Calculator Inputs
Use rectangular input for z. Then choose a z to z mapping rule.
Formula used
Complex input: z = x + iy
Complex output: w = f(z) = u + iv
Magnitude: |w| = √(u² + v²)
Argument: θ = atan2(v, u)
Polar form: w = |w|(cos θ + i sin θ)
Linear map: w = A z + B
Mobius map: w = (A z + B) / (C z + D)
How to use this calculator
- Enter the real part and imaginary part of z.
- Select the z to z function you want to test.
- Enter A, B, C, and D when your chosen function needs them.
- Set the integer power, precision, and angle unit.
- Press the calculate button to show the mapped result.
- Use CSV or PDF options when you need a saved record.
Example data table
| Input z | Function | Coefficients | Output idea |
|---|---|---|---|
| 2 + 1i | w = A z + B | A = 1 + 2i, B = 3 - i | Rotates, scales, then shifts z. |
| 1 - 3i | w = z² | No extra coefficient | Squares the complex input. |
| 4 + 2i | w = 1/z | z must not be zero | Returns the reciprocal mapping. |
| 2 + 0.5i | w = (A z + B)/(C z + D) | A, B, C, D enabled | Applies a fractional linear map. |
Understanding complex z to z mapping
A z to z function accepts one complex number and returns another complex number. The input lives on the complex plane. The output also lives on the same plane. This makes the tool useful for algebra, signals, controls, geometry, and complex analysis.
Every complex number has two common forms. Rectangular form writes z as x plus iy. Polar form writes z with a radius and an angle. Rectangular form is easy for addition. Polar form is useful for powers, rotations, and scaling. This calculator shows both forms for the mapped result.
Linear functions are the simplest maps. A z plus B can scale, rotate, and shift a point. If A has a magnitude greater than one, the point moves farther from the origin. If A has an angle, the point rotates. B then translates the result.
The Mobius function is more advanced. It uses a ratio of two linear expressions. This type of map can convert lines and circles into other lines or circles. It is common in geometry, filter design, and conformal mapping. The denominator must not become zero.
Powers change distance and angle in a strong way. Squaring doubles the argument and squares the magnitude. Integer powers repeat this pattern. Negative powers first create a reciprocal effect. So z cannot be zero when a negative power is used.
Exponential and logarithmic maps connect rectangular and polar behavior. The exponential uses growth from the real part and rotation from the imaginary part. The principal logarithm returns the natural log of magnitude and the principal angle. It cannot accept zero.
Trigonometric complex functions are also included. Complex sine and cosine use hyperbolic terms. They can grow much faster than real sine and cosine. This matters in engineering models, wave equations, and analytic function studies.
For careful work, always read the domain note. Some functions accept every finite z. Others exclude zero or special denominator values. Also choose enough decimal precision. More digits help when comparing small changes, checking transformations, or preparing reports.
Domain behavior is also important. A function can stretch one region. It can compress another region quickly. It can preserve angles near regular points. It can create sharp changes near poles. Branch cuts can also change interpretation. Testing several inputs gives a better picture. One point rarely tells the full story. Use values near zero and far away. Try negative coordinates and pure imaginary values. Then compare how the output moves. This habit helps you catch singularities. It also helps estimate stability. Use the saved result for review. Each record lists formulas and steps. It also lists polar data and domain notes clearly online.
This calculator is not only for one answer. It is also a learning tool. Change z slightly and compare outputs. Try pure real, pure imaginary, and mixed values. Watch how magnitude and angle respond. Those patterns explain what the function does to the plane.
FAQs
What is a z to z function?
It is a complex function. It takes a complex input z and returns a complex output w. Both values are usually shown on the complex plane.
What does z = x + iy mean?
The x value is the real part. The y value is the imaginary part. The symbol i means the square root of negative one.
What is the output w?
The output w is the mapped complex number. The calculator shows it in rectangular form, polar form, magnitude, and angle.
When should I use the linear option?
Use it for scaling, rotation, and translation. The coefficient A controls scaling and rotation. The coefficient B shifts the final point.
What is a Mobius function?
It is a fractional linear map. It has the form (A z + B)/(C z + D). It is useful for advanced complex plane transformations.
Why can some inputs fail?
Some functions have domain limits. Reciprocal and logarithm cannot use z = 0. Mobius maps fail when the denominator becomes zero.
What does magnitude mean?
Magnitude is the distance from the origin to the complex output. It is calculated with the square root of real squared plus imaginary squared.
What does argument mean?
Argument is the angle of the output point from the positive real axis. It can be displayed in degrees or radians.
Why use decimal precision?
Precision controls how many decimal places appear. Higher precision helps with small values, engineering checks, and repeated comparisons.
Can I save the result?
Yes. You can download a CSV result. You can also print the page or save it as a PDF from your browser.
Is this useful for learning transformations?
Yes. Try different z values and functions. Compare the output magnitude and angle. This shows how each function changes the complex plane.