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ComplexVectorPlot—Wolfram Documentation

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ComplexVectorPlot[f,{z,z_min,z_max}]

generates a vector plot of the vector field {Re[f],Im[f]} over the complex rectangle with corners z_min and z_max.

• ComplexVectorPlot is also known as field plot and direction plot.
• ComplexVectorPlot displays a vector field by drawing arrows. By default, the direction of the vector is indicated by the direction of the arrow, and the magnitude is indicated by its color.
• ComplexVectorPlot omits any arrows for which f does not evaluate to a complex number.
• ComplexVectorPlot[f,{z,n}] is equivalent to ComplexVectorPlot[f,{z,-n-n I,n+n I}].
• ComplexVectorPlot treats the variable z as local, effectively using Block.
• ComplexVectorPlot has attribute HoldAll and evaluates f only after assigning specific numerical values to z. In some cases, it may be more efficient to use Evaluate to evaluate f symbolically first.
• ComplexVectorPlot has the same options as Graphics, with the following additions and changes: [List of all options]
• The individual arrows are scaled to fit inside bounding circles around each point.
• VectorPoints{c₁,c₂,…} uses the complex numbers c₁, c₂, etc.… as the points at which to draw the arrows.
• VectorScaling scales the magnitudes of the vectors into the range of arrow sizes s_min to s_max given by VectorSizes.
• VectorScaling->Automatic will scale the arrow lengths depending on the vector magnitudes:
• The arrow markers given by VectorMarkers are drawn inside a box whose width and length are in the proportion specified by VectorAspectRatio.
• Common markers include:
• VectorColorFunction->None will draw the arrows with the style specified by VectorStyle.
• The arguments supplied to functions in RegionFunction and VectorColorFunction are z and f.

Plot the vector field for with color indicating the vector magnitude:

Include a legend for the vector magnitudes:

Use a drop-shaped marker to represent the vectors:

Plot a vector field with vectors placed with specified densities:

Sample the vector field on a regular grid of points:

Sample the vector field on an irregular mesh:

Specify how many vector points to use in each direction:

Plot the vectors that go through a set of seed points:

Plot vectors over a specified region:

Plot the fields for a function and its conjugate:

Use Evaluate to evaluate the vector field symbolically before numeric assignment:

Plot a vector field with automatically scaled arrows:

Plot a vector field with arrows of specified size:

Draw the arrows starting from the sample points:

Draw the arrows without the arrowheads:

Use drop-like shapes instead of arrows:

Change the overall shape of the markers:

Change the default color function:

Include a legend:

Vary the arrow sizes instead of the colors:

Set the style for multiple vector fields:

Apply vector style:

Set the style for multiple vector fields:

Use a theme with simple ticks and grid lines:

Use a colored background:

By default, extremely short and extremely long vectors are displayed:

Use ClippingStyleNone to remove extreme vectors from the plot:

Style the clipped vectors:

Style the short and long clipped vectors differently:

Show where the vector field function is sampled:

Count the number of times the vector field function is evaluated:

Include a legend for the vector norms:

Use the expressions in the legend for multiple vector functions:

Specify the legend labels for multiple functions:

Control the placement of the legend:

Use a legend with placeholders:

The full plot range is used by default:

Specify an explicit limit for both real and imaginary ranges:

Specify an explicit real range:

Specify an explicit imaginary range:

Specify different real and imaginary ranges:

Use a theme:

Change the vector style:

Show the region defined by a region function:

The boundaries of full rectangular regions are not shown:

Use None to not show the boundary:

Omit the interior filling as well:

Specify a style for the boundary:

Specify a style for full rectangular regions:

Show the region defined by a region function:

The interiors of full rectangular regions are not shown:

Use None to not show the interior filling:

Omit the boundary curve as well:

Specify a style for the interior filling:

Specify a style for full rectangular region:

Restrict the plotting region based on :

Restrict the plotting region based on :

The default ratio of the width to the length of the vector marker is 1/4:

Modify the ratio of the width to the length of the vector marker:

Vectors are colored according to their norms by default:

Choose the color scheme for coloring vectors by their norms:

Use any named color gradient from ColorData:

Color the vectors according to the real part of its location:

Color the vectors according to the real part of the function:

Vectors are drawn as arrows by default:

Use a named appearance to draw the vectors:

Use different markers for different vector fields:

By default, markers are centered on vector points:

Start the vectors at the points:

End the vectors at the points:

Use automatically determined vector points:

Use symbolic names to specify the set of field vectors:

Create a hexagonal grid of field vectors with the same number of arrows in the real and imaginary directions:

Create a hexagonal grid of field vectors with a different number of arrows in the real and imaginary directions:

Specify a list of points for showing field vectors:

By default, vector ranges are determined automatically:

Plot vectors with magnitudes between 0.2 and 2:

Plot vectors with magnitudes between 0.2 and 2 with scaled arrow lengths:

Style the clipped vectors:

Plot scaled vectors with all lengths:

Increase the lengths of the smaller vectors:

By default, VectorScaling is None:

Use automatic scaling to scale the length of vectors:

Vector markers have automatically scaled lengths to prevent any vectors from being too small and to keep them from overlapping:

Specify a minimum and maximum scaled vector size:

Set the style for the displayed vectors:

Set the style for multiple functions:

Use Arrowheads to specify an explicit style of the arrowheads:

Specify both arrow tail and head:

Graphics primitives without Arrowheads are scaled based on the vector scale:

Change the scaling using the VectorScaling option:

For a complex function f, plot {Re[f],Im[f]}:

The vector length increases with Abs[f] and the orientation is determined by Arg[f]:

Identify poles and zeros. Poles are visible at and :

The zeros at and are more readily visible if the vectors are scaled:

Vectors in the field rotate twice along the unit circle surrounding the zero of the function at the origin, which implies that has a double zero at the origin:

The function has a pole of order 2 at since has a double zero:

Specify a direction field and several solutions for the complex initial value problem , :

The Pólya field of an analytic function is both divergence and curl free:

Examine partial sums of an infinite series:

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