Calculator inputs
Use the exact pendulum modulus and select the Jacobian family you want to inspect. The result appears above this form after submission.
Example data table
| Family | k | K | K′ | Principal zero | Principal pole |
|---|---|---|---|---|---|
| sn | 0.40 | 1.639999866 | 2.359263555 | 0 | 0 + 2.359263555i |
| cn | 0.60 | 1.750753803 | 1.995302778 | 1.750753803 + 0i | 0 + 1.995302778i |
| dn | 0.80 | 1.995302778 | 1.750753803 | 1.995302778 + 1.750753803i | 0 + 1.750753803i |
Formula used
Exact pendulum displacement:
θ(t) = 2 asin(k · sn(ωt + u₀, k))
Quarter periods:
K(k) = π / (2 AGM(1, √(1-k²)))
K′(k) = π / (2 AGM(1, k))
Zero and pole lattices:
sn(u,k): zeros2mK + 2niK′, poles2mK + (2n+1)iK′cn(u,k): zeros(2m+1)K + 2niK′, poles2mK + (2n+1)iK′dn(u,k): zeros(2m+1)K + (2n+1)iK′, poles2mK + (2n+1)iK′
Here, m and n are integers defining congruent lattice points.
How to use this calculator
- Choose the elliptic family. Use sn for direct pendulum displacement work.
- Enter a modulus k between 0 and 1.
- Set lattice index windows for m and n.
- Set the real-axis plot range and sample density.
- Optionally define ω and u₀ for time mapping.
- Press Find poles and zeros.
- Review the summary cards, coordinate tables, and Plotly graphs.
- Export the current table as CSV or PDF.
Frequently asked questions
1) What does the modulus k control?
It sets the pendulum amplitude and changes both quarter periods. Larger values move the system farther from small-angle harmonic behavior.
2) Why are K and K′ important?
They generate the repeating lattice. Real translations depend on K, while imaginary translations depend on K′.
3) Which family matches the exact pendulum angle?
The exact angle formula uses sn inside the inverse sine. The cn and dn families remain useful companion functions for derivatives and identities.
4) Do these poles occur in ordinary physical time?
No. They are complex-plane singularities of the analytic continuation. Real pendulum motion remains finite for 0 < k < 1.
5) Why might dn show no real zeros?
For real arguments with 0 < k < 1, dn stays positive. Its zeros appear only after moving into the complex lattice.
6) What does the phase shift u₀ change?
It shifts the real-axis zero locations and the time mapping. The lattice formulas themselves remain unchanged.
7) Why is the graph numerical?
The page uses a stable real-axis ODE integration for plotting. That keeps the single-file calculator lightweight and portable.
8) What gets exported to CSV and PDF?
Both downloads include the current family, modulus, quarter periods, and every listed pole and zero from the visible result.