Sinhc function

In mathematics, the Sinhc function is defined as^[1]

${\displaystyle \operatorname {Sinhc} (z)={\frac {\sinh(z)}{z}}}$

Sinhc 2D plot

Sinhc'(z) 2D plot

File:Sinhc integral.png

Sinhc integral 2D plot

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First-order derivative

• ${\displaystyle {\frac {1-\sinh(z))^{2}}{z}}-{\frac {\sinh(z)}{z^{2}}}}$

Real part of derivative

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

Imaginary part of derivative

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

absolute value of derivative

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

In terms of other special functions

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

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

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

Series expansion

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

Gallery

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

References

1. Weisstein, Eric W. "Sinhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SinhcFunction.html