Use the following functions defined in the GraphicsFunctions.mthZG4ZTI utility file to plot expressions having complex values:


To superimpose plots of the real and imaginary parts of an expression w that depends on a single real-valued variable, issue the Options > Plot Real and Imaginary Parts command3TKJ2T in a 2D-plot window, and then plot w.  For example, turn plot real and imaginary parts mode on, and then plot the expression

ASIN(x)


To superimpose plots of the absolute value (magnitude) and phase angle of an expression w that depends on a single real-valued variable, plot the expressions ABS(w) and PHASE(w) in a 2D-plot window.  For example, plot the expressions

ABS(ASIN(x))

PHASE(ASIN(x))


To parametrically plot the real and imaginary parts of an expression w that depends on a single real-valued variable, issue the Options > Trace Plots command1FN.FSI or press the F3 key in a 2D-plot window, and plot the vector [RE(w), IM(w)].  Then as the cross is moved along the curve, the cross coordinates displayed on the status bar show the real and imaginary parts of w as a function of the parameter values displayed on the title bar of the 2D-plot window.  For example, to plot the sine of z = t·(1+i) for t = -2·π to 2·π, turn trace mode on and plot the expression

[RE(SIN(t·(1 + #i))), IM(SIN(t·(1 + #i)))]


To superimpose surface plots of the real and imaginary parts of an expression w that depends on a single complex-valued variable, substitute the expression x+i·y for the variable in the expressions RE(w) and IM(w), and then plot the resulting expressions in a 3D-plot window.  In order to distinguish between the plots, use the Insert > Plot command72HLS1 to plot the imaginary surface and select a different color scheme than that used for the real surface.  For example, plot the expressions

RE(EXP(x + #i·y)/5)

IM(EXP(x + #i·y)/5)


To superimpose surface plots of the absolute value (magnitude) and phase angle of an expression w that depends on a single complex-valued variable, substitute the expression x + i·y for the variable in the expressions ABS(w) and PHASE(w), and then plot the resulting expressions in a 3D-plot window.  In order to distinguish between the plots, use the Insert > Plot command72HLS1 to plot the phase surface and select a different color scheme than that used for the absolute value surface.  For example, plot the expression

ABS(EXP(x + #i·y)/5)

PHASE(EXP(x + #i·y)/5)


To plot rays in a 2D-plot window whose length and direction represent the absolute value (magnitude) and phase angle of an expression w that depends on a single complex-valued variable at a rectangular array of points in the complex plane, use the RAYS function as described below.  The length and direction of rays between array points can be estimated using mental interpolation.  Also, the real and imaginary parts of w can be estimated from the length and direction of rays using mental conversion.


RAYS(w, z, z00, zmn, m, n) approximates to a matrix that plots in a 2D-plot window as a rectangular array of rays (i.e. line segments) whose length and direction represent the absolute value and phase angle of the expression w that depends on the variable z.  z varies from z00 at one corner of the array through zmn at the diagonally opposite corner, with m panels in the x direction and n panels in the y direction.  Before plotting an expression involving the RAYS function, use the 2D-plot window’s Options > Display > Points command7CKH0ZT to select connected small size points, and use the Options > Approximate Before Plotting commandEVZ132 to approximate expressions before plotting them.  If the rays are too short to discern accurately or so long that they confusingly cross each other, multiply w by a suitable numerical scale factor.  For example, to see how exp(z) varies exponentially with the real part of z and sinusoidally with the imaginary part of z, issue the above Options commands and then plot the expression

RAYS(EXP(z)/10, z, -1 - 2·pi·#i, 1 + 2·pi·#i, 12, 24)


To show in a 2D-plot window how a rectangular array of values in the z-plane maps into a distorted grid in the w-plane of a complex-valued expression w that depends on a complex-valued variable z, use the HORIZONTALS function as described below.  The grid points in the w-plane can be joined by line segments to help visualize the underlying grid topology.  By noting the order in which grid points are drawn and counting them, each grid point in the w-plane can be associated with a coordinate value in the z-plane.  Also, the values of w between grid points can be estimated using mental interpolation.


HORIZONTALS(w, z, z00, zmn, m, n) approximates to a matrix that plots in a 2D-plot window as an array of dots, with each row of w values joined by segments.  z varies from z00 at one corner of the grid through zmn at the diagonally opposite corner, with m panels in the x direction and n panels in the y direction.  Before plotting an expression involving the HORIZONTALS function, use the 2D-plot window’s Options > Display > Points command7CKH0ZT to select connected medium size points, and use the Options > Approximate Before Plotting commandEVZ132 to approximate expressions before plotting them.  To join the dot columns by segments, approximate and plot the coprojection of the matrix (i.e. plot COPROJECTION(HORIZONTALS(w, z, z00, zmn, m, n)) ).  For example, issue the above Options commands and then plot the expressions

HORIZONTALS(EXP(z), z, -1-pi·#i, 1+pi·#i, 12, 24)

COPROJECTION(HORIZONTALS(EXP(z), z, -1-pi·#i, 1+pi·#i, 12, 24))


To show in a 2D-plot window how a regular polar array of values in the z-plane maps into a distorted grid in the w-plane of a complex-valued expression w that depends on a complex-valued variable z, use the ARCS function as described below.  The grid points in the w-plane can be joined by line segments along lines of constant radius and/or lines of constant phase in the z-plane to help visualize the underlying grid topology.  By noting the order in which grid points are drawn and counting them, each grid point in the w-plane can be associated with a coordinate value in the z-plane.  Also, the values of w between grid points can be estimated using mental interpolation.


ARCS(w, z, r0, rm, m, θ0, θn, n) approximates to a matrix that plots in a 2D-plot window as an array of dots, with each row of w values joined by segments.  z=r·exp(i·θ) varies from r=r0 through rm and θ=θ0 through θn, with m panels in the r direction and n panels in the θ direction.  Before plotting an expression involving the ARCS function, use the 2D-plot window’s Options > Display > Points command7CKH0ZT to select connected medium size points, and use the Options > Approximate Before Plotting commandEVZ132 to approximate expressions before plotting them.  To join the dots in the radial direction by segments, approximate and plot the coprojection of the matrix (i.e. plot COPROJECTION(ARCS(w, z, r0, rm, m, θ0, θn, n)) ).  For example, issue the above Options commands and then plot the expressions

ARCS(EXP(z), z, 0, pi, 4, -pi, pi, 24)

COPROJECTION(ARCS(EXP(z), z, 0, pi, 4, -pi, pi, 24))


Other Graphics FunctionsZG4ZTI 

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