Plot the absolute value of a complex function colored by its argument.
By default, the aspect ratio of 2D plots is set to aspect="equal".
Depending on the provided range, this function will produce different
types of plots:
Line plot over the reals.
Image plot over the complex plane if threed=False. This is also
known as Domain Coloring. Use the coloring keyword argument to
select a different color scheme. At the moment, only HSV coloring is
supported.
If threed=True, plot a 3D surface of the absolute value over the
complex plane, colored by its argument. Use the coloring keyword
argument to select a different color scheme. At the moment, only HSV
coloring is supported.
Typical usage examples are in the followings:
Plotting a single expression with a single range.
plot_complex(expr, range, **kwargs)
Plotting a single expression with the default range (-10, 10).
plot_complex(expr, **kwargs)
Plotting multiple expressions with a single range.
plot_complex(expr1, expr2, …, range, **kwargs)
Plotting multiple expressions with multiple ranges.
Represent the complex function to be plotted. It can be a:
Symbolic expression.
Numerical function of one variable, supporting vectorization.
In this case the following keyword arguments are not supported:
params.
range3-element tuple
Denotes the range of the variables. For example:
(z,-5,5): plot a line over the reals from point -5 to
5
(z,-5+2*I,5+2*I): plot a line from complex point
(-5+2*I) to (5+2*I). Note the same imaginary part
for the start/end point. Also note that we can specify the
ranges by using standard Python complex numbers, for example
(z,-5+2j,5+2j).
(z,-5-3*I,5+3*I): surface or contour plot of the
complex function over the specified domain.
labelstr, optional
The name of the complex function to be eventually shown on the
legend. If none is provided, the string representation of the
function will be used.
rendering_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines, surfaces or images.
Refer to the plotting library (backend) manual for more
informations.
adaptivebool, optional
If True, creates line plots by using an adaptive algorithm.
Use adaptive_goal and loss_fn to further customize the output.
Image and surface plots do not use an adaptive algorithm.
Default to False, which uses a uniform sampling strategy.
adaptive_goalcallable, int, float or None
Controls the “smoothness” of the evaluation. Possible values:
None (default): it will use the following goal:
lambdal:l.loss()<0.01
number (int or float). The lower the number, the more
evaluation points. This number will be used in the following goal:
lambdal:l.loss()<number
callable: a function requiring one input element, the learner. It
must return a float number. Refer to [2] for more information.
aspect(float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend’s documentation to find out the
possible values.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
labelstr or list/tuple, optional
The label to be shown in the legend or colorbar in case of a line plot.
If not provided, the string representation of expr will be used.
The number of labels must be equal to the number of expressions.
loss_fncallable or None
The loss function to be used by the adaptive learner.
Possible values:
None (default): it will use the default_loss from the
adaptive module.
callable : Refer to [2] for more information. Specifically,
look at adaptive.learner.learner1D to find more loss functions.
modulesstr, optional
Specify the modules to be used for the numerical evaluation. Refer to
lambdify to visualize the available options. Default to None,
meaning Numpy/Scipy will be used. Note that other modules might
produce different results, based on the way they deal with branch
cuts.
n1, n2int, optional
Number of discretization points in the real/imaginary-directions,
respectively, when adaptive=False. For line plots, default to 1000.
For surface/contour plots (2D and 3D), default to 300.
nint or two-elements tuple (n1, n2), optional
If an integer is provided, set the same number of discretization
points in all directions. If a tuple is provided, it overrides
n1 and n2. It only works when adaptive=False.
paramsdict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn’t support the
adaptive algorithm (meaning it will use adaptive=False).
Learn more by reading the documentation of iplot.
rendering_kwdict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines, surfaces or images.
Refer to the plotting library (backend) manual for more informations.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of series generated by the plotting function.
showboolean, optional
Default to True, in which case the plot will be shown on the screen.
size(float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to None, meaning
the size will be set by the backend.
threedboolean, optional
It only applies to a complex function over a complex range. If False,
a 2D image plot will be shown. If True, 3D surfaces will be shown.
Default to False.
coloringstr or callable
Choose between different domain coloring options. Default to "a".
Refer to [1] for more information.
"a": standard domain coloring using HSV, showing the argument
of the complex function.
"b": enhanced domain coloring using HSV, showing iso-modulus
and is-phase lines.
"c": enhanced domain coloring using HSV, showing iso-modulus
lines.
"d": enhanced domain coloring using HSV, showing iso-phase
lines.
"e": alternating black and white stripes corresponding to
modulus.
"f": alternating black and white stripes corresponding to
phase.
"g": alternating black and white stripes corresponding to
real part.
"h": alternating black and white stripes corresponding to
imaginary part.
"i": cartesian chessboard on the complex points space. The
result will hide zeros.
"j": polar Chessboard on the complex points space. The result
will show conformality.
The user can also provide a callable, f(w), where w is an
[n x m] Numpy array (provided by the plotting module) containing
the results (complex numbers) of the evaluation of the complex
function. The callable should return:
imgndarray [n x m x 3]
An array of RGB colors (0 <= R,G,B <= 255)
colorscalendarray [N x 3] or None
An array with N RGB colors, (0 <= R,G,B <= 255).
If colorscale=None, no color bar will be shown on the plot.
phaseresint
Default value to 20. It controls the number of iso-phase and/or
iso-modulus lines in domain coloring plots.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabel, zlabelstr, optional
Labels for the x-axis, y-axis or z-axis, respectively.
zlabel is only available for 3D plots.
xscale, yscale‘linear’ or ‘log’, optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to 'linear'.
xlim, ylim, zlim(float, float), optional
Denotes the x-axis limits, y-axis limits or z-axis limits,
respectively, (min,max). zlim is only available for 3D plots.
Interactive-widget plot of a Fourier Transform. Refer to the interactive
sub-module documentation to learn more about the params dictionary.
This plot illustrates:
the use of prange (parametric plotting range).
for plot_complex, symbols going into prange must be real.
the use of the params dictionary to specify sliders in
their basic form: (default, min, max).
fromsympyimport*fromspbimport*x,k,a,b=symbols("x, k, a, b")c=symbols("c",real=True)f=exp(-x**2)*(Heaviside(x+a)-Heaviside(x-b))fs=fourier_transform(f,x,k)plot_complex(fs,prange(k,-c,c),params={a:(1,-2,2),b:(-2,-2,2),c:(4,0.5,4)},label="Arg(fs)",xlabel="k",yscale="log",ylim=(1e-03,10),use_latex=False)
Domain coloring plot. To improve the smoothness of the results, increase
the number of discretization points and/or apply an interpolation (if the
backend supports it):
>>> plot_complex(gamma(z),(z,-3-3j,3+3j),... {"interpolation":"spline36"},# passed to matplotlib's imshow... coloring="b",n=500,grid=False)Plot object containing:[0]: complex domain coloring: gamma(z) for re(z) over (-3.0, 3.0) and im(z) over (-3.0, 3.0)
Interactive-widget domain coloring plot. Refer to the interactive
sub-module documentation to learn more about the params dictionary.
This plot illustrates:
the use of prange (parametric plotting range).
the use of the params dictionary to specify sliders in
their basic form: (default, min, max).
fromsympyimport*fromspbimport*z,u,a,b=symbols("z, u, a, b")plot_complex(sin(u*z),prange(z,-a-b*I,a+b*I),{"interpolation":"spline36"},use_latex=False,coloring="b",n=250,grid=False,params={u:(0.5,0,2),a:(pi,0,2*pi),b:(pi,0,2*pi),})
The name associated to the list of the complex numbers to be
eventually shown on the legend. Default to empty string.
rendering_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations. Note that
the same options will be applied to all series generated for the
specified expression.
aspect(float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend’s documentation to find out the
possible values.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
is_pointboolean
If True, a scatter plot will be produced. Otherwise a line plot will
be created. Default to True.
is_filledboolean, optional
Default to True, which will render empty circular markers. It only
works if is_point=True.
If False, filled circular markers will be rendered.
labelstr or list/tuple, optional
The name associated to the list of the complex numbers to be
eventually shown on the legend. The number of labels must be equal to
the number of series generated by the plotting function.
paramsdict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn’t support the
adaptive algorithm (meaning it will use adaptive=False).
Learn more by reading the documentation of iplot.
rendering_kwdict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of series generated by the plotting function.
showboolean
Default to True, in which case the plot will be shown on the screen.
size(float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to None, meaning
the size will be set by the backend.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, optional
Labels for the x-axis or y-axis, respectively.
xscale, yscale‘linear’ or ‘log’, optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to 'linear'.
xlim, ylim(float, float), optional
Denotes the x-axis limits or y-axis limits, respectively,
(min,max).
Plot the real part, the imaginary parts, the absolute value and the
argument of a complex function. By default, only the real and imaginary
parts will be plotted. Use keyword argument to be more specific.
By default, the aspect ratio of 2D plots is set to aspect="equal".
Depending on the provided expression, this function will produce different
types of plots:
line plot over the reals.
surface plot over the complex plane if threed=True.
contour plot over the complex plane if threed=False.
Typical usage examples are in the followings:
Plotting a single expression with a single range.
plot_real_imag(expr, range, **kwargs)
Plotting a single expression with the default range (-10, 10).
plot_real_imag(expr, **kwargs)
Plotting multiple expressions with a single range.
plot_real_imag(expr1, expr2, …, range, **kwargs)
Plotting multiple expressions with multiple ranges.
(z,-5,5): plot a line over the reals from point -5 to
5
(z,-5+2*I,5+2*I): plot a line from complex point
(-5+2*I) to (5+2*I). Note the same imaginary part
for the start/end point. Also note that we can specify the
ranges by using standard Python complex numbers, for example
(z,-5+2j,5+2j).
(z,-5-3*I,5+3*I): surface or contour plot of the
complex function over the specified domain using a rectangular
discretization.
labelstr, optional
The name of the complex function to be eventually shown on the
legend. If none is provided, the string representation of the
function will be used.
rendering_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations. Note that
the same options will be applied to all series generated for the
specified expression.
absboolean, optional
If True, plot the modulus of the complex function. Default to False.
adaptivebool, optional
If True, creates line plots by using an adaptive algorithm.
Use adaptive_goal and loss_fn to further customize the output.
Image and surface plots do not use an adaptive algorithm.
Default to False, which uses a uniform sampling strategy.
adaptive_goalcallable, int, float or None
Controls the “smoothness” of the evaluation. Possible values:
None (default): it will use the following goal:
lambdal:l.loss()<0.01
number (int or float). The lower the number, the more
evaluation points. This number will be used in the following goal:
lambdal:l.loss()<number
callable: a function requiring one input element, the learner. It
must return a float number. Refer to [3] for more information.
argboolean, optional
If True, plot the argument of the complex function. Default to False.
aspect(float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend’s documentation to find out the
possible values.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
detect_polesboolean, optional
Chose whether to detect and correctly plot poles. Defaulto to False.
It only works with line plots. To improve detection, increase the
number of discretization points if adaptive=False and/or change
the value of eps.
epsfloat, optional
An arbitrary small value used by the detect_poles algorithm.
Default value to 0.1. Before changing this value, it is better to
increase the number of discretization points.
imagboolean, optional
If True, plot the imaginary part of the complex function.
Default to True.
labellist/tuple, optional
The labels to be shown in the legend. If not provided, the string
representation of expr will be used. The number of labels must be
equal to the number of series generated by the plotting function.
loss_fncallable or None
The loss function to be used by the adaptive learner.
Possible values:
None (default): it will use the default_loss from the
adaptive module.
callable : Refer to [3] for more information. Specifically,
look at adaptive.learner.learner1D to find more loss functions.
modulesstr, optional
Specify the modules to be used for the numerical evaluation. Refer to
lambdify to visualize the available options. Default to None,
meaning Numpy/Scipy will be used. Note that other modules might
produce different results, based on the way they deal with branch
cuts.
n1, n2int, optional
Number of discretization points in the real/imaginary-directions,
respectively, when adaptive=False. For line plots, default to 1000.
For surface/contour plots (2D and 3D), default to 300.
nint or two-elements tuple (n1, n2), optional
If an integer is provided, set the same number of discretization
points in all directions. If a tuple is provided, it overrides
n1 and n2. It only works when adaptive=False.
paramsdict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn’t support the
adaptive algorithm (meaning it will use adaptive=False).
Learn more by reading the documentation of iplot.
rendering_kwdict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of series generated by the plotting function.
realboolean, optional
If True, plot the real part of the complex function. Default to True.
showboolean, optional
Default to True, in which case the plot will be shown on the screen.
size(float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to None, meaning
the size will be set by the backend.
surface_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of surfaces. Refer to the
plotting library (backend) manual for more informations.
threedboolean, optional
It only applies to a complex function over a complex range. If False,
contour plots will be shown. If True, 3D surfaces will be shown.
Default to False.
use_cmboolean, optional
If False, surfaces will be rendered with a solid color.
If True, a color map highlighting the elevation will be used.
Default to True.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
wireframeboolean, optional
Enable or disable a wireframe over the 3D surface. Depending on the
number of wireframe lines (see wf_n1 and wf_n2), activating
thisoption might add a considerable overhead during the plot’s
creation. Default to False (disabled).
wf_n1, wf_n2int, optional
Number of wireframe lines along the x and y ranges, respectively.
Default to 10. Note that increasing this number might considerably
slow down the plot’s creation.
wf_npointint or None, optional
Number of discretization points for the wireframe lines. Default to
None, meaning that each wireframe line will have n1 or n2
number of points, depending on the line direction.
wf_rendering_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of wireframe lines.
xlabel, ylabel, zlabelstr, optional
Labels for the x-axis, y-axis or z-axis, respectively.
zlabel is only available for 3D plots.
xscale, yscale‘linear’ or ‘log’, optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to 'linear'.
xlim, ylim, zlim(float, float), optional
Denotes the x-axis limits, y-axis limits or z-axis limits,
respectively, (min,max). zlim is only available for 3D plots.
Plot the real and imaginary parts of a function over reals:
>>> plot_real_imag(sqrt(x),(x,-3,3))Plot object containing:[0]: cartesian line: re(sqrt(x)) for x over (-3.0, 3.0)[1]: cartesian line: im(sqrt(x)) for x over (-3.0, 3.0)
>>> plot_real_imag(sqrt(x),(x,-3,3),real=False,imag=False,abs=True,arg=True)Plot object containing:[0]: cartesian line: abs(sqrt(x)) for x over (-3.0, 3.0)[1]: cartesian line: arg(sqrt(x)) for x over (-3.0, 3.0)
3D plot of the real and imaginary part of the principal branch of a
function over a complex range. Note the jump in the imaginary part: that’s
a branch cut. The rectangular discretization is unable to properly capture
it, hence the near vertical wall. Refer to plot3d_parametric_surface
for an example about plotting Riemann surfaces and properly capture
the branch cuts.
>>> plot_real_imag(sqrt(x),(x,-3-3j,3+3j),n=100,threed=True,... use_cm=True)Plot object containing:[0]: complex cartesian surface: re(sqrt(x)) for re(x) over (-3.0, 3.0) and im(x) over (-3.0, 3.0)[1]: complex cartesian surface: im(sqrt(x)) for re(x) over (-3.0, 3.0) and im(x) over (-3.0, 3.0)
3D plot of the absolute value of a function over a complex range:
>>> plot_real_imag(sqrt(x),(x,-3-3j,3+3j),... n=100,real=False,imag=False,abs=True,threed=True)Plot object containing:[0]: complex cartesian surface: abs(sqrt(x)) for re(x) over (-3.0, 3.0) and im(x) over (-3.0, 3.0)
Interactive-widget plot. Refer to the interactive sub-module documentation
to learn more about the params dictionary. This plot illustrates:
the use of prange (parametric plotting range).
the use of the params dictionary to specify sliders in
their basic form: (default, min, max).
fromsympyimport*fromspbimport*x,u,a,b=symbols("x, u, a, b")plot_real_imag(sqrt(x)*exp(u*x),prange(x,-3*a-b*3j,3*a+b*3j),backend=PB,aspect="cube",wireframe=True,wf_rendering_kw={"line_width":1},params={u:(0.25,0,1),a:(1,0,2),b:(1,0,2)},n=25,threed=True,use_latex=False,use_cm=True)
Plot the vector field [re(f), im(f)] for a complex function f
over the specified complex domain. By default, the aspect ratio of 2D
plots is set to aspect="equal".
Typical usage examples are in the followings:
Plotting a vector field of a complex function.
plot_complex_vector(expr, range, **kwargs)
Plotting multiple vector fields with different ranges and custom labels.
Denotes the range of the variables. For example
(z,-5-3*I,5+3*I). Note that we can specify the range
by using standard Python complex numbers, for example
(z,-5-3j,5+3j).
labelstr, optional
The name of the complex expression to be eventually shown on the
legend. If none is provided, the string representation of the
expression will be used.
aspect(float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend’s documentation to find out the
possible values.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
contours_kwdict
A dictionary of keywords/values which is passed to the backend
contour function to customize the appearance. Refer to the plotting
library (backend) manual for more informations.
n1, n2int
Number of discretization points for the quivers or streamlines in the
x/y-direction, respectively. Default to 25.
nint or two-elements tuple (n1, n2), optional
If an integer is provided, set the same number of discretization
points in all directions for quivers or streamlines. If a tuple is
provided, it overrides n1 and n2. It only works when
adaptive=False. Default to 25.
ncint
Number of discretization points for the scalar contour plot.
Default to 100.
paramsdict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn’t support the
adaptive algorithm (meaning it will use adaptive=False).
Learn more by reading the documentation of iplot.
quiver_kwdict
A dictionary of keywords/values which is passed to the backend
quivers-plotting function to customize the appearance. Refer to the
plotting library (backend) manual for more informations.
scalarboolean, Expr, None or list/tuple of 2 elements
Represents the scalar field to be plotted in the background of a 2D
vector field plot. Can be:
True: plot the magnitude of the vector field. Only works when a
single vector field is plotted.
False/None: do not plot any scalar field.
Expr: a symbolic expression representing the scalar field.
list/tuple: [scalar_expr, label], where the label will be
shown on the colorbar.
Remember: the scalar function must return real data.
Default to True.
showboolean
The default value is set to True. Set show to False and
the function will not display the plot. The returned instance of
the Plot class can then be used to save or display the plot
by calling the save() and show() methods respectively.
size(float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to None, meaning
the size will be set by the backend.
streamlinesboolean
Whether to plot the vector field using streamlines (True) or quivers
(False). Default to False.
stream_kwdict
A dictionary of keywords/values which is passed to the backend
streamlines-plotting function to customize the appearance. Refer to
the plotting library (backend) manual for more informations.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, optional
Labels for the x-axis or y-axis, respectively.
xscale, yscale‘linear’ or ‘log’, optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to 'linear'.
xlim, ylim, zlim(float, float), optional
Denotes the x-axis limits ory-axis limits, respectively,
(min,max).
Overlay the quiver plot to a domain coloring plot. By setting n=26
(even number) in the complex vector plot, the quivers won’t to cross
the branch cut.
Interactive-widget plot. Refer to the interactive sub-module documentation
to learn more about the params dictionary. This plot illustrates:
the use of prange (parametric plotting range).
the use of the params dictionary to specify sliders in
their basic form: (default, min, max).
fromsympyimport*fromspbimport*z,u,a,b=symbols("z u a b")plot_complex_vector(log(gamma(u*z)),prange(z,-5*a-b*5j,5*a+b*5j),params={u:(1,0,2),a:(1,0,2),b:(1,0,2)},n=20,grid=False,use_latex=False,quiver_kw=dict(color="orange",headwidth=4))