Solve residue theorem problems using poles, orders, and coefficients. Inspect values before exporting detailed results. Plot residue contributions and verify contour integral totals easily.
Accepted complex input forms: 3+2i, 4-5i, -2i, i, and 7.
| Method | Main Inputs | Residue Result | Contour Integral Result |
|---|---|---|---|
| Simple pole | g(z0)=2+3i, h′(z0)=4-i | 0.294118 + 0.823529i | -5.174388 + 1.847996i |
| Higher-order pole | m=2, derivative value=6-2i | 6 - 2i | 12.566371 + 37.699112i |
| Laurent coefficient | a(-1)=-1+0.5i | -1 + 0.5i | -3.141593 - 6.283185i |
| Contour sum | z1=1+2i, z2=-0.5+0.75i, z3 outside | 0.5 + 2.75i | -17.27876 + 3.141593i |
When f(z)=g(z)/h(z) and h(z0)=0 with h′(z0) not zero:
Res(f,z0) = g(z0) / h′(z0)
For a pole of order m at z0:
Res(f,z0) = (1 / (m-1)!) × dm-1/dzm-1[(z-z0)mf(z)] evaluated at z0
If the Laurent series is Σ an(z-z0)n, then:
Res(f,z0) = a-1
For a positively oriented simple closed contour:
∮ f(z)dz = 2πi × Σ residues inside the contour
Clockwise orientation changes the sign. Repeated encirclement scales the result by the winding number.
This calculator helps estimate contour integrals from residue inputs without needing a symbolic engine inside the page. It supports simple poles, higher-order poles, Laurent coefficients, and direct residue summation. That makes it practical for students, instructors, analysts, and anyone checking complex integration work quickly.
The result panel summarizes the computed residue set, totals the interior contributions, and converts them into the contour integral. It also reports magnitude-based measures such as mean magnitude, largest magnitude, and summed magnitudes. Those extra summaries are useful when you want a compact review of how strongly each pole contributes.
The contour sum mode is especially helpful for worked examples, lecture notes, and validation tasks. You can enter several poles at once, mark whether each lies inside the contour, and immediately see how the theorem changes with orientation or winding number. The Plotly graph then provides a quick visual comparison of residue magnitudes across poles.
Because this page accepts complex values directly, it works well after you have already simplified algebra elsewhere. For instance, you may compute g(z0), h′(z0), or the transformed derivative term by hand and then use this tool to finish the numerical evaluation cleanly. The export buttons also make it easier to archive results for assignments, reports, or revision sheets.
It computes residues, sums interior residues, and returns the contour integral from the residue theorem. It also reports magnitude-based summary values and a graph.
No. For higher-order poles, you supply the already evaluated derivative expression at the pole. The page then applies the factorial scaling and theorem steps.
Use forms such as 3+2i, 4-5i, -2i, i, or 7. Spaces are removed automatically before calculation.
Its residue is listed, but it is not included in the interior sum. Therefore it does not affect the final contour integral.
The residue theorem assumes positive, counterclockwise orientation. Reversing the direction changes the sign of the contour integral.
It multiplies the residue-theorem total. A contour encircling the same poles twice gives twice the integral, assuming the same orientation.
It shows the magnitude of each residue entry. That makes it easier to compare which poles contribute more strongly in absolute size.
Yes. CSV downloads the summary and residue tables. PDF captures the visible result area into a downloadable document.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.