--- title: "AbsArgPlot" language: "en" type: "Symbol" summary: "AbsArgPlot[f, {x, xmin, xmax}] generates a plot of Abs[f] colored by Arg[f] as a function of x \\[Element] \\[DoubleStruckCapitalR] from xmin to xmax. AbsArgPlot[{f1, f2, ...}, {x, xmin, xmax}] plots several functions. AbsArgPlot[{..., w[fi], ...}, ...] plots fi with features defined by the symbolic wrapper w. AbsArgPlot[..., {x} \\[Element] reg] takes the variable x to be in the geometric region reg." canonical_url: "https://reference.wolfram.com/language/ref/AbsArgPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Complex Visualization" link: "https://reference.wolfram.com/language/guide/ComplexVisualization.en.md" - title: "Function Visualization" link: "https://reference.wolfram.com/language/guide/FunctionVisualization.en.md" - title: "Functions of Complex Variables" link: "https://reference.wolfram.com/language/guide/FunctionsOfComplexVariables.en.md" related_functions: - title: "ReImPlot" link: "https://reference.wolfram.com/language/ref/ReImPlot.en.md" - title: "ComplexPlot" link: "https://reference.wolfram.com/language/ref/ComplexPlot.en.md" - title: "ComplexListPlot" link: "https://reference.wolfram.com/language/ref/ComplexListPlot.en.md" - title: "ComplexPlot3D" link: "https://reference.wolfram.com/language/ref/ComplexPlot3D.en.md" - title: "AbsArg" link: "https://reference.wolfram.com/language/ref/AbsArg.en.md" - title: "ReIm" link: "https://reference.wolfram.com/language/ref/ReIm.en.md" - title: "Plot" link: "https://reference.wolfram.com/language/ref/Plot.en.md" - title: "ParametricPlot" link: "https://reference.wolfram.com/language/ref/ParametricPlot.en.md" --- # AbsArgPlot AbsArgPlot[f, {x, xmin, xmax}] generates a plot of Abs[f] colored by Arg[f] as a function of x∈\[DoubleStruckCapitalR] from xmin to xmax. AbsArgPlot[{f1, f2, …}, {x, xmin, xmax}] plots several functions. AbsArgPlot[{…, w[fi], …}, …] plots fi with features defined by the symbolic wrapper w. AbsArgPlot[…, {x}∈reg] takes the variable x to be in the geometric region reg. ## Details and Options * ``AbsArgPlot`` evaluates ``f`` at different values of ``x`` to create smooth curves of the form ``{x, Abs[f[x]]}`` colored by ``{x, Arg[f[x]]}``. * Gaps are left at any ``x`` where the ``fi`` evaluate to non-numeric values. * The region ``reg`` can be any ``RegionQ`` object in 1D. [image] * ``AbsArgPlot`` treats the variable ``x`` as local, effectively using ``Block``. * ``AbsArgPlot`` has the attribute ``HoldAll`` and evaluates ``f`` only after assigning specific numerical values to ``x``. * In some cases, it may be more efficient to use ``Evaluate`` to evaluate ``f`` symbolically before specific numerical values are assigned to ``x``. * Wrappers apply to both ``Re[f]`` and ``Im[f]``. * The following wrappers ``w`` can be used for the ``fi``: | | | | ------------------------------------ | ------------------------------------------------------ | | Annotation[fi, label] | provide an annotation for the fi | | Button[fi, action] | evaluate action when the curve for fi is clicked | | Callout[fi, label] | label the function with a callout | | Callout[fi, label, pos] | place the callout at relative position pos | | EventHandler[fi, events] | define a general event handler for fi | | Highlighted[fi, effect] | dynamically highlight fi with an effect | | Highlighted[fi, Placed[effect, pos]] | statically highlight fi with an effect at position pos | | Hyperlink[fi, uri] | make the function a hyperlink | | Labeled[fi, label] | label the function | | Labeled[fi, label, pos] | place the label at relative position pos | | Legended[fi, label] | identify the function in a legend | | PopupWindow[fi, cont] | attach a popup window to the function | | StatusArea[fi, label] | display in the status area on mouseover | | Style[fi, styles] | show the function using the specified styles | | Tooltip[fi, label] | attach a tooltip to the function | | Tooltip[fi] | use functions as tooltips | * Wrappers ``w`` can be applied at multiple levels: | | | | ---------- | ----------------------- | | w[fi] | wrap the fi | | w[{f1, …}] | wrap a collection of fi | | w1[w2[…]] | use nested wrappers | * ``Callout``, ``Labeled`` and ``Placed`` can use the following positions ``pos``: | | | | --------------------------- | ---------------------------------------------------------- | | Automatic | automatically placed labels | | Above, Below, Before, After | positions around the curve | | x | near the curve at a position x | | Scaled[s] | scaled position s along the curve | | {s, Above}, {s, Below}, … | relative position at position s along the curve | | {pos, epos} | epos in label placed at relative position pos of the curve | * ``AbsArgPlot`` has the same options as ``Graphics``, with the following additions and changes: [] | | | | | :-------------------- | :---------------- | :------------------------------------------------------- | | AspectRatio | 1 / GoldenRatio | ratio of height to width | | Axes | True | whether to draw axes | | ClippingStyle | None | what to draw where curves are clipped | | ColorFunction | Automatic | how to determine the coloring of curves | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | points in x to exclude | | ExclusionsStyle | None | what to draw at excluded points | | Filling | None | filling to insert under each curve | | FillingStyle | Automatic | style to use for filling | | LabelingSize | Automatic | maximum size of callouts and labels | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | None | how many mesh points to draw on each curve | | MeshFunctions | {#1&} | how to determine the placement of mesh points | | MeshShading | None | how to shade regions between mesh points | | MeshStyle | Automatic | the style for mesh points | | Method | Automatic | the method to use for refining curves | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotHighlighting | Automatic | highlighting effect for curves | | PlotLabel | None | overall label for the plot | | PlotLabels | None | labels to use for curves | | PlotLegends | None | legends for curves | | PlotPoints | Automatic | initial number of sample points | | PlotRange | {Full, Automatic} | the range of y or other values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotStyle | Automatic | graphics directives to specify the style for each curve | | PlotTheme | \$PlotTheme | overall theme for the plot | | RegionFunction | (True&) | how to determine whether a point should be included | | ScalingFunctions | None | how to scale individual coordinates | | WorkingPrecision | MachinePrecision | the precision used in internal computations | * Possible settings for ``ClippingStyle`` are: | | | | | ------- | --------- | ----------------------------------------- | | [image] | Automatic | use a dotted line for the clipped portion | | [image] | None | omit the clipped portion of the curve | | [image] | style | use style for the clipped portion | * With the default settings ``Exclusions -> Automatic`` and ``ExclusionsStyle -> None``, ``AbsArgPlot`` breaks curves at discontinuities and singularities it detects. ``Exclusions -> None`` joins across discontinuities and singularities. * ``Exclusions -> {x1, x2, …}`` is equivalent to ``Exclusions -> {x == x1, x == x2, …}``. * ``PlotLabels -> "Expressions"`` uses the ``fi`` as the label text. * ``AbsArgPlot`` initially evaluates ``f`` at a number of equally spaced sample points specified by ``PlotPoints``. Then it uses an adaptive algorithm to choose additional sample points, subdividing a given interval at most ``MaxRecursion`` times. [image] * Since only a finite number of sample points are used, it is possible for ``AbsArgPlot`` to miss features of ``f``. Increasing the settings for ``PlotPoints`` and ``MaxRecursion`` will often catch such features. * Themes that affect curves include: | | | | | ------- | ------------- | ----------------- | | [image] | "ThinLines" | thin plot lines | | [image] | "MediumLines" | medium plot lines | | [image] | "ThickLines" | thick plot lines | * The arguments supplied to functions in ``MeshFunctions`` and ``RegionFunction`` are ``x``, ``y``. Functions in ``ColorFunction`` are by default supplied with scaled versions of these arguments. * ``ScalingFunctions -> "scale"`` scales the $y$ coordinate; ``ScalingFunctions -> {"scalex", "scaley"}`` scales both the $x$ and $y$ coordinates. * Possible highlighting effects for ``Highlighted`` and ``PlotHighlighting`` include: | | | | | ------- | ------------------- | ----------------------------------------------------------------------------- | | [image] | style | highlight the indicated curve | | [image] | "Ball" | highlight and label the indicated point in a curve | | [image] | "Dropline" | highlight and label the indicated point in a curve with droplines to the axes | | [image] | "XSlice" | highlight and label all points along a vertical slice | | [image] | "YSlice" | highlight and label all points along a horizontal slice | | [image] | Placed[effect, pos] | statically highlight the given position pos | * Highlight position specifications ``pos`` include: | | | | --------------- | -------------------------------------------- | | x, {x} | effect at {x, y} with y chosen automatically | | {x, y} | effect at {x, y} | | {pos1, pos2, …} | multiple positions posi | ### List of all options | | | | | ---------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | 1 / GoldenRatio | ratio of height to width | | Axes | True | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | ClippingStyle | None | what to draw where curves are clipped | | ColorFunction | Automatic | how to determine the coloring of curves | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | Epilog | {} | primitives rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | points in x to exclude | | ExclusionsStyle | None | what to draw at excluded points | | Filling | None | filling to insert under each curve | | FillingStyle | Automatic | style to use for filling | | FormatType | TraditionalForm | the default format type for text | | Frame | False | whether to put a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame ticks | | FrameTicksStyle | {} | style specifications for frame ticks | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelingSize | Automatic | maximum size of callouts and labels | | LabelStyle | {} | style specifications for labels | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | None | how many mesh points to draw on each curve | | MeshFunctions | {#1&} | how to determine the placement of mesh points | | MeshShading | None | how to shade regions between mesh points | | MeshStyle | Automatic | the style for mesh points | | Method | Automatic | the method to use for refining curves | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotHighlighting | Automatic | highlighting effect for curves | | PlotLabel | None | overall label for the plot | | PlotLabels | None | labels to use for curves | | PlotLegends | None | legends for curves | | PlotPoints | Automatic | initial number of sample points | | PlotRange | {Full, Automatic} | the range of y or other values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotStyle | Automatic | graphics directives to specify the style for each curve | | PlotTheme | \$PlotTheme | overall theme for the plot | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RegionFunction | (True&) | how to determine whether a point should be included | | RotateLabel | True | whether to rotate y labels on the frame | | ScalingFunctions | None | how to scale individual coordinates | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | | WorkingPrecision | MachinePrecision | the precision used in internal computations | --- ## Examples (150) ### Basic Examples (6) Plot the modulus of a complex function of a real variable colored by its argument: ```wl In[1]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}] Out[1]= [image] ``` --- Plot several functions: ```wl In[1]:= AbsArgPlot[{Sin[I x] + Sin[Pi x], 2Exp[I π x]}, {x, -2, 2}] Out[1]= [image] ``` --- Label each curve: ```wl In[1]:= AbsArgPlot[{Sin[I x] + Sin[Pi x], 2Exp[I π x]}, {x, -2, 2}, PlotLabels -> "Expressions"] Out[1]= [image] ``` --- Include a legend: ```wl In[1]:= AbsArgPlot[{Sin[I x] + Sin[Pi x], 2Exp[I π x]}, {x, -2, 2}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Fill below a curve: ```wl In[1]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}, Filling -> Axis] Out[1]= [image] ``` --- Compare ``Plot`` and ``AbsArgPlot`` : ```wl In[1]:= { {Plot[Abs[Sin[x] + Sin[I Sin[3x]]], {x, -Pi, Pi}], Plot[Arg[Sin[x] + Sin[I Sin[3x]]], {x, -Pi, Pi}, ColorFunction -> Hue]}, AbsArgPlot[Sin[x] + Sin[I Sin[3x]], {x, -Pi, Pi}]} Out[1]= [image] ``` ### Scope (18) #### Sampling (8) More points are sampled where the function changes quickly: ```wl In[1]:= AbsArgPlot[Sin[x] + Sin[I Sin[3x]], {x, 0, 4Pi}] Out[1]= [image] ``` --- The plot range is selected automatically: ```wl In[1]:= AbsArgPlot[1 + Exp[-Abs[x]]Sin[I Sin[5x]], {x, -Pi, Pi}] Out[1]= [image] ``` --- The curve is split when there are discontinuities in the function: ```wl In[1]:= AbsArgPlot[I + Floor[Sqrt[x]], {x, -100, 100}] Out[1]= [image] ``` --- Use ``Exclusions```` -> ````None`` to draw connected curves: ```wl In[1]:= AbsArgPlot[I + Floor[Sqrt[x]], {x, -100, 100}, Exclusions -> None] Out[1]= [image] ``` --- Use ``PlotPoints`` and ``MaxRecursion`` to control adaptive sampling: ```wl In[1]:= Grid@Table[AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}, PlotPoints -> pp, MaxRecursion -> mr], {mr, {0, 1, 2}}, {pp, {15, 30}}] Out[1]= | | | | ------- | ------- | | [image] | [image] | | [image] | [image] | | [image] | [image] | ``` --- The domain can be specified by a region: ```wl In[1]:= \[ScriptCapitalD] = ImplicitRegion[-2 ≤ x ≤ 1 / 2∨1 ≤ x ≤ 2, {x}]; In[2]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x}∈\[ScriptCapitalD]] Out[2]= [image] ``` --- Specify a domain using a ``MeshRegion`` : ```wl In[1]:= \[ScriptCapitalD] = MeshRegion[{{-2}, {-1}, {-1 / 2}, {1 / 2}, {1}, {2}}, Line[{{1, 2}, {3, 4}, {5, 6}}]]; In[2]:= AbsArgPlot[Sin[x^2] - I, {x}∈\[ScriptCapitalD]] Out[2]= [image] ``` --- Plot over an infinite domain with automatic ticks: ```wl In[1]:= AbsArgPlot[{Sin[x] / x, 1 / x, -1 / x}, {x, -Infinity, Infinity}] Out[1]= [image] ``` #### Labeling and Legending (5) The ``Automatic`` legend shows the range of ``Arg`` values: ```wl In[1]:= AbsArgPlot[{ArcCos[x], ArcSin[x]}, {x, -3, 3}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Explicitly label the individual curves: ```wl In[1]:= AbsArgPlot[{ArcCos[x], ArcSin[x]}, {x, -3, 3}, PlotLabels -> Automatic] Out[1]= [image] ``` --- Identify curves with wrappers: ```wl In[1]:= AbsArgPlot[{Callout[ArcCos[x]], Tooltip[ArcSin[x]]}, {x, -3, 3}] Out[1]= [image] ``` --- Curves usually have interactive callouts showing the coordinates when you mouse over them: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -5, 5}] Out[1]= [image] ``` --- Choose from multiple interactive highlighting effects: ```wl In[1]:= {AbsArgPlot[{x^2 - 4, Sinc[x]}, {x, -3, 3}, PlotHighlighting -> "Dropline"], AbsArgPlot[{x^2 - 4, Sinc[x]}, {x, -3, 3}, PlotHighlighting -> "XSlice"]} Out[1]= {[image], [image]} ``` #### Presentation (5) Provide explicit styling to different curves: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotStyle -> {Thick, Dashed}] Out[1]= [image] ``` --- Add labels and a legend: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, AxesLabel -> {x, ""}, PlotLabels -> Automatic, PlotLegends -> Automatic] Out[1]= [image] ``` --- Create filled plots: ```wl In[1]:= AbsArgPlot[Sin[x] + Sin[I Sin[3x]], {x, -Pi, Pi}, Filling -> {1 -> Axis}] Out[1]= [image] ``` --- Use a plot theme: ```wl In[1]:= AbsArgPlot[{ArcSin[x], 3Exp[I x] + 0.5Exp[5I x]}, {x, -3, 3}, PlotTheme -> "Business"] Out[1]= [image] ``` --- Use ``ScalingFunctions`` to scale the axes: ```wl In[1]:= AbsArgPlot[x + I Exp[x], {x, -5, 5}, ScalingFunctions -> "Log"] Out[1]= [image] ``` ### Options (113) #### AspectRatio (4) By default, ``AbsArgPlot`` uses a fixed ratio of the height to the width of the plot: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}] Out[1]= [image] ``` --- Make the height the same as the width with ``AspectRatio -> 1`` : ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> 1] Out[1]= [image] ``` --- ``AspectRatio -> Automatic`` determines the ratio from the plot ranges: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> Automatic] Out[1]= [image] ``` --- ``AspectRatio -> Full`` adjusts the height and width to tightly fit inside other constructs: ```wl In[1]:= plot = AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> Full]; In[2]:= {Framed[Pane[plot, {50, 100}]], Framed[Pane[plot, {100, 100}]], Framed[Pane[plot, {100, 50}]]} Out[2]= [image] ``` #### Axes (4) By default, ``Axes`` is drawn for ``AbsArgPlot`` : ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}] Out[1]= [image] ``` --- Use ``Frame`` instead of axes: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, Axes -> False, Frame -> True] Out[1]= [image] ``` --- Use ``AxesOrigin`` to specify where the axes intersect: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesOrigin -> {0, 0}] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {AbsArgPlot[x^2 - 4, {x, -4, 4}, Axes -> {True, False}], AbsArgPlot[x^2 - 4, {x, -4, 4}, Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (3) No axes labels are drawn by default: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesLabel -> "Abs[F[x]]"] Out[1]= [image] ``` --- Use specific labels for each axis: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesLabel -> {"x", "Abs[F[x]]"}] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesOrigin -> {2, 2}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesStyle -> {Directive[Thick, Brown], Directive[Thick, Blue]}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesStyle -> Blue, TicksStyle -> Red] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AxesStyle -> Blue, LabelStyle -> Red] Out[1]= [image] ``` #### ClippingStyle (2) Omit clipped regions of the plot: ```wl In[1]:= AbsArgPlot[Exp[I x] / (x^2 - 25)^2, {x, -10, 10}, ClippingStyle -> None] Out[1]= [image] ``` --- Show clipped regions with black lines: ```wl In[1]:= AbsArgPlot[Exp[I x] / (x^2 - 25)^2, {x, -10, 10}, ClippingStyle -> Black] Out[1]= [image] ``` #### ColorFunction (4) Color by the scaled argument: ```wl In[1]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}] Out[1]= [image] ``` --- Use a named color gradient: ```wl In[1]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}, ColorFunction -> "AvocadoColors"] Out[1]= [image] ``` --- ``ColorFunction`` has higher priority than ``PlotStyle`` : ```wl In[1]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}, ColorFunction -> "AvocadoColors", PlotStyle -> Directive[Red, Dashed]] Out[1]= [image] ``` --- Highlight part of the plot: ```wl In[1]:= AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -2, 2}, ColorFunction -> Function[{arg}, If[arg ≤ 0, Red, Black]], PlotStyle -> Thick, ColorFunctionScaling -> False] Out[1]= [image] ``` #### ColorFunctionScaling (1) No argument scaling on the left; automatic scaling on the right: ```wl In[1]:= Table[AbsArgPlot[Exp[I x], {x, -Pi, Pi}, AxesOrigin -> {0, 0}, ColorFunctionScaling -> cf], {cf, {False, True}}] Out[1]= {[image], [image]} ``` #### Exclusions (3) Use automatic methods for computing exclusions: ```wl In[1]:= AbsArgPlot[x + Floor[x^2]I, {x, -3, 3}] Out[1]= [image] ``` --- Exclusions are automatically computed for both ``Abs`` and ``Arg`` : ```wl In[1]:= AbsArgPlot[Exp[I SquareWave[x / 2]]SawtoothWave[x / 4], {x, -10, 10}] Out[1]= [image] ``` --- Indicate that no exclusions should be computed: ```wl In[1]:= AbsArgPlot[x + Floor[x^2]I, {x, -3, 3}, Exclusions -> None] Out[1]= [image] ``` #### ExclusionStyle (1) Use gray lines to connect portions of the curve and black points to indicate exclusions: ```wl In[1]:= AbsArgPlot[LogGamma[x], {x, -5, 5}, ExclusionsStyle -> {Gray, Black}] Out[1]= [image] ``` #### Filling (4) Use symbolic or explicit values: ```wl In[1]:= Table[AbsArgPlot[Sin[I x] + Sin[Pi x], {x, -3, 3}, Filling -> f], {f, {Axis, Top, Bottom, 3}}] Out[1]= [image] ``` --- Filling uses transparent colors if more than one curve is specified: ```wl In[1]:= AbsArgPlot[{Sin[I x] + Sin[Pi x], 2Exp[I π x]}, {x, -2, 2}, Filling -> Axis] Out[1]= [image] ``` --- Fill between curve 1 and the $x$ axis: ```wl In[1]:= AbsArgPlot[{ArcSin[x], 3Exp[I x] + 0.5Exp[5I x]}, {x, -3, 3}, Filling -> {1 -> Axis}] Out[1]= [image] ``` --- Fill between curves 1 and 2: ```wl In[1]:= AbsArgPlot[{Sin[I x] + Sin[Pi x], 2Exp[I π x]}, {x, -2, 2}, Filling -> {1 -> {2}}] Out[1]= [image] ``` #### FillingStyle (2) Use different fill colors: ```wl In[1]:= Table[AbsArgPlot[{ArcSin[x], 3Exp[I x] + 0.5Exp[5I x]}, {x, -3, 3}, Filling -> {1 -> Axis}, FillingStyle -> c], {c, {Red, Green, Blue, Yellow}}] Out[1]= [image] ``` --- Fill between two curves with red below the second curve and blue above: ```wl In[1]:= AbsArgPlot[{ArcSin[x], Exp[I x] + 0.5Exp[5I x]}, {x, -3, 3}, Filling -> {1 -> {2}}, FillingStyle -> {Red, Blue}] Out[1]= [image] ``` #### Frame (3) Draw a frame around the plot: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True] Out[1]= [image] ``` --- Draw a frame on the left and right edges: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> {{True, True}, {False, False}}] Out[1]= [image] ``` --- Draw a frame on the top and bottom edges: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> {{False, False}, {True, True}}] Out[1]= [image] ``` #### FrameLabel (2) Frame labels are placed on the bottom and left frame edges by default: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameLabel -> {"x", "y"}] Out[1]= [image] ``` --- Place labels on each of the edges in the frame: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameLabel -> {{"left", "right"}, {"bottom", "top"}}] Out[1]= [image] ``` #### FrameStyle (2) Specify the style of the frame: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameStyle -> Directive[Black, Thick]] Out[1]= [image] ``` --- Specify the style for each frame edge: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameStyle -> {{Directive[Black, Thick], Directive[Red]}, {Directive[Gray, Thick], Directive[Blue]}}] Out[1]= [image] ``` #### FrameTicks (8) ``FrameTicks`` are placed automatically by default: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True] Out[1]= [image] ``` --- Use a frame with no ticks: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> None] Out[1]= [image] ``` --- Use frame ticks on the $x$ axes but not the $y$ axes: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{False, False}, {True, True}}] Out[1]= [image] ``` --- Place frame ticks at specific positions: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{{0, .1, .2}, {0, .1, .2}}, {{-.2, .1, .2}, {-.2, .1, .2}}}] Out[1]= [image] ``` --- Draw frame ticks at the specified positions with specific labels: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{{{0, a}, {.1, b}, {.2, c}}, {0, .1, .2}}, {{{-.2, -c}, {.1, b}, {.2, c}}, {-.2, .1, .2}}}] Out[1]= [image] ``` --- Specify the lengths for frame ticks as a fraction on graphics size: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{{{0, a, .2}, {.1, b, .16}, {.2, c, .11}}, Automatic}, {Automatic, Automatic}}] Out[1]= [image] ``` Use different sizes in the positive and negative directions for each frame tick: ```wl In[2]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{{{0, a, .2}, {.1, b, .16}, {.2, c, .11}}, {{0, a, {.2, 0}}, {.1, b, {.16, 0}}, {.2, c, {.11, 0}}}}, {Automatic, Automatic}}] Out[2]= [image] ``` --- Specify a style for each frame tick: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{{{0, a, .2, Directive[Red]}, {.1, b, .16, Directive[Red, Thick, Dashed]}, {.2, c, .11, Directive[Red, Thick]}}, Automatic}, {Automatic, Automatic}}] Out[1]= [image] ``` --- Construct a function that places frame ticks at the midpoint and extremes of the frame edge: ```wl In[1]:= minMeanMax[min_, max_] := {{min, min}, {(max + min) / 2, (max + min) / 2}, {max, max}} In[2]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, FrameTicks -> {{Automatic, minMeanMax}, {Automatic, Automatic}}, PlotRangePadding -> None] Out[2]= [image] ``` #### FrameTicksStyle (3) By default, the frame ticks and frame tick labels use the same styles as the frame: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, Frame -> True, FrameStyle -> Directive[Red]] Out[1]= [image] ``` --- Specify an overall style for the ticks, including the labels: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, Frame -> True, FrameTicks -> All, FrameTicksStyle -> {Directive[Black, Thick], Directive[Blue, Thick]}] Out[1]= [image] ``` --- Use a different style for the different frame edges: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Frame -> True, Frame -> True, FrameTicks -> All, FrameTicksStyle -> {{Black, Blue}, {Red, Green}}] Out[1]= [image] ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> Tiny], AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> 150], AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> {Automatic, 150}], AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> UpTo[200]], AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> {200, 200}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> Full, ImageSize -> {200, 200}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> {UpTo[150], UpTo[100]}], AbsArgPlot[x^2 - 4, {x, -4, 4}, AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= {[image], [image]} ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> Full, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[AbsArgPlot[x^2 - 4, {x, -4, 4}, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` #### MaxRecursion (1) Each level of ``MaxRecursion`` adaptively subdivides the initial mesh into a finer mesh: ```wl In[1]:= Table[AbsArgPlot[I + Sin[1 / x], {x, 0.001, .1}, MaxRecursion -> i, Mesh -> All, Ticks -> None], {i, {0, 3, 6}}] Out[1]= {[image], [image], [image]} ``` #### Mesh (3) Show the initial and final sampling meshes: ```wl In[1]:= {AbsArgPlot[Cos[x] + I ArcCosh[x], {x, 0, 2Pi}, Mesh -> Full, MeshStyle -> AbsolutePointSize[3]], AbsArgPlot[Cos[x] + I ArcCosh[x], {x, 0, 2Pi}, Mesh -> All, MeshStyle -> AbsolutePointSize[3]]} Out[1]= [image] ``` --- Use 10 mesh points evenly spaced in the $x$ direction: ```wl In[1]:= AbsArgPlot[Cos[x] + I ArcCosh[x], {x, 0, 2Pi}, Mesh -> 10, MeshStyle -> AbsolutePointSize[5]] Out[1]= [image] ``` --- Use an explicit list of values for the mesh in the $x$ direction: ```wl In[1]:= AbsArgPlot[Cos[x] + I ArcCosh[x], {x, 0, 2Pi}, Mesh -> {Range[0, 2Pi, Pi / 4]}, MeshStyle -> AbsolutePointSize[5], Ticks -> {Range[0, 2Pi, Pi / 4], Automatic}] Out[1]= [image] ``` #### MeshFunctions (2) Use a mesh evenly spaced in the $x$ and $y$ directions: ```wl In[1]:= Table[AbsArgPlot[2Exp[I x] + 0.5Exp[3I x], {x, 0, 2Pi}, MeshFunctions -> {f}, Mesh -> 10, MeshStyle -> Black], {f, {Function[{x, y}, x], Function[{x, y}, y]}}] Out[1]= [image] ``` --- Show 10 mesh levels in the $x$ direction (black) and 6 in the $y$ direction (red): ```wl In[1]:= AbsArgPlot[2Exp[I x] + 0.5Exp[3I x], {x, 0, 2Pi}, MeshFunctions -> {#1&, #2&}, Mesh -> {10, 6}, MeshStyle -> {Black, Red}] Out[1]= [image] ``` #### MeshShading (2) Remove segments of the curve: ```wl In[1]:= AbsArgPlot[2Exp[I x] + 0.5Exp[3I x], {x, 0, 2Pi}, Mesh -> 15, MeshShading -> {None, Automatic}, MeshStyle -> Black] Out[1]= [image] ``` --- ``MeshShading`` has higher priority than ``PlotStyle`` for styling: ```wl In[1]:= AbsArgPlot[2Exp[I x] + 0.5Exp[3I x], {x, 0, 2Pi}, Mesh -> 15, MeshShading -> {None, Automatic}, MeshStyle -> Black, PlotStyle -> Directive[Black, Dashed]] Out[1]= [image] ``` #### MeshStyle (2) Use a black mesh in the $x$ direction: ```wl In[1]:= AbsArgPlot[2Exp[I x] + 0.5Exp[3I x], {x, 0, 2Pi}, Mesh -> 10, MeshStyle -> Black] Out[1]= [image] ``` --- Use a black mesh in the $x$ direction and a red mesh in the $y$ direction: ```wl In[1]:= AbsArgPlot[2Exp[I x] + 0.5Exp[3I x], {x, 0, 2Pi}, Mesh -> 10, MeshFunctions -> {#1&, #2&}, MeshStyle -> {Directive[PointSize[Large], Black], Red}] Out[1]= [image] ``` #### PerformanceGoal (2) Generate a higher-quality plot: ```wl In[1]:= AbsArgPlot[1 + Exp[I / x], {x, 0, 1}, PerformanceGoal -> "Quality"]//Timing Out[1]= {0.124801, [image]} ``` --- Emphasize performance, possibly at the cost of quality: ```wl In[1]:= AbsArgPlot[1 + Exp[I / x], {x, 0, 1}, PerformanceGoal -> "Speed"]//Timing Out[1]= {0.0468003, [image]} ``` #### PlotHighlighting (8) Plots have interactive coordinate callouts with the default setting ``PlotHighlighting -> Automatic`` : ```wl In[1]:= AbsArgPlot[Cos[x], {x, 0, 10}] Out[1]= [image] ``` Use ``PlotHighlighting -> None`` to disable the highlighting for the entire plot: ```wl In[2]:= AbsArgPlot[Cos[x], {x, 0, 10}, PlotHighlighting -> None] Out[2]= [image] ``` --- Move the mouse over the curve to highlight it with a ball and label: ```wl In[1]:= AbsArgPlot[E^Cos[x], {x, 0, 10}, PlotHighlighting -> "Ball"] Out[1]= [image] ``` --- Move the mouse over the curve to highlight it with a label and droplines to the axes: ```wl In[1]:= AbsArgPlot[E^Cos[x] - 1, {x, 0, 10}, PlotHighlighting -> "Dropline"] Out[1]= [image] ``` --- Move the mouse over the plot to highlight it with a slice showing $y$ values corresponding to the $x$ position: ```wl In[1]:= AbsArgPlot[{E^Cos[x] - 1, Sin[x]}, {x, 0, 10}, PlotHighlighting -> "XSlice"] Out[1]= [image] ``` --- Move the mouse over the plot to highlight it with a slice showing $x$ values corresponding to the $y$ position: ```wl In[1]:= AbsArgPlot[E^Cos[x] - 1, {x, 0, 10}, PlotHighlighting -> "YSlice"] Out[1]= [image] ``` --- Use a component that shows the points on the curve closest to the $x$ position of the mouse cursor: ```wl In[1]:= AbsArgPlot[{E^Cos[x] - 1, Sin[x]}, {x, 0, 10}, PlotHighlighting -> "XNearestPoint"] Out[1]= [image] ``` Specify the style for the points: ```wl In[2]:= AbsArgPlot[{E^Cos[x] - 1, Sin[x]}, {x, 0, 10}, PlotHighlighting -> {"XNearestPoint", <|"Style" -> Black|>}] Out[2]= [image] ``` --- Use a component that shows the coordinates on the curve closest to the mouse cursor: ```wl In[1]:= AbsArgPlot[{E^Cos[x] - 1, Sin[x]}, {x, 0, 10}, PlotHighlighting -> "XYLabel"] Out[1]= [image] ``` Use ``Callout`` options to change the appearance of the label: ```wl In[2]:= AbsArgPlot[E^Cos[x] - 1, {x, 0, 10}, PlotHighlighting -> {"XYLabel", <|"Appearance" -> "Corners", "CalloutMarker" -> "Circle"|>}] Out[2]= [image] ``` --- Combine components to create a custom effect: ```wl In[1]:= AbsArgPlot[E^Cos[x] - 1, {x, 0, 10}, PlotHighlighting -> {{"XNearestPoint", <|"Style" -> Black|>}, {"XYLabel", <|"Appearance" -> "Corners", "CalloutMarker" -> "Circle"|>}}] Out[1]= [image] ``` #### PlotLabel (1) Add an overall label to the plot: ```wl In[1]:= AbsArgPlot[ArcSin[x], {x, -3, 3}, PlotLabel -> TraditionalForm[Abs[ArcSin[x]]]] Out[1]= [image] ``` #### PlotLabels (6) Specify text to label curves: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLabels -> {"arcsine", "arccosine"}] Out[1]= [image] ``` --- Modify the appearance of the labels: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLabels -> Placed[{"arcsine", "arccosine"}, Above]] Out[1]= [image] ``` --- Place the labels differently for each curve: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLabels -> {Placed["arcsine", Below], Placed["arccosine", Above]}] Out[1]= [image] ``` --- ``PlotLabels```` -> ````"Expressions"`` uses functions as curve labels: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLabels -> "Expressions"] Out[1]= [image] ``` --- Use callouts to identify the curves: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLabels -> {Callout["arcsine", Above], Callout["arccosine", Below]}] Out[1]= [image] ``` --- Use ``None`` to not add a label: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLabels -> {"arcsine", None}] Out[1]= [image] ``` #### PlotLegends (2) Create a legend for the argument color: ```wl In[1]:= AbsArgPlot[{ArcSin[x], ArcCos[x]}, {x, -3, 3}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Create multiple legends: ```wl In[1]:= AbsArgPlot[{Exp[x], -2Exp[x], 4Exp[Pi I x]}, {x, -1, 1}, PlotLegends -> {Automatic, LineLegend[{Red, Cyan}, {"real, positive", "real, negative"}]}] Out[1]= [image] ``` #### PlotPoints (1) Use more initial points to get smoother curves: ```wl In[1]:= Table[AbsArgPlot[PolyGamma[I x], {x, 0, 6}, PlotPoints -> i, MaxRecursion -> 0], {i, {10, 15, 25, 40}}] Out[1]= {[image], [image], [image], [image]} ``` #### PlotRange (1) The plot range is selected automatically: ```wl In[1]:= ft = FourierTransform[Exp[-x]UnitStep[x], x, k]; In[2]:= AbsArgPlot[ft, {k, -20, 20}] Out[2]= [image] ``` Focus on a specified range of $y$ values: ```wl In[3]:= AbsArgPlot[ft, {k, -20, 20}, PlotRange -> {0, 0.5}] Out[3]= [image] ``` #### PlotStyle (2) ``PlotStyle`` can be used to style curves: ```wl In[1]:= AbsArgPlot[{I + ArcSinh[x], ArcCosh[x]}, {x, -3, 3}, PlotStyle -> {Thick, Directive[Blue, Dashed]}] Out[1]= [image] ``` --- The coloring of the argument takes precedence over colors specified with ``PlotStyle`` : ```wl In[1]:= AbsArgPlot[{I + ArcSinh[x], ArcCosh[x]}, {x, -3, 3}, PlotStyle -> {Red, Blue}] Out[1]= [image] ``` #### PlotTheme (1) Use a theme: ```wl In[1]:= AbsArgPlot[{I + ArcSinh[x], ArcCosh[x]}, {x, -3, 3}, PlotTheme -> "Marketing"] Out[1]= [image] ``` #### RegionFunction (1) Show the curve where $0 Function[{x, y}, 0 < Mod[x, 4] < 2]] Out[1]= [image] ``` #### ScalingFunctions (6) By default, plots have linear scales in each direction: ```wl In[1]:= AbsArgPlot[x + I Exp[x], {x, -5, 5}] Out[1]= [image] ``` --- Use a logarithm to scale the modulus but leave the argument (color) unscaled: ```wl In[1]:= AbsArgPlot[x + I Exp[x], {x, -5, 5}, ScalingFunctions -> "Log"] Out[1]= [image] ``` --- Use different scales in the $x$ and $y$ directions: ```wl In[1]:= AbsArgPlot[x + I Exp[x], {x, -5, 5}, ScalingFunctions -> {"Reverse", "Log"}] Out[1]= [image] ``` --- Domains that contain infinite values are scaled automatically: ```wl In[1]:= AbsArgPlot[Sinc[x], {x, -Infinity, Infinity}] Out[1]= [image] ``` --- Use ``"Reverse"`` scale in an infinite domain: ```wl In[1]:= AbsArgPlot[Sinc[x], {x, 0, Infinity}, ScalingFunctions -> {"Reverse", None}] Out[1]= [image] ``` --- Use ``Interval`` to focus on a region of interest in an infinite domain: ```wl In[1]:= AbsArgPlot[x ^ 2 * Sin[1 / x], {x, -Infinity, Infinity}, ScalingFunctions -> {{"Infinite", Interval[{-1, 1}]}, None}] Out[1]= [image] ``` #### Ticks (9) Ticks are placed automatically on each axis: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}] Out[1]= [image] ``` --- Draw axes but no tick marks: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> None] Out[1]= [image] ``` --- Use ticks on the $x$ axis but not the $y$ axis: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {Automatic, None}] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {{-.2, .1, .2}, {0, .1, .2}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with the specified labels: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {{{-.2, "-a"}, {.1, "b"}, {.2, "a"}}, {{0, 0}, {.1, "c"}, {.2, "d"}}}] Out[1]= [image] ``` --- Use specific ticks on one axis and automatic ticks on the other: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {{{-.2, "-a"}, {.1, "b"}, {.2, "a"}}, Automatic}] Out[1]= [image] ``` --- Specify the lengths for ticks as a fraction on graphics size: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {{{-.2, "-a", .05}, {.1, "b", .05}, {.2, "a", .05}}, {{0, "-d", .05}, {.13, "c", .25}, {.2, "d", .4}}}] Out[1]= [image] ``` Use different sizes in the positive and negative directions for each tick: ```wl In[2]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {{{-.2, "-a", .05}, {.1, "b", {.05, .1}}, {.2, "a", .05}}, {{0, "-d", .05}, {.13, "c", {.25, .1}}, {.2, "d", {.4, .1}}}}] Out[2]= [image] ``` --- Specify a style for each tick: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {{{-.2, "-a", .05, Directive[Red]}, {.1, "b", .05, Directive[Black]}, {.2, "a", .15, Directive[Thick, Black]}}, {{0, "-d", .05}, {.13, "c", .25, Directive[Thick, Green]}, {.21, "d", .42, Directive[Blue]}}}] Out[1]= [image] ``` --- Construct a function that places ticks at the midpoint and extremes of the axis: ```wl In[1]:= minMeanMax[min_, max_] := {{min, min}, {(max + min) / 2, (max + min) / 2}, {max, max}} In[2]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, Ticks -> {Automatic, minMeanMax}] Out[2]= [image] ``` #### TicksStyle (4) By default, the ticks and tick labels use the same style as the axis: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, AxesStyle -> Red] Out[1]= [image] ``` --- Specify an overall ticks style, including the tick labels: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, Axes -> True, TicksStyle -> Directive[Red]] Out[1]= [image] ``` --- Specify ticks style for each of the axes: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, TicksStyle -> {Directive[Red], Directive[Blue]}] Out[1]= [image] ``` --- Use a different style for the tick labels and tick marks: ```wl In[1]:= AbsArgPlot[x Sin[1 / x], {x, -.25, .25}, TicksStyle -> Directive[Red, Thick], LabelStyle -> Blue] Out[1]= [image] ``` ### Applications (3) Plot Fourier transforms: ```wl In[1]:= f1 = Exp[-x^2]UnitStep[x]; f2 = Exp[-x^2](UnitStep[x + 1] - UnitStep[x - 2]); ft1 = FourierTransform[f1, x, k]; ft2 = FourierTransform[f2, x, k]; In[2]:= AbsArgPlot[{ft1, ft2}, {k, -20, 20}, PlotRange -> All, ScalingFunctions -> "Log", PlotLabels -> {ℱ[f1], ℱ[f2]}] Out[2]= [image] ``` --- Plot the solution of a complex differential equation with initial conditions: ```wl In[1]:= sol = NDSolve[{y''[x] - I Sin[y[x]] / 2 == 0, y[0] == 2, y'[0] == I}, y[x], {x, 0, 60}][[1]]; ``` The colors are rescaled since the argument never exceeds 1 in magnitude: ```wl In[2]:= AbsArgPlot[Evaluate[y[x] /. sol], {x, 0, 60}, PlotRange -> All, ColorFunction -> (Hue[(#/Pi)]&), ColorFunctionScaling -> False, PlotLegends -> Automatic] Out[2]= [image] ``` --- Graph special functions: ```wl In[1]:= AbsArgPlot[EllipticE[x], {x, -5, 5}, Exclusions -> None, PlotLegends -> Automatic] Out[1]= [image] ``` ### Properties & Relations (8) ``AbsArgPlot`` is a special case of ``Plot`` : ```wl In[1]:= {Plot[Abs[ArcSin[x]], {x, -3, 3}], AbsArgPlot[ArcSin[x], {x, -3, 3}]} Out[1]= {[image], [image]} ``` --- ``ComplexPlot`` shows the argument and magnitude of a function using color: ```wl In[1]:= ComplexPlot[Log[z], {z, 3}] Out[1]= [image] ``` --- Use ``ComplexPlot3D`` to use the $z$ axis for the magnitude: ```wl In[1]:= ComplexPlot3D[Log[z], {z, 3}] Out[1]= [image] ``` --- Use ``ReImPlot`` to plot real and imaginary components over the real numbers: ```wl In[1]:= ReImPlot[Log[x], {x, -3, 3}] Out[1]= [image] ``` --- Use ``ComplexListPlot`` to show the location of complex numbers in the plane: ```wl In[1]:= ComplexListPlot[RandomComplex[{-3 - 3I, 3 + 3I}, 100]] Out[1]= [image] ``` --- ``ComplexContourPlot`` plots curves over the complexes: ```wl In[1]:= ComplexContourPlot[Abs[Log[z]] == 1, {z, 3}] Out[1]= [image] ``` --- ``ComplexRegionPlot`` plots regions over the complexes: ```wl In[1]:= ComplexRegionPlot[Abs[Log[z]] < 1, {z, 3}] Out[1]= [image] ``` --- ``ComplexStreamPlot`` and ``ComplexVectorPlot`` treat complex numbers as directions: ```wl In[1]:= ComplexStreamPlot[Log[z], {z, 3}] Out[1]= [image] In[2]:= ComplexVectorPlot[Log[z], {z, 3}] Out[2]= [image] ``` ### Possible Issues (2) An apparent lack of color change does not mean that the argument value does not change: ```wl In[1]:= AbsArgPlot[x + (-2^x + 1 - I / 10 - x)UnitStep[x], {x, -1, 1}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Mesh points that coincide with argument exclusions may be missing: ```wl In[1]:= {AbsArgPlot[Exp[I x], {x, 0, 2Pi}, Mesh -> 15, MeshStyle -> Black], AbsArgPlot[Exp[I x], {x, 0, 2Pi}, Mesh -> 15, MeshStyle -> Black, Exclusions -> None]} Out[1]= {[image], [image]} ``` ## See Also * [`ReImPlot`](https://reference.wolfram.com/language/ref/ReImPlot.en.md) * [`ComplexPlot`](https://reference.wolfram.com/language/ref/ComplexPlot.en.md) * [`ComplexListPlot`](https://reference.wolfram.com/language/ref/ComplexListPlot.en.md) * [`ComplexPlot3D`](https://reference.wolfram.com/language/ref/ComplexPlot3D.en.md) * [`AbsArg`](https://reference.wolfram.com/language/ref/AbsArg.en.md) * [`ReIm`](https://reference.wolfram.com/language/ref/ReIm.en.md) * [`Plot`](https://reference.wolfram.com/language/ref/Plot.en.md) * [`ParametricPlot`](https://reference.wolfram.com/language/ref/ParametricPlot.en.md) ## Related Guides * [Complex Visualization](https://reference.wolfram.com/language/guide/ComplexVisualization.en.md) * [Function Visualization](https://reference.wolfram.com/language/guide/FunctionVisualization.en.md) * [Functions of Complex Variables](https://reference.wolfram.com/language/guide/FunctionsOfComplexVariables.en.md) ## History * [Introduced in 2019 (12.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn120.en.md) \| [Updated in 2021 (13.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn130.en.md) ▪ [2023 (13.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn133.en.md)