Sinhc function

In mathematics, the Sinhc function appears frequently in papers about optical scattering,[1] Heisenberg Spacetime[2] and hyperbolic geometry.[3] It is defined as[4][5]

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

It is a solution of the following differential equation:

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

Sinhc 2D plot

Sinhc'(z) 2D plot

Sinhc integral 2D plot

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First-order derivative

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

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

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