Laguerre Polynomial Calculator
Free online laguerre polynomial calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：拉盖尔多项式在线计算器
拉盖尔多项式 Ln 在半条轴 [0, ∞) 上带着权重 e^(−x) 正交：∫ Ln·Lm·e^(−x) dx = δ(n,m)（标准归一化下范数恰为 1）。递推 (n+1)Ln+1 = (2n+1−x)Ln − n·Ln−1 从 L0 = 1、L1 = 1−x 出发。
它是氢原子的骨架：径向波函数由关联拉盖尔多项式构造，主量子数 n 决定多项式阶数。数值积分里伽辽金法用 Ln 的根做节点，处理指数衰减型被积函数时比高斯-勒让德更高效。
参考值
L4(1) = −0.625，L5(1) = −7/15 = −0.4666666666666667
正交性：∫ L2·L3·e^(−x) dx = 0，∫ L2²·e^(−x) dx = 1（0 到 60 截断误差 1e−13 级）
L3(2) = 1 − 6 + 6 − 8/6 = −1/3
How to use the Laguerre Polynomial Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Laguerre Polynomial Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-28
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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