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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 41 "The modular surface of a c
omplex function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 31 "In the case of a real function " }{XPPEDIT 18 0 "y = f(x)
;" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 77 " we are used to visualizing t
he graph of the function as the set of points ( " }{XPPEDIT 18 0 "x,f(
x);" "6$%\"xG-%\"fG6#F#" }{TEXT -1 51 " ) in the plane.  However, with
 a complex function " }{XPPEDIT 18 0 "w = f(z);" "6#/%\"wG-%\"fG6#%\"z
G" }{TEXT -1 26 " a point on the graph has " }{TEXT 256 5 "four " }
{TEXT -1 12 "coordinates:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x+I*y) =
 u(x,y)+I*v(x,y);" "6#/-%\"fG6#,&%\"xG\"\"\"*&%\"IGF)%\"yGF)F),&-%\"uG
6$F(F,F)*&F+F)-%\"vG6$F(F,F)F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "Each point has coordinates ( " }{XPPEDIT 18 0 "x,y,u(x,y)
,v(x,y);" "6&%\"xG%\"yG-%\"uG6$F#F$-%\"vG6$F#F$" }{TEXT -1 70 " ).  We
 would need four-dimensional eyeglasses to see such a function!" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "However,
 there is some hope.  One type of graph of a complex function that is \+
sometimes useful is called the " }{TEXT 257 21 "modular surface of f.
" }{TEXT -1 14 "  Every point " }{XPPEDIT 18 0 "w = f(z);" "6#/%\"wG-%
\"fG6#%\"zG" }{TEXT -1 38 "  can be written in exponential form: " }
{XPPEDIT 18 0 "w = abs(f(z))*exp(I*Arg(f(z)));" "6#/%\"wG*&-%$absG6#-%
\"fG6#%\"zG\"\"\"-%$expG6#*&%\"IGF--%$ArgG6#-F*6#F,F-F-" }{TEXT -1 32 
".  To make the modular surface, " }{TEXT 258 9 "throw out" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Arg(f(z));" "6#-%$ArgG6#-%\"fG6#%\"zG" }{TEXT 
-1 29 " and just use the modulus of " }{XPPEDIT 18 0 "f(z);" "6#-%\"fG
6#%\"zG" }{TEXT -1 19 ".  For each point (" }{XPPEDIT 18 0 "x,y;" "6$%
\"xG%\"yG" }{TEXT -1 32 ") just construct a point above (" }{XPPEDIT 
18 0 "x,y;" "6$%\"xG%\"yG" }{TEXT -1 12 ") at height " }{XPPEDIT 18 0 
"abs(f(z)) = abs(f(x+I*y));" "6#/-%$absG6#-%\"fG6#%\"zG-F%6#-F(6#,&%\"
xG\"\"\"*&%\"IGF1%\"yGF1F1" }{TEXT -1 35 ", to form the modular surfac
e of f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
49 "Thus, the modular surface of f is the graph of ( " }{XPPEDIT 18 0 
"x,y,abs(f(x+I*y));" "6%%\"xG%\"yG-%$absG6#-%\"fG6#,&F#\"\"\"*&%\"IGF,
F$F,F," }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 118 "Your assignment is to produce a graph of the modu
lar surface for your function, drawn either using Maple or by hand.  \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 35 "Maple processes the complex
 number " }{XPPEDIT 18 0 "x+iy;" "6#,&%\"xG\"\"\"%#iyGF%" }{TEXT -1 
13 " in the form " }{MPLTEXT 0 21 8 "x + I*y." }{TEXT -1 14 "  Notice \+
that " }{TEXT 259 1 "i" }{TEXT -1 43 " is always capitalized!  Try the
 following:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z := 3 + 2*I;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "I*z;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 4 "z^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "sqrt(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(sq
rt(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "abs(z);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "abs(sqrt(z));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 25 "Notice that the function " }{MPLTEXT 0 
21 6 "abs(z)" }{TEXT -1 11 " gives you " }{XPPEDIT 18 0 "abs(z);" "6#-
%$absG6#%\"zG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 58 "To graph the modular surface, we want to \+
get a 3d plot of " }{XPPEDIT 18 0 "abs(f(x+I*y));" "6#-%$absG6#-%\"fG6
#,&%\"xG\"\"\"*&%\"IGF+%\"yGF+F+" }{TEXT -1 25 ".  Here's how I did it
..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "In
 this example, I made the modular surface for " }{XPPEDIT 18 0 "f(z) =
 1/(1-z^6);" "6#/-%\"fG6#%\"zG*&\"\"\"F),&F)F)*$F'\"\"'!\"\"F-" }
{TEXT -1 10 ".  I used " }{MPLTEXT 0 21 11 "smartplot3d" }{TEXT -1 54 
" to make the graph because it's more interactive than " }{MPLTEXT 0 
21 6 "plot3d" }{TEXT -1 42 ".  (but you can do it however you like...)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "smartplot3d( abs( 1 / (1 - (x+I*y)^6) ) );" }}{PARA 
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