Sinhc函數

數學上的Sinhc函數定義如下^[1][2][3]

${\displaystyle \operatorname {Sinhc} (z)={\frac {\sinh(z)}{z}}}$，它是下列微分方程的一個解：

${\displaystyle w(z)z-2\,{\frac {d}{dz}}w(z)-z{\frac {d^{2}}{dz^{2}}}w(z)=0}$

Sinhc 2D plot

Sinhc'(z) 2D plot

Sinhc integral 2D plot

複域虛部

• ${\displaystyle \operatorname {Im} \left({\frac {\sinh(x+iy)}{x+iy}}\right)}$

複域實部

• ${\displaystyle \operatorname {Re} \left({\frac {\sinh \left(x+iy\right)}{x+iy}}\right)}$

絕對值

• ${\displaystyle \left|{\frac {\sinh(x+iy)}{x+iy}}\right|}$

一階導數

• ${\displaystyle {\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}}$

導數實部

• ${\displaystyle -\operatorname {Re} \left(-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right)}$

導數虛部

• ${\displaystyle -\operatorname {Im} \left(-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right)}$

絕對值

• ${\displaystyle \left|-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right|}$

表示為其他特殊函數

• ${\displaystyle \operatorname {Sinhc} (z)={\frac {{\rm {KummerM}}\left(1,\,2,\,2\,z\right)}{{\rm {e}}^{z}}}}$

• ${\displaystyle \operatorname {Sinhc} (z)={\frac {{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{{\rm {e}}^{z}}}}$

• ${\displaystyle \operatorname {Sinhc} (z)=1/2\,{\frac {{\rm {WhittakerM}}\left(0,\,1/2,\,2\,z\right)}{z}}}$

級數展開

${\displaystyle \operatorname {Sinhc} z\approx (1+{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}+{\frac {17}{315}}{z}^{6}+{\frac {62}{2835}}{z}^{8}+{\frac {1382}{155925}}{z}^{10}+{\frac {21844}{6081075}}{z}^{12}+{\frac {929569}{638512875}}{z}^{14}+O\left({z}^{16}\right))}$

圖集

Sinhc abs complex 3D

Sinhc Im complex 3D plot

Sinhc Re complex 3D plot

Sinhc'(z) Im complex 3D plot

Sinhc'(z) Re complex 3D plot

Sinhc'(z) abs complex 3D plot

Sinhc abs plot

Sinhc Im plot

Sinhc Re plot

Sinhc'(z) Im plot

Sinhc'(z) abs plot

Sinhc'(z) Re plot

參見

• Tanhc函數
• Coshc函數
• Tanc函數
• 雙曲正弦積分函數