Complex Analysis Playground

Experiment with complex functions, contour integrals, and series expansions. This is your space to explore and apply everything you've learned!

Welcome to the Playground!

You've journeyed through 9 phases of complex analysis. Now it's time to put your knowledge to work!

In this playground, you can:

Plot any complex function using domain coloring - type in your own formulas!
Draw custom contours and compute their integrals using the residue theorem
Explore series expansions for functions around different centers

This is where theory meets practice. Experiment, make discoveries, and deepen your understanding!

1. Complex Function Plotter

Visualize any complex function using domain coloring. Type in expressions like z^2, exp(z), sin(z), or 1/(z^2+1).

Enter a complex function (in terms of z):

Supported: z, z^n, exp(z), sin(z), cos(z), log(z), sqrt(z), conj(z), 1/z, 1/(z^2+1), etc.

Quick presets:

x range: [-2, 2]

y range: [-2, 2]

Domain Coloring Visualization:

Hue = phase (argument), Brightness = magnitude. Black = zero, white rings = poles.

💡 How to use:

Type any complex function in terms of z
Click presets for common functions
Adjust the viewing window to zoom in/out
Look for zeros (black points) and poles (white centers)
Notice how colors swirl around singularities - the number of rotations = pole/zero order!

2. Custom Contour Integrator

Draw your own integration paths and compute ∮ f(z) dz numerically. Test the Cauchy theorem: integrals around singularities are non-zero, while integrals avoiding them are zero!

Function to integrate:

Load preset path:

Click to draw contour (green = start, red dots = singularities):

Click to add points to your contour. Blue path shows your integration path.

🔄 How to use:

Enter a function in terms of z
Click on the canvas to add points to your contour
Use "Close Path" to connect back to the start
Click "Compute Integral" to evaluate ∮ f(z) dz
Test Cauchy's theorem: Paths that don't enclose singularities should give ≈ 0
For 1/z around origin: Should get 2πi (residue theorem!)

3. Series Calculator

Compute Taylor and Laurent series coefficients for complex functions. Explore how series converge in different regions of the complex plane.

Function f(z):

Expansion center (Re):

Expansion center (Im):

Series type:

Taylor Series (analytic functions)Laurent Series (includes negative powers)

Quick presets:

📊 How to use:

Enter a complex function
Choose the expansion center z₀ (often 0)
Select Taylor (for analytic functions) or Laurent (for functions with singularities)
Click "Compute Coefficients" to see the series expansion
Taylor: f(z) = Σ aₙ(z-z₀)ⁿ (n ≥ 0)
Laurent: f(z) = Σ aₙ(z-z₀)ⁿ (n ∈ ℤ, includes negative powers)

🎓

Congratulations!

You've completed the Complex Analysis module

You've mastered one of the most beautiful and powerful areas of mathematics. From basic complex arithmetic to residue theory, from conformal mappings to real-world applications - you now have the tools to solve problems that would be intractable with real analysis alone.

Keep exploring, keep experimenting, and remember: complex analysis is the language of nature! 🌟

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