Calculator Inputs
Example Data Table
| Function | Center a | Point z | Order N | Use case |
|---|---|---|---|---|
| e^z | 0 + 0i | 0.5 + 0.3i | 8 | Classic Maclaurin approximation in the complex plane |
| sin(z) | 0.2 + 0.1i | 0.6 + 0.4i | 10 | Oscillatory analytic expansion around a nonzero center |
| 1 / (1 - z) | 0 + 0i | 0.4 + 0.2i | 12 | Geometric-series behavior with convergence radius checking |
| log(1 + z) | 0 + 0i | 0.3 + 0.2i | 9 | Principal branch expansion away from the branch point |
Formula Used
For an analytic function, the Taylor polynomial about a complex center a is TN(z) = Σ[n=0..N] f(n)(a)(z-a)n / n!. Each coefficient is cn = f(n)(a)/n!.
This calculator evaluates derivatives at the chosen center, builds every term, forms each partial sum, compares the approximation with the true function value, and reports convergence information.
Choose a supported analytic function, center a, point z, and order N to generate coefficients, terms, and approximation error.
How to Use This Calculator
- Select an analytic function from the dropdown list.
- Enter the real and imaginary parts of the expansion center a.
- Enter the real and imaginary parts of the evaluation point z.
- Choose the truncation order N between 0 and 30.
- Click Compute Taylor Series to show the result above the form.
- Review the coefficient table, partial sums, error values, and complex-plane plots.
- Use the CSV button for term export or the PDF button for a print-ready summary.
Frequently Asked Questions
1) What does this calculator compute?
It builds a complex Taylor polynomial around a selected center, evaluates partial sums at a target point, reports coefficients, and measures approximation error.
2) Why is the center a important?
The center determines the derivatives used in the coefficients and changes the convergence region. A better center often improves local accuracy near the target point.
3) What is the radius of convergence?
It is the distance from the center to the nearest singularity or branch obstruction. Inside that radius, the Taylor series converges to the analytic function.
4) Why can error remain large?
Error may remain large when the point lies near or beyond the convergence boundary, or when the chosen order is too low for the requested accuracy.
5) Which functions are supported here?
This version supports e^z, sin(z), cos(z), sinh(z), cosh(z), 1/(1-z), and log(1+z) on the principal branch where applicable.
6) What does the complex-plane plot show?
It tracks how successive partial sums move in the complex plane and compares their endpoint with the exact function value at the chosen point.
7) Can this replace symbolic algebra software?
It is designed for fast applied analysis and teaching. It supports selected analytic functions rather than arbitrary symbolic expressions with automatic differentiation.
8) When should I increase the order N?
Increase N when the approximation error is still too high, especially if the evaluation point is farther from the center but still inside the convergence region.