{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 "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -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 "Tim es" 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 43 " \+ PVInt - numerical principal value integral" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling sequence:\n" }{MPLTEXT 0 21 19 "PVint(f, x=a..b, p)" } {TEXT -1 0 "" }{MPLTEXT 0 21 0 "" }}{PARA 0 "" 0 "" {TEXT 26 11 "Param eters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 1 "f" }{TEXT -1 45 " - the integrand, expression in one variable " }{MPLTEXT 0 21 1 "x" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 1 "x" }{TEXT -1 7 " - name" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {MPLTEXT 0 21 4 "a, b" }{TEXT -1 51 " - real constants, the endpoints \+ of the integration" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 0 21 1 "p" }{TEXT -1 74 " - real constant, location of a singularity in the i nterval of integration" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 12 "Descri ption:" }}{PARA 15 "" 0 "" {TEXT -1 106 "This function produces a nume rical approximation for the Cauchy principal value integral of an expr ession " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 60 " over \+ an interval that may contain one known singularity of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 30 "The Cauchy principal value of " }{XPPEDIT 18 0 "Int(f(x),x=a..b )" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 9 ", where " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 22 " has a singulari ty at " }{XPPEDIT 18 0 "x=p" "6#/%\"xG%\"pG" }{TEXT -1 17 " in the int erval " }{XPPEDIT 18 0 "``(a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 16 ", is defined as " }{XPPEDIT 18 0 "Limit(``(Int(f(x),x = a .. p-epsilon) +Int(f(x),x = p+epsilon .. b)),epsilon = 0,right);" "6#-%&LimitG6%-%!G 6#,&-%$IntG6$-%\"fG6#%\"xG/F0;%\"aG,&%\"pG\"\"\"%(epsilonG!\"\"F6-F+6$ -F.6#F0/F0;,&F5F6F7F6%\"bGF6/F7\"\"!%&rightG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 10 "Note that " }{MPLTEXT 0 21 37 "int(f(x),x=a..b ,CauchyPrincipalValue)" }{TEXT -1 42 " is the symbolic version of this function." }}{PARA 15 "" 0 "" {TEXT -1 80 "This function works by fir st finding the singular part of the Laurent series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=p" "6#/%\"xG%\"pG" }{TEXT -1 80 ", and integrating it (symbolically). If this is finite, the difference between " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 73 " and its singular part is then integrated n umerically over the interval. " }}{PARA 15 "" 0 "" {TEXT -1 29 "This f unction is part of the " }{TEXT 256 22 "Maple Advisor Database" } {TEXT -1 9 " library." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples :" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "PVInt(ta n(x), x = 0 .. 2, Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'4rrw) !#5" }}}{PARA 0 "" 0 "" {TEXT -1 49 "This one could have been integrat ed symbolically." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int(tan( x),x=0..2,CauchyPrincipalValue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ -%#lnG6#,$-%$cosG6#\"\"#!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'3rrw)!#5" }}} {PARA 0 "" 0 "" {TEXT -1 19 "The singularity at " }{XPPEDIT 18 0 "x = \+ Pi/2" "6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 52 " would cause ord inary numerical integration to fail." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(Int(tan(x),x=0..2));" }}{PARA 8 "" 1 "" {TEXT -1 59 "Error, (in evalf/int) numeric exception: invalid operation\n" } }}{PARA 0 "" 0 "" {TEXT -1 101 "The next one can not be integrated sym bolically: presumably there is no closed form for the integral." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int(1/(x+sin(x)),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&\"\"\"F',&%\"xGF'-%$sinG6#F )F'!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "PVInt(1/(x+si n(x)),x=-1..2,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^:&39%!#5" }} }{PARA 0 "" 0 "" {TEXT -1 22 "The next one diverges." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "PVInt(1/(sin(x)^3+x^4), x=-1..2, 0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 107 "The location of the singularity must be known exactly. A floating-point approximation would not be enough." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p0:= fsolve(exp(x)-3*x,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0G$\"+nGh!>'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "PVInt(1/(exp(x)-3*x),x=0..1,p0);" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in evalf/int) unable to handle singularity\n " }}}{PARA 0 "" 0 "" {TEXT -1 100 "Since it depends on finding a finit e number of terms with negative exponents in the Laurent series, " } {MPLTEXT 0 21 5 "PVInt" }{TEXT -1 145 " will not work with singulariti es that are essential or non-isolated. For example, the following pri ncipal value integral is 0 by symmetry, but " }{MPLTEXT 0 21 5 "PVInt " }{TEXT -1 20 " can't evaluate it. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "PVInt(exp(1/x)-exp(-1/x),x=-1..1,0);" }}{PARA 8 "" 1 "" {TEXT -1 64 "Error, (in numapprox[laurent]) unable to compute Laure nt series\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 10 "See also: " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "int[nume ric]" 2 "int,numeric" "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Maple Advisor Database " }{TEXT -1 16 " R. Israel, 2000" }}}}{MARK "2 16 1 \+ 0" 63 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }