18 0 " n" "6#%\"nG" }{TEXT -1 15 " is an integer." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "expand(ln(x*y)),expands(ln(x*y)); \n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$-%#lnG6#*&%\"xG\"\"\"%\"yGF(,&-F$6#F'F(-F$6#F)F( " }}}{PARA 0 "" 0 "" {TEXT -1 85 "The square root of a product. This \+ is not always valid, as there may be a factor of " }{XPPEDIT 18 0 "-1 " "6#,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "expand(sqrt(x*y)),expands(sqrt(x*y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$-%%sqrtG6#*&%\"xG\"\"\"%\"yGF)F)*&-F%6#F(F)-F%6# F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 62 "A power of a power. Again, not \+ always valid (e.g. try it for " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\" \"\"!\"\"" }{TEXT -1 2 ")." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "expand((x^(3/2))^(2/3)), expands((x^(3/2))^(2/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$)*$)%\"xG#\"\"$\"\"#\"\"\"#F*F)F+F'" }}}{PARA 0 "" 0 "" {TEXT -1 23 "An expandable arctan. " }{MPLTEXT 0 21 6 "expand " }{TEXT -1 56 " won't do anything with this, even under conditions th at" }}{PARA 0 "" 0 "" {TEXT -1 23 "guarantee its validity." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "assume(x>0,x<1/4,y>0,y<1/4);\nexpan d(arctan((x+y)/(1-x*y))), expands(arctan((x+y)/(1-x*y)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'arctanG6#*&,&%#x|irG\"\"\"%#y|irGF)F),&F)F)* &F(F)F*F)!\"\"F-,&-F$6#F*F)-F$6#F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 13 " A case where " }{MPLTEXT 0 21 6 "expand" }{TEXT -1 13 " expands but " }{MPLTEXT 0 21 7 "expands" }{TEXT -1 40 " does not. I think it is bet ter to use " }{MPLTEXT 0 21 3 "exp" }{TEXT -1 24 " rather than a power of " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 58 " here, to av oid complications with multivalued functions. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "assume(x,real); expand(exp(2*Pi*I*x)), expands(e xp(2*Pi*I*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$)!\"\",$%#x|irG\"\"# -%$expG6#*(^#F'\"\"\"%#PiGF-F&F-" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "See also:" }}{PARA 0 "" 0 "" {HYPERLNK 17 "assume" 2 "assume" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "combine" 2 "combine" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "expand" 2 "expand" "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 26 23 "Maple Advis or Database " }{TEXT -1 15 "R. Israel 2000 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "2 7 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }