Tanhc函数

Tanhc函数定义如下^[1]

$t a n h c (z) = \frac{\tanh (z)}{z}$

复域虚部

$𝐼 𝑚 (\frac{\tanh (x + i y)}{x + i y})$

复域实部

$𝑅 𝑒 (\frac{\tanh (x + i y)}{x + i y})$

复域绝对值

$| \frac{\tanh (x + i y)}{x + i y} |$

一阶微商

$\frac{1 - {(\tanh (z))}^{2}}{z} - \frac{\tanh (z)}{z^{2}}$

微商实部

$- 𝑅 𝑒 (- \frac{1 - {(\tanh (x + i y))}^{2}}{x + i y} + \frac{\tanh (x + i y)}{{(x + i y)}^{2}})$

微商虚部

$- 𝐼 𝑚 (- \frac{1 - {(\tanh (x + i y))}^{2}}{x + i y} + \frac{\tanh (x + i y)}{{(x + i y)}^{2}})$

微商绝对值

$| - \frac{1 - {(\tanh (x + i y))}^{2}}{x + i y} + \frac{\tanh (x + i y)}{{(x + i y)}^{2}} |$

积分函数

$\int_{0}^{z} \frac{\tanh (x)}{x} d x$

用其他特殊函数表示

$t a n h c (z) = 2 \frac{K u m m e r M (1, 2, 2 z)}{(2 i z + π) K u m m e r M (1, 2, i π - 2 z) e^{2 z - 1 / 2 i π}}$
$t a n h c (z) = 2 \frac{𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, \sqrt{2} \sqrt{z})}{(2 i z + π) 𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, \sqrt{2} \sqrt{1 / 2 i π - z}) e^{2 z - 1 / 2 i π}}$
$t a n h c (z) = \frac{i W h i t t a k e r M (0, 1 / 2, 2 z)}{W h i t t a k e r M (0, 1 / 2, i π - 2 z) z}$
$t a n h c (z) = \frac{i (e^{2 z} - 1)}{(e^{i π - 2 z} - 1) e^{2 z - 1 / 2 i π} z}$

级数展开

$t a n h c \approx (1 - \frac{1}{3} z^{2} + \frac{2}{15} z^{4} - \frac{17}{315} z^{6} + \frac{62}{2835} z^{8} - \frac{1382}{155925} z^{10} + \frac{21844}{6081075} z^{12} - \frac{929569}{638512875} z^{14} + O (z^{16}))$

$\int_{0}^{z} \frac{\tanh (x)}{x} d x = (z - \frac{1}{9} z^{3} + \frac{2}{75} z^{5} - \frac{17}{2205} z^{7} + \frac{62}{25515} z^{9} - \frac{1382}{1715175} z^{11} + O (z^{13}))$