Calculator Input
Formula Used
For a complex function f(z) = u(x, y) + iv(x, y), the Cauchy Riemann equations are:
ux = vy
and
uy = -vx.
This calculator estimates partial derivatives with the central difference method:
ux ≈ [u(x+h,y)-u(x-h,y)] / 2h
and
uy ≈ [u(x,y+h)-u(x,y-h)] / 2h.
The same method is applied to v(x, y). The calculator then compares both equations with your tolerance.
How to Use This Calculator
- Enter the real component
u(x, y). - Enter the imaginary component
v(x, y). - Choose the point where the equations should be tested.
- Set a derivative step size and tolerance.
- Press calculate to view partial derivatives and equation errors.
- Use the graph to inspect local behavior near the point.
- Download the result as CSV or PDF for records.
Example Data Table
| u(x, y) | v(x, y) | Point | Expected Result |
|---|---|---|---|
x^2 - y^2 |
2*x*y |
(1, 1) |
Passes. This represents z^2. |
exp(x)*cos(y) |
exp(x)*sin(y) |
(0, 0) |
Passes. This represents e^z. |
x^2 + y^2 |
x-y |
(1, 2) |
Usually fails. |
Understanding the Cauchy Riemann Equations
Why These Equations Matter
Complex functions behave differently from ordinary two variable functions. A function may look smooth in real variables. Yet it may fail to be complex differentiable. The Cauchy Riemann equations give a fast test for that behavior. They compare how the real and imaginary parts change near a point. When both equations hold, the function has a strong analytic signal. Extra continuity conditions are still important in formal proofs.
What This Tool Measures
This calculator accepts u(x, y) and v(x, y). It treats them as parts of f(z). Then it estimates four partial derivatives. These are ux, uy, vx, and vy. The calculator compares ux with vy. It also compares uy with negative vx. The difference becomes an error score. A small score means the equations nearly hold.
Using Numerical Results Carefully
Numerical differentiation is practical. It is also sensitive. Very large step sizes can blur local changes. Very tiny step sizes can create rounding noise. A step near 0.0001 often works well. Some functions need adjustment. Try several step sizes when results seem unclear. A pass here is an estimate. It is not a complete symbolic proof.
Graph and Interpretation
The graph maps the local Cauchy Riemann error. Darker or lower regions suggest better agreement. Large peaks suggest failure or unstable derivatives. Use the graph with the table of derivative values. Together they show local structure. For analytic functions, the error often stays low nearby. For nonanalytic functions, the graph changes sharply. This helps students check examples. It also helps teachers demonstrate complex differentiability.
Best Practice
Enter expressions with explicit multiplication. Write 2*x*y instead of 2xy. Use radians for trigonometric functions. Keep the selected point inside the expression domain. Avoid log of negative values. Avoid division by zero. Review both equations before drawing conclusions. Use exports when you need a clean record.
FAQs
What are the Cauchy Riemann equations?
They are two equations that link the partial derivatives of real and imaginary parts of a complex function. They are ux = vy and uy = -vx.
Does passing this test prove analyticity?
Passing numerically suggests the equations hold at the selected point. A full proof may also require derivative continuity in a neighborhood.
Why does the calculator use a step size?
The step size controls numerical differentiation. It compares nearby values to estimate slopes. Good choices reduce rounding and truncation errors.
Which expressions can I enter?
You can use x, y, numbers, pi, e, powers, arithmetic operators, and common functions such as sin, cos, exp, log, sqrt, and abs.
Why should I use explicit multiplication?
The parser reads clear operators. Use 2*x*y instead of 2xy. This avoids ambiguity and helps the expression evaluate correctly.
What does the graph show?
The graph shows local equation error around the selected point. Smaller values mean the Cauchy Riemann equations agree more closely nearby.
What is the estimated complex derivative?
When the equations hold, f'(z) can be estimated as ux + i vx. The calculator also reports magnitude and angle.
Can this handle every complex function?
It handles many explicit real component formulas. It does not replace symbolic algebra or rigorous proof for singularities, branch cuts, or discontinuities.