BesselH0

Real and imaginary parts of $H_0(x)$ for $x>0$ compared to BesselJ1 (green)

$u+\mathrm i v=H_0(x+\mathrm i y)$

BesselH0 $=H_0$ is the Cylindric function H (called also the Hankel function) of zero order.

BesselH0 is related with $J_0=$BesselJ0 and $J_0=$BesselY0 with simple relation

$H_0(z)=J_0(z)+\mathrm i Y_0(z)$

In particular, for $x>0$, the relations $\Im(J_0(x))=0$ and $\Im(Y_0(x))=0$ hold, and, therefore,

$\Re(H_0(x))=J_0(x)$

$\Im(H_0(x))=Y_0(x)$

The explicit plot of real and imaginary parts of BesselH0 versus real positive argument are shown in the upper right corner.

Below, the complex map of $H_0$ is plotted, $u+\mathrm i v=H_0(x+\mathrm i y)$.

References

http://mathworld.wolfram.com/BesselFunction.html

http://en.wikipedia.org/wiki/Bessel_function