Controls

Signal Configuration

Waveform Type

Frequency: 1.0 Hz

Amplitude: 1.0

Sampling Parameters

Time Window: 4.0 s

Samples: 128

Winding Analysis

Watch the golden center of mass jump when winding frequency matches signal frequency!

Winding Freq: 1.0 Hz

0.1 10

Progress: 100%

Signal Configuration

Waveform Type

Frequency: 2.0 Hz

Damping σ: 0.5

s-Plane Exploration

σ (Real): 0.0

ω (Imaginary): 1.0

Show ROC (Region of Convergence)

3D Visualization

Display Mode

Clip Height: 10

Show Poles & Zeros

Integral Animation

Speed: 3.0s

Loop Animation

Enable Output Chaining

Chain Mode: Chain Depth: 1

Enable Algebraic Chaining

Dynamic Plotting

Dynamic Plotting Studio Integers → selected function → map

Build a discrete complex plot

Choose a sequence, write its general term, then map, add, or multiply.

What you are building Map

Select a preset or configure the pipeline below

Choose the sequence

Define d_j and how many terms to generate

Sequence Terms (N)

First term (a) Common difference (Δ)

First term (a) Common ratio (r)

First denominator (a) Denominator difference (Δ)

Minimum Maximum (optional)

Include signed associates (±p)

Starting search radius

Expands automatically until all N values are found.

Bound type Associates

Include conjugates

Points — one per line or semicolon-separated (re,im or a+bi)

Source formula d_j =

Use j = 0, 1, 2, ...

Filter — keep when (optional)

Source rule

First terms 1, 2, 3, 4, 5, …

Place it on the input plane

Usually z_j = d_j; change it only when you want a transformed input path

Input point z_j =

Usually leave this as d. Example: d * exp(i*k*j) creates a spiral.

Write the general term

Build a_j, then define how each symbol changes with j

For each input point, compute

Current general term

Product-term composer

Each factor can sit in the numerator or denominator, carry a power, or be wrapped in a function.

Complete formula

Edit the equation directly, or use the factor composer above.

Start from an expression

d source value j index z input point s free parameter i imaginary unit

Equation guide

Write ordinary mathematical expressions. New symbols automatically get their own sequence rule.

Values you already have

j: term index: 0, 1, 2, ...
c: current input-plane parameter
d: current source value d_j
z: current input-plane point z_j
i, pi, e: mathematical constants

Writing expressions

+ - * / ^ ! are supported. Multiplication may be implicit: 2j, 3i, 2(n+1).

Use conditions as isPrime(j) ? 1/j : 0.

Complex functions

sin cos tan exp ln sqrt abs arg conj re im zeta selected

Also: floor ceil round mod gcd factorial isPrime min max complex.

Defined behavior

Division by zero, invalid factorials, overflow, and undefined function values are marked undefined instead of breaking the plot.

Equations are bounded in size and complexity so live editing remains responsive.

How each symbol changes

At term j, every symbol below supplies its j-th value. Change the sequence independently for each symbol.

Combine the terms

Choose the mathematical object to draw

⊙ Map Σ Sum Π Product

When a term is undefined Product visualization

Free parameter

Evaluate the construction at this complex value.

s = + i

Examples Ready-made constructions Browse

Choose an example, then change any part of it.

Named parameters

Accumulation Playback

Watch terms accumulate one by one. The slider controls how many terms are visible.

Showing 50 / 50 terms

Visible count Terms / second

Loop playback

Navigate through complex plane

Size: 0.55

Opacity: 0.90

Speed: 1.10

Trail: 0

Real part of z (a₀): 0.80

Imaginary part of z (b₀): 0.50

Radius (R): 1.0

Semi-axis A: 1.5

Semi-axis B: 0.7

Param A: 1.0

Param B: 0.5

Image Upload Bring a source image into the z-plane and map it like any other input shape.

Choose Source Image PNG, JPG, GIF, or WEBP

Resolution (pts): 300

Image Base Size: 2.0

Opacity: 1.0

Video Upload Map video frames through the active complex function with controllable playback, FPS sampling, and scale.

Choose Source Video MP4, WEBM, MOV, OGG, or any browser-playable video

No video loaded.

Resolution (pts): 300

Processing FPS: 60

Video Base Size: 2.0

Opacity: 1.0

Horizontal Strip Parameters:

y₁: -0.5

y₂: 0.5

Angular Sector Parameters:

θ₁ (deg): 0

θ₂ (deg): 45

R_min: 0.5

R_max: 2.0

Möbius Parameters (w = (az+b)/(cz+d)):

Re(a): 1.0

Im(a): 0.0

Re(b): 0.0

Im(b): 0.0

Re(c): 0.0

Im(c): 0.0

Re(d): 1.0

Im(d): 0.0

Polynomial: w = a₀zⁿ + ... + a_n-1z + a_n

Degree (n): 2

Fractional Power: w = zⁿ

Power (n): 0.5

Visualization

Grid Density: 15

Probe (r_local): 0.2

Z-Plane Zoom: 1.00x

W-Plane Zoom: 1.00x

Analysis

Zeros & Poles Mark detected roots and singularities. Critical Points Show where f'(z) vanishes or breaks. Radial Steps Draw discrete sampling circles on the input plane. Cauchy Integral Contour image, winding, and residue readout. Taylor Series Local polynomial expansion around z0.

Discrete steps: 200

Local Approximation

Expand Around z0

Taylor series is local. By default the expansion stays centered at the origin. Pin a custom center when you want to inspect a different neighborhood.

z0 = 0

Taylor Series Terms 3

Use Custom Expansion Center

Surface & Projection

Domain Coloring Hue for argument, lightness bands for magnitude. Riemann Sphere Project the mapped output onto the sphere. Riemann Surface Explore branch sheets as an interactive WebGL surface.

Surface Height

Visible Sheets: 5

Branch Center k: 0

Height Scale: 1.00

Height Clip: 8.0

Surface Grid

GPU surface ready

Domain Palette: Analytic Base

Brightness: 1.0

Contrast: 1.0

Saturation: 1.0

Lightness Cycles: 1.00

3D Sphere Use Three.js for an interactive w-sphere. Split View Keep the z-plane beside the w-sphere. Show Transformation Animate grid wrapping into the sphere.

Sphere Opacity: 0.10

Surface Grid Opacity: 0.00

Vector Field

Vector Field Sample arrows from f(z), 1/f(z), or f'(z). Streamlines Trace integral curves through the active field. Particle Motion Animate particles drifting through the field.

Vector Field Settings Choose the source function and tune arrow rendering.

Vector Function:

Vector Scale: 0.10

Arrow Thickness: 1.5

Arrow Head Size: 6

Streamline Settings Control seeding, thickness, and integration quality.

Integration Step Size: 0.06

Max Length (steps): 400

Streamline Thickness: 1.5

Seed Density Factor: 0.8

Particle Settings Tune density, speed, and particle lifetime.

Particle Density: 150

Particle Speed: 0.04

Particle Max Lifetime (frames): 300

z-plane (Input)

z-plane (Input: `z = a + ib`)

Refining...

Stereographic Projection

Plane (z) Riemann Sphere

Arrow keys to navigate

w-plane (Output)

w-plane (Output: `w = cos(z)`)

Refining...

Stereographic Projection

Plane (w) Riemann Sphere

Expand Around z0

z-plane (Input: z = a + ib)

Stereographic Projection

w-plane (Output: w = cos(z))

Stereographic Projection

3D Surface: |F(s)| Magnitude

Themes

Custom Grid Colors

z-plane (Input: `z = a + ib`)

w-plane (Output: `w = cos(z)`)