Plot a complex functions with streamlines placed with specified densities:

Plot the streamlines that go through a set of complex seed points:

Use both automatic and explicit seeding with styles for explicitly seeded streamlines:

Plot streamlines over a specified complex region:

Plot two functions together:

Use a specific number of mesh lines:

Specify specific mesh lines:

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

Specify different dashings and arrowheads by setting to StreamScale:

Plot the streamlines with arrows colored according to the modulus of the function:

Apply a variety of streamline markers:

Use a theme with axes and a different default color:

Override the style from the theme:

Change the color function:

Specify a uniform color for the streamlines:

Specify mesh lines with different styles:

Specify global mesh line styles:

Shade mesh regions cyclically:

Apply a variety of styles to region boundaries:

Add a legend indicating the modulus of the function:

Use the functions as legend labels:

Use explicit labels for each vector field:

Use a colored background:

Show where the vector field function is sampled:

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

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

No legends are included, by default:

Include a legend that indicates the modulus of the function:

Include a legend to distinguish two functions:

Control the placement of the legend:

Use the functions as the legend text:

Use placeholder text:

Change the appearance of the legend:

The full plot range is used by default:

Specify an explicit limit for both the and ranges:

Specify an explicit range:

Specify an explicit range:

Specify different and ranges:

Use a theme with simpler ticks and brighter colors:

Use a theme with automatic legends and dense streamlines:

Change the stream styles:

By default, region boundaries are styled automatically:

Apply a variety of styles to region boundaries:

By default, regions are filled:

Show no filling:

Choose a different filling:

Plot streamlines only over a disk:

Plot streamlines only over regions where the modulus of the function exceeds a given threshold:

Use a logical combination of conditions:

Color streamlines according to the modulus of the function:

Use any named color gradient from ColorData:

Use ColorData for predefined color gradients:

Specify a color function that blends two colors by :

Use StreamColorFunctionScalingFalse to get unscaled values:

By default, scaled values are used:

Use StreamColorFunctionScalingFalse to get unscaled values:

Explicitly specify the scaling for each color function argument:

Streamlines are drawn as arrows by default:

Use a named appearance to draw the streamlines:

Use different markers for different vector fields:

Use named styles:

Named arrow styles:

Named dot styles:

Named pointer styles:

Named dart styles:

Specify a specific maximum number of streamlines:

Use symbolic names to specify the number of streamlines:

Use both automatic and explicit seeding with styles for explicitly seeded streamlines:

Specify the minimum distance between streamlines:

Specify the minimum distance between streamlines at the start and end of a streamline:

Create full streamlines without segmentation:

Use curves for streamlines:

Use symbolic names to control the lengths of streamlines:

Specify segment lengths:

Specify an explicit dashing pattern for streamlines:

Specify the number of points rendered on each streamline segment:

Specify absolute aspect ratios relative to the longest line segment:

Specify relative aspect ratios relative to each line segment:

Scale the length of the arrows by the :

StreamColorFunction has precedence over StreamStyle for colors:

Use StreamColorFunctionNone to specify colors with StreamStyle:

Apply a variety of styles to the streamlines:

Specify a custom arrowhead:

Set the style for multiple functions:

Plot a function with a simple zero:

Shift the function to the left by 1:

Plot a function with a double zero:

Plot a square root function:

Plot a trigonometric function:

Plot a transcendental function:

Plot a function with a simple pole:

Plot a function with a double pole:

Streamlines that diverge from a point indicate a simple zero:

Streamlines can also converge at a simple zero:

With , the real vector field corresponding to the complex function is , and the trajectories that follow the field satisfy the differential equation . The implicit solution is for real , which corresponds to a family of circles that are tangent to the real axis at the origin:

In polar coordinates, the trajectories are for any real :

More generally, for where is an integer, the streamlines follow for constant :

Near a zero of order , the streamlines form loops that start and end at the zero in directions:

Near a pole of order , the streamlines converge to the pole from directions and diverge from the pole from directions:

The function has simple zeros at and , poles of order 1 at , and a pole of order 2 at :

Near an essential singularity, the streamlines vary wildly:

Plot a function and its derivatives:

Generate a Pólya plot:

Let be a complex potential for an ideal fluid flow. Then is the velocity potential, is the stream function, and the fluid velocity field is . By the Cauchy–Riemann equations, , so you can generate a stream plot with the conjugate of . Show streamlines for flow around a cylinder with circulation:

Show streamlines for flow external to a corner: