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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 60 "Closures of branch cuts f
or elementary functions in Maple 7." }}{PARA 19 "" 0 "" {TEXT -1 32 "R
obert M. Corless\nAugust 1, 2001" }}{PARA 0 "" 0 "" {TEXT -1 293 "This
 worksheet looks at the closures of the Maple implementations of the e
lementary functions on their branch cuts, by experiment.  Done during \+
the Celebration of Light, Vancouver; reflections of fireworks bouncing
 off the glass buildings in front of me, thunder and cheers echoing al
l around." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
184 "I emphasize that this worksheet explores Maple's implementations:
 other computer algebra systems may differ, not usually in the placeme
nt of the branch cut but perhaps on the closures." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plots[setoptions](axes=BOX
ED,colour=BLACK,scaling=CONSTRAINED);\nplots[setoptions3d](axes=BOXED,
style=PATCHNOGRID);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "The routi
ne \"compar\" takes a function \"f\", a point \"a\", and a direction \+
\"dir\" (meant to be a signed zero).  If f(a+dir)-f(a) is zero then f \+
is continuous at a in the direction dir.  The routine returns four lis
ts, checking out +/-a +/- dir." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 202 "compar := proc( f, a, dir )\n  print( evalf([a,dir,f(a+dir)-f
(a)]),\n         evalf([a,-dir,f(a-dir)-f(a)]),\n         evalf([-a,di
r,f(-a+dir)-f(-a)]),\n         evalf([-a,-dir,f(-a-dir)-f(-a)]) )\nend
 proc;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 13 "CCC functions" }}{PARA 0 "" 0 "" {TEXT 
-1 74 "CCC means ``counter-clockwise closure'' and in these cases the \+
rule holds." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "arcsin, arccos, ar
ctanh" }}{PARA 0 "" 0 "" {TEXT -1 157 "The branch cuts for these three
 functions are the segments of the real axis larger than 1 in magnitud
e (separately verified by plotting the imaginary parts)." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "compar( arcsin, 2., 0.*I );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot3d( Im(arcsin(x+I*y)), x
=-2..2, y=-1..1, colour=Re(arcsin(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "compar( arccos, 2, 0.*I );" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 72 "plot3d( Im(arccos(x+I*y)), x=-2..2, y=-1..1, \+
colour=Re(arccos(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "compar( arctanh, 2, 0.*I );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 74 "plot3d( Im(arctanh(x+I*y)), x=-2..2, y=-1..1, colour=Re(arctan
h(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "The following hand-coded call to \+
\"plot\" indicates graphically which side the cuts are closed on, by t
he presence of tick-marks." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
156 "plot( [[t,0,t=-3..-1],[t,0,t=1..3],\n       seq([-(1+i/N),t,t=0..
0.2], i=0..2*N),\n       seq([(1+i/N),-t,t=0..0.2], i=0..2*N)], \n    \+
   view=[-3..3,-3..3] );" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "arctan, arccot, arcsinh" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "The branch cuts for these three f
unctions are the segments of imaginary axis larger than 1 in magnitude
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "compar( arctan, 2*I, 0
. ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot3d( Re(arctan(
x+I*y)), x=-1..1,y=-3..3, colour=Im(arctan(x+I*y)) );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "compar( arccot, 2*I, 0. );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot3d( Re(arccot(x+I*y)), x
=-1..1,y=-3..3, colour=Im(arccot(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "compar( arcsinh, 2*I, 0. );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 73 "plot3d( Re(arcsinh(x+I*y)), x=-1..1,y=-3..3, c
olour=Im(arcsinh(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "N := 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "plot( [[0,t,
t=-3..-1],[0,t,t=1..3],\n       seq([-t,-(1+i/N),t=0..0.2], i=0..2*N),
\n       seq([t,(1+i/N),t=0..0.2], i=0..2*N)], \n       view=[-3..3,-3
..3] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 10 "Exceptions" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 14 "arccsc, arcsec" }}{PARA 0 "" 0 "" {TEXT -1 98 "These two \+
functions have branch cuts on the real line between -1 and 1, and a cl
osure switch at 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "compar
( arccsc, 1/2, 0.*I );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "p
lot3d( Im(arccsc(x+I*y)), x=-2..2, y=-1..1, colour=Re(arccsc(x+I*y)) )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "compar( arcsec, 1/2, 0
.*I );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot3d( Im(arcsec
(x+I*y)), x=-2..2, y=-1..1, colour=Re(arcsec(x+I*y)) );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 149 "plot( [[t,0,t=-1..0],[t,0,t=0..1],\n       seq([-(
i/N),-t,t=0..0.15], i=0..N),\n       seq([(i/N),t,t=0..0.15], i=0..N)]
, \n       view=[-2..2,-2..2] );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "arccsc(0); arccsc(0.*I); arccsc(-0.*I); arccsc(0.+0.*
I); arccsc(-0.+0.*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ar
csec(0);" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 7 "arccoth" }}{PARA 0 "" 0 "" {TEXT -1 88 "This is the same a
s the above two, except the closure is different: it is closed on top.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "compar( arccoth, 1/2, 0.*I );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "compar( arccoth, 2, 0.*I ):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "plot3d( Im(arccoth(x+I*y)), x=-2..2, y=-1..1, co
lour=Re(arccoth(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "N := 5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "plot( [[t,0,t=-1..1]
,[t,0,t=0..1],\n        seq([(i/N),t,t=0..0.15], i=-N..N)], \n       v
iew=[-2..2,-2..2] );" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "arccsch
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "This \+
is closed counterclockwise about each singularity, and has a closure s
witch at 0." }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "compar( arccsch, I/2, 0. );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "plot3d( Re(arccsch(x+I*y)), x=-1..1, y=-2..2, colour=
Im(arccsch(x+I*y)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N :=
 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "plot( [[0,t,t=-1..0
],[0,t,t=0..1],\n       seq([t,-(i/N),t=0..0.15], i=0..N),\n       seq
([-t,(i/N),t=0..0.15], i=0..N)], \n       view=[-2..2,-2..2] );" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "arccosh and arcsech" }}{PARA 0 "
" 0 "" {TEXT -1 64 "These are the true oddballs.  The branch cuts are \+
not symmetric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "compar( arccosh, 2, 0.*I );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "compar( arccosh, 1/2, 0.*I );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot3d( Im(arccosh(x+I*y)), x=-2..2
, y=-1..1, orientation=[-70,60], colour=Re(arccosh(x+I*y)) );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 181 "plot( [[t,0,t=-2..1],\n       seq( [-1-i/N,t
,t=0..0.15], i=0..N ),\n       seq( [-i/N, t, t=0..0.15], i=0..N ),\n \+
      seq( [i/N,t,t=0..0.15], i=0..N ) ],\n       view=[-2..2,-2..2] )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "compar( arcsech, 2, 0.
*I );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "compar( arcsech, 1
/2, 0.*I );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot3d( Im(a
rcsech(x+I*y)), x=-2..2, y=-1..1, orientation=[-70,60], colour=Re(arcs
ech(x+I*y)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "plot( [[t,0,t=-2..0],[t,0,t=1..2]
,\n       seq( [-1-i/N,-t,t=0..0.15], i=0..N ),\n       seq( [-i/N, -t
, t=0..0.15], i=0..N ),\n       seq( [1+i/N,-t,t=0..0.15], i=0..N ) ],
\n       view=[-2..2,-2..2] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 \+
4 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }