Chi函数

Chi 函数定义如下^[1]^[2]

$𝐶 ℎ 𝑖 (z) = \int_{0}^{z} \frac{\cosh (t)}{t} d t$

$C h i (z)$ 是下列三阶非线性常微分方程的一个解:

$z \frac{d}{d z} w (z) - 2 \frac{d^{2}}{d z^{2}} w (z) - z \frac{d^{3}}{d z^{3}} w (z) = 0$

即：

$w (z) =_𝐶 1 +_𝐶 2 𝐶 ℎ 𝑖 (z) +_𝐶 3 𝑆 ℎ 𝑖 (z)$

对称性

$C h i (- z) = C h i (z)$

$\frac{- 1}{2} \sqrt{π} G_{1, 3}^{2, 0} (- 1 / 4 z^{2} |_{0, 0, 1 / 2}^{1}) - 1 / 2 i π$

$𝐶 ℎ 𝑖 (z) = (γ + \ln (z) + \frac{1}{4} z^{2} + \frac{1}{96} z^{4} + \frac{1}{4320} z^{6} + \frac{1}{322560} z^{8} + \frac{1}{36288000} z^{10} + \frac{1}{5748019200} z^{12} + \frac{1}{1220496076800} z^{14} + O (z^{16}))$

↑ Abramowitz, M. and Stegun, I. A. (Eds.). "Sine and Cosine Integrals." §5.2 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231-233, 1972.
↑ Sloane, N. J. A. Sequence A061079 in "The On-Line Encyclopedia of Integer Sequences