Legendre Polynomial Calculator
Free online legendre 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：勒让德多项式在线计算器
勒让德多项式 Pn 是球面几何的通用语：单位球上的拉普拉斯方程分离变量后得到它，引力势、静电多极展开、球谐函数全都建立在它上面。递推 (n+1)Pn+1 = (2n+1)x·Pn − n·Pn−1 从 P0 = 1、P1 = x 出发。
它在 [−1, 1] 上带单位权重正交：∫ Pm·Pn dx = 2δ(m,n)/(2n+1)。高斯-勒让德求积的节点正是 Pn 的根——这正是本站多个积分工具内部使用的 GL16 求积的数学来源。
参考值
P5(0.5) = 0.08984375，P6(0.5) = 0.3232421875
P4(−1) = 1（Pn(±1) = (±1)ⁿ）
正交性：∫ P3·P4 dx = 0，∫ P5² dx = 2/11 = 0.18181818181818182
How to use the Legendre 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 Legendre 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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