Laurent Series Online Calculator

Calculator Inputs

Function model

Variable

Center a

Amplitude A

Parameter k

Shift c

Pole order m

Number of terms

Decimal precision

Shifted rational region

Custom terms

Example Data Table

Model	Center	Parameters	Region	Expected note
A exp(k/(z-a))	0	A=1, k=1	0 < \|z\| < infinity	Essential singularity with residue 1
A/((z-a)-c)	1	A=2, c=3	\|z-1\| < 3	Regular geometric style expansion
A/(z-a)^m	-2	A=4, m=2	0 < \|z+2\| < infinity	Pole of order 2

Formula Used

The calculator uses t = z - a, where a is the chosen center. It then applies standard Laurent templates for each selected model.

A exp(k/t) = A Σ k^n t^-n / n!
A sin(k/t) = A Σ (-1)^j k^(2j+1) t^-(2j+1) / (2j+1)!
A cos(k/t) = A Σ (-1)^j k^(2j) t^-2j / (2j)!
A/(t-c) = -(A/c) Σ (t/c)^n for |t| < |c|
A/(t-c) = A Σ c^n t^-(n+1) for |t| > |c|

The residue is the coefficient of t^-1. Negative powers form the principal part. Nonnegative powers form the regular part.

How To Use This Calculator

Select a function model that matches your problem.
Enter the center a for expansion around z = a.
Set A, k, c, or m where the model needs them.
Choose the number of terms and decimal precision.
For shifted rational functions, select inside or outside region.
Press the calculate button and review the result above the form.
Download the coefficient table as CSV or save the result as PDF.

Laurent Series Guide

A Laurent series rewrites a complex function around a chosen center. It allows negative powers. That makes it useful near isolated singularities. A Taylor series only uses zero and positive powers. A Laurent series can show both the principal part and the analytic part.

Why This Calculator Helps

This calculator is built for study, checking, and quick explanation. It handles common model functions used in complex analysis. You can expand inverse exponential, inverse trigonometric, shifted rational, geometric, logarithmic, pole, and custom coefficient forms. The output separates negative powers from regular terms. It also estimates residue and pole behavior where the selected model supports it.

Understanding The Center

The center is the point around which the expression is expanded. The calculator uses t equal to z minus the center. This keeps every term clear. When the center changes, the same function may have a different series shape. Always choose the center that matches your problem statement.

Principal And Regular Parts

Negative powers form the principal part. They reveal removable singularities, poles, or essential singularities. The coefficient of t to the power negative one is the residue. Positive powers and the constant term form the regular part. These terms behave like a Taylor expansion around the same center.

Convergence And Annulus

Laurent expansions are valid in annular regions. An annulus may be inside a nearest singularity, outside a shifted singularity, or punctured around the center. This tool reports the standard region for the chosen model. Treat the note as a guide. For complicated functions, confirm singularities separately.

Good Study Workflow

Start with a simple term count. Review the leading negative powers. Check whether the residue is zero. Then raise the term count for more detail. Use the CSV export for tables. Use the PDF export for notes, assignments, and comparison work.

Practical Limits

The calculator uses model based formulas. It does not replace a full symbolic algebra system. Custom mode lets you enter known coefficients directly. That is useful after hand derivation. For rigorous proofs, write the annulus, assumptions, and coefficient formula beside the computed result.

Result Review

Compare displayed coefficients with class formulas. Small differences often come from term limits, rounded decimals, or selected inner and outer regions.

FAQs

What is a Laurent series?

A Laurent series is a power series that may include negative powers. It is commonly used to study complex functions near singularities.

What is the principal part?

The principal part is the sum of all negative power terms. It shows the singular behavior near the selected expansion center.

How is the residue found?

The residue is the coefficient of t^-1, where t equals z minus the chosen center. The result table highlights that coefficient.

Can this solve every function?

No. This calculator uses common model formulas. Use custom mode when you already have coefficients from manual work or another symbolic method.

What does the annulus mean?

The annulus is the region where the displayed expansion is valid. It depends on the center and the nearest relevant singularities.

Why choose inside or outside region?

Shifted rational functions have different Laurent expansions in different regions. The inside region gives regular powers. The outside region gives negative powers.

What should I enter in custom mode?

Enter pairs as power:coefficient. Example entries are -2:3, -1:5, 0:1, and 1:-2. Separate pairs with commas or new lines.

Why are results rounded?

The decimal precision field controls displayed rounding. Increase precision for more digits, but remember that formulas remain model based.