Bickley Function Calculator
Free online bickley function 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.
Bickley-Naylor 函数 Kiₙ(x) = ∫₀^∞ e^(−x cosh t)·sechⁿt dt 由 W. G. Bickley 在 1930 年代为反应堆屏蔽的中子输运引入，是第三类修正 Bessel 函数 Ki₀ = K₀（第二类修正 Bessel）的逐层积分族：Ki₁(x) = ∫ₓ^∞ K₀(t)dt，Ki₂ 是 Ki₁ 的再积分。恒星大气辐射传输（H 我函数的精确角积分）与平板屏蔽计算都靠它。
积分实现两条路径：Simpson 16000 段直接积到 t = 28（e^(−28·1) ≈ 7e−13 后尾部可忽略）与 8 段 16 点 Gauss-Legendre 复合，一致到 1e-9。三个精确锚点 Ki₁(0) = π/2、Ki₂(0) = 1、Ki₃(0) = π/4 恰好是 sech 与 sech² 的经典积分值，为 x = 0 提供硬校验。
参考值
Ki₁(1) = 0.3282864781711172，Ki₂(1) = 0.2736207520261150，Ki₃(1) = 0.2378450821928554
Ki₁(0) = π/2 = 1.5707963267948966，Ki₂(0) = 1，Ki₃(0) = π/4 = 0.7853981633974483
Kiₙ 逐层关系：Kiₙ′(x) = −Kiₙ₋₁(x)（n 大于 1）
How to use the Bickley Function 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 Bickley Function 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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