Advanced Contour Integral Calculator

Calculator Inputs

Contour type

Orientation

Samples

Center Re(c)

Center Im(c)

Circle radius

Ellipse radius x

Ellipse radius y

Rectangle width

Rectangle height

Start Re(z₀)

Start Im(z₀)

End Re(z₁)

End Im(z₁)

Integrand: f(z) = A·zⁿ + B/(z−P) + C·e^(kz) + D·sin(mz) + E

A real

A imaginary

Power n

B real

B imaginary

Pole Re(P)

Pole Im(P)

C real

C imaginary

Exponential scale k

D real

D imaginary

Sine scale m

E real

E imaginary

How to Use This Calculator

Choose a contour type: circle, ellipse, rectangle, or line segment.
Enter contour geometry values. Closed contours also use orientation.
Define the integrand coefficients A, B, C, D, and E.
Set the pole location P for the rational term B/(z−P).
Pick a sample count. Higher values improve numerical accuracy.
Submit the form. The numerical integral appears above the form.
Review the residue-based exact check for supported closed contours.
Use the CSV and PDF buttons to export your summary and sampled data.

Example Data Table

This example uses a circle centered at 1 + 0i with radius 2 and integrand f(z) = 1/(z−1).

Contour	Center	Radius	B	Pole	Expected Exact Integral
Circle, CCW	1 + 0i	2	1 + 0i	1 + 0i	0 + 6.28318531i

Formula Used

Integrand family: f(z) = A·zⁿ + B/(z − P) + C·e^kz + D·sin(mz) + E

Numerical contour integral: the calculator samples the contour, builds midpoint values, and sums f(z_mid)·Δz across all segments.

Closed contour residue check: the polynomial, exponential, sine, and constant terms are entire. Their closed integrals vanish. The rational term contributes 2πi·B times the winding number around the pole.

Error estimate: for closed contours, the tool compares the numerical result with the residue-based exact value and reports the absolute error.

Notes

Use more samples near tight curves or near poles.
Closed contours support a residue-based exact comparison.
Open line segments compute only the numerical line integral.
If the path approaches the pole, stability may worsen.
Plotly shows the contour in the complex plane.

FAQs

1) What does this calculator compute?

It numerically evaluates contour integrals over parameterized paths in the complex plane. It also reports geometric metrics and an exact residue check for supported closed contours.

2) Which contour shapes are supported?

You can use circles, ellipses, rectangles, and open line segments. Closed shapes can be traversed clockwise or counterclockwise to change the winding sign.

3) Which integrands are supported?

The calculator supports A·zⁿ, B/(z−P), C·e^(kz), D·sin(mz), and a constant term E. Complex coefficients are allowed for A, B, C, D, and E.

4) Why is there an exact result for some cases?

For closed contours, the entire terms contribute zero by complex analysis. Only the simple pole term contributes, so the exact value becomes 2πi times its residue and winding number.

5) Why can the numerical answer differ slightly?

The path is approximated by finitely many segments. Larger sample counts usually reduce error. Very sharp bends or paths near poles may need substantially finer sampling.

6) What does the minimum distance to the pole mean?

It is the smallest sampled distance between the contour and the pole location. Small values warn that the rational term may vary rapidly and reduce numerical stability.

7) What is exported in the CSV file?

The CSV contains summary metrics first, then sampled midpoint data for each stored step, including z midpoint values, integrand values, segment increments, and contribution terms.

8) Can I use this for study and verification?

Yes. It works well for checking residue ideas, winding effects, and numerical convergence. Increase samples and compare the exact closed-contour check whenever possible.