Calculator Inputs
Example data table
| Case | Integrand | Contour | Expected result | Reason |
|---|---|---|---|---|
| 1 | 1 / z | Circle centered at 0, radius 2 | 2πi | The contour winds once around the pole. |
| 2 | e^z | Any closed ellipse | 0 | The function is entire on and inside the contour. |
| 3 | 1 / (z - 1) | Rectangle enclosing z = 1 | 2πi | Cauchy’s integral theorem gives one full residue contribution. |
| 4 | z^3 | Closed circle, any radius | 0 | Polynomials are entire, so closed contour integrals vanish. |
Formula used
The calculator uses the parameterized contour integral formula
∫C f(z) dz = ∫αβ f(z(t)) z′(t) dt.
For circles and ellipses, the path is parameterized directly. For rectangles, the contour is split into four line segments. The numerical value is estimated with a composite trapezoidal rule, which improves as the segment count increases.
When the selected integrand has a simple pole, the page also compares the numerical answer with 2πi × winding number. Entire functions such as polynomials, sine, cosine, and exponential should return zero on closed contours.
How to use this calculator
- Choose an integrand preset such as 1 / z, 1 / (z - a), or e^z.
- Select a contour shape and enter the needed geometric values.
- Set the traversal direction and choose a segment count for numerical accuracy.
- Use 0 to 360 degrees for one counterclockwise closed revolution.
- Press Compute Contour Integral to show the result above the form.
- Review the sampled data, analytic check, and error estimate.
- Use the export buttons to save the current result as CSV or PDF.
FAQs
1. What does this calculator estimate?
It estimates complex contour integrals by parameterizing a path in the complex plane and numerically integrating f(z)z′(t). It also reports magnitude, argument, path length, and pole-based checks when available.
2. Why do entire functions often return zero?
If the contour is closed and the function is analytic everywhere on and inside it, Cauchy-Goursat implies the integral is zero. Numerical results may show tiny residual values because the page uses finite discretization.
3. When should I use more segments?
Increase the segment count when the path is large, the pole is close to the contour, or the function varies rapidly. More segments usually reduce discretization error, though computation will take slightly longer.
4. What does the winding number mean here?
The winding number measures how many times the closed contour wraps around a selected pole. For 1/(z-a), the expected integral is 2πi multiplied by that integer, provided the pole is not on the path.
5. Can I use open paths?
Yes. Choose the line segment contour for an open path. The calculator still computes the integral numerically, but closed-contour checks like residue and Cauchy-Goursat comparisons will usually not apply.
6. Why is the result unstable near a pole?
If the contour passes through or extremely close to a singularity, the integral may be undefined or highly sensitive. The page warns when the estimated pole distance is very small.
7. Does direction change the answer?
Yes. Reversing traversal changes dz to -dz, so the integral changes sign. A clockwise loop around a simple pole gives the negative of the corresponding counterclockwise result.
8. What are the exports used for?
CSV export is helpful for spreadsheets, lab notes, and reproducibility. PDF export creates a compact report containing the current configuration, computed integral, and sampled points for sharing or printing.