Configure the Complex Mapping
The page uses one main content column. The input fields switch to 3, 2, and 1 columns by screen size.
Formula Used
A complex number is written as z = x + iy, where x is the real part and y is the imaginary part. This tool evaluates one selected mapping family over a rectangular grid in the complex plane.
- Polynomial: f(z) = a·zn + b
- Exponential: f(z) = exp(a·z) + b
- Complex sine: f(z) = sin(a·z + b) + c
- Reciprocal: f(z) = a / (z + b) + c
- Möbius transform: f(z) = (a·z + b) / (c·z + d)
The tool also computes the magnitude and phase: |f(z)| = √(Re(f(z))² + Im(f(z))²) and arg(f(z)) = atan2(Im(f(z)), Re(f(z))).
How to Use This Calculator
- Select a function family that matches the complex mapping you want to study.
- Enter the complex coefficients a, b, c, and d as separate real and imaginary parts.
- Set the input-plane boundaries for the real and imaginary axes.
- Choose a grid resolution. Larger values provide more detail but take more processing.
- Enter a sample point z to get one direct mapped result.
- Press Plot Complex Function to calculate values and draw the graphs.
- Review the summary, magnitude surface, output-plane scatter, and example data table.
- Use the export buttons to download CSV data or a quick PDF report.
1. What does this tool actually plot?
It samples many input points z in a selected rectangular region, evaluates f(z), and visualizes the output through a magnitude surface and output-plane scatter plot.
2. Why are some points undefined?
Undefined points usually appear when the denominator becomes zero or extremely small, especially in reciprocal or Möbius mappings. Those points behave like poles or singularities.
3. What is the meaning of magnitude?
Magnitude measures the distance of f(z) from the origin in the output complex plane. Larger values indicate stronger amplification or stretching at that sampled point.
4. What is phase in this calculator?
Phase is the angle of the output complex number relative to the positive real axis. It helps reveal rotation behavior introduced by the selected mapping.
5. Which function family should I choose first?
Start with the polynomial option for basic transformations, then try exponential or sine for periodic growth behavior, and Möbius for conformal-style fractional mappings.
6. Does a higher grid resolution improve accuracy?
Yes. A larger grid produces a denser sampling and smoother plots, but it also increases computation time and the amount of exportable data.
7. Can I use negative powers in the polynomial map?
Yes. Negative integer powers are allowed. They effectively introduce reciprocal behavior, so values near zero can become very large or undefined.
8. What is included in the CSV and PDF exports?
The CSV contains sampled points with input, output, magnitude, and phase. The PDF summarizes the chosen function, sample result, and key magnitude statistics.