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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "                    " }
{TEXT 256 32 "Complex Transcendental Functions" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 28 "     Load the plots package." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 32 "P
art 1: The exponential function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 14 "1.   Increase " }{XPPEDIT 18 0 "n" "6#%\"
nG" }{TEXT -1 94 " one step at a time to see how the polynomial approx
imation compares with the actual function." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "n:=1;\nP:=z->sum
(z^k/k!,k=0..n);\ngraph1a:=conformal(exp(z),z=-2-2*I..2+2*I, grid=[16,
8], scaling=constrained):\ngraph1b:=conformal(P(z),z=-2-2*I..2+2*I, gr
id=[16,8], color=blue, scaling = constrained):\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "display(\{graph1a,graph1b\});" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "2.  \+
Look at the conformal plot of the exponential function alone. Watch th
e scale on the plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 83 "d:=0.5;\nconformal(exp(z), z =-d - d*I..d
 + d*I,numxy=[50,50], scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "3.  Now lo
ok at the graph of the magnitude of the exponential function." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "complexplot3d(exp(z), z=-1 - I..1 + I);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "4.  The \+
real and imaginary parts of the exponential function" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "u:=(x,y)
->Re(exp(x+I*y));\nv:=(x,y)->Im(exp(x+I*y));\nplot3d(u(x,y),x=-6..6, y
=-6..6);\nplot3d(v(x,y),x=-6..6, y=-6..6);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "5.  The exp
onential of a sum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "z1:=1+I; z2:=2-3*I;\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 72 "n:=10;\nsum((z1+z2)^k/k!,k=0..n)-sum(z1^k/k!,k
=0..n)*sum(z2^k/k!,k=0..n):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "\nev
alf(abs(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "6.  Write your explanation here. You do n
ot need " }{TEXT 258 7 "Maple's" }{TEXT -1 35 " symbolic capability fo
r this step." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "7.  Writ
e your explanation here. You do not need " }{TEXT 262 7 "Maple's" }
{TEXT -1 35 " symbolic capability for this step." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 259 45 "Part 2: The complex sine and cosine func
tions" }}{PARA 0 "" 0 "" {TEXT 260 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 234 "1 & 2.  Look at the conformal plot of the sine function.
 Be sure to click each time on the 1:1 button to obtain the proper asp
ect ratio. Watch the scale on the plot. For comparison we give the con
formal plot for the identity function." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "d:=0.5;\nconformal(z, z
 =-d - d*I..d + d*I);\nconformal(sin(z), z =-d - d*I..d + d*I);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 55 "3 & 4.  The graph of the magnitude of the sine function" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "complexplot3d(sin(z), z=-3 - I..3 + I);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "5 & 6.
  The real and imaginary parts of the sine and cosine functions" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "u:=(x,y)->Re(sin(x+I*y));\nv:=(x,y)->Im(sin(x+I*y));\n" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "plot3d(u(x,y),x=-6..6, y=-2..2);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot3d(v(x,y),x=-6..6, y=-2..2);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 90 "7.  Symbolic descriptions of the real and imaginary par
ts of the sine and cosine functions" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "evalc(Re(sin(x+I*y)));evalc(Im(sin(x+I*y)));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalc(Re(cos(x+I*y)));evalc(
Im(cos(x+I*y)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here are plots of the real hyperbo
lic functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 45 "plot(sinh(t),t=-3..3);\nplot(cosh(t),t=-3..3);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 261 15 "Part 3: Summary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0 0" 12 }
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