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$(z)=H_0(z)\!=$HankelH1$[0,z]$ 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, $u+\mathrm i v=H_0(x+\mathrm i y)$.

Asymptotic expansions

TeXForm[Expand[Series[(HankelH1[0, x]) (Pi I x/2)^(1/2), {x, Infinity, 5}]]]

does

$e^{i x} \left(1-\frac{i}{8 x}-\frac{9}{128 x^2}+\frac{75

  i}{1024 x^3}+\frac{3675}{32768 x^4}-\frac{59535
  i}{262144
  x^5}+O\left(\left(\frac{1}{x}\right)^6\right)\right)$

Square of BesselH0

Some expansions of square of BesselH0 look counter-intuitive and require analysis. One example ix copy pasted below.

TeXForm[Expand[Series[(HankelH1[0, x])^2 Pi I x/2, {x, Infinity, 5}]]]

does

$e^{2 i x} \left(1-\frac{i}{4 x}-\frac{5}{32 x^2}+\frac{21

  i}{128 x^3}+\frac{507}{2048 x^4}-\frac{4035 i}{8192
  x^5}+O\left(\left(\frac{1}{x}\right)^6\right)\right)$

and

TeXForm[Expand[Series[(HankelH1[0, x^(1/2)])^2 Pi I x^(1/2)/2, {x, Infinity, 2}]]]

does

$e^{2 i \sqrt{x}} \left(1-\frac{1}{4} i

  \sqrt{\frac{1}{x}}-\frac{5}{32 x}+\frac{21}{128} i
  \left(\frac{1}{x}\right)^{3/2}+\frac{381}{16384
  x^2}-\frac{675 i
  \left(\frac{1}{x}\right)^{5/2}}{65536}+O\left(\left(\frac{1}{x}\right)^3\right)\right)

$

At least one of these two asymptotic expansions seems to be wrong. Perhaps, the last one, because

TeXForm[Expand[Series[(HankelH1[0, x^(1/2)])^2 Pi I x^(1/2)/2, {x, Infinity, 3}]]]

does

$e^{2 i \sqrt{x}} \left(1-\frac{1}{4} i

  \sqrt{\frac{1}{x}}-\frac{5}{32 x}+\frac{21}{128} i
  \left(\frac{1}{x}\right)^{3/2}+\frac{507}{2048
  x^2}-\frac{5025 i
  \left(\frac{1}{x}\right)^{5/2}}{131072}-\frac{44325}{2
  097152 x^3}+\frac{275625 i
  \left(\frac{1}{x}\right)^{7/2}}{16777216}+O\left(\left
  (\frac{1}{x}\right)^4\right)\right)$

Correcting the two terms (that showes deviation in the previous example); however, two new wrong terms are added. The control on the precision seems to be lost, because

TeXForm[Expand[Series[(HankelH1[0, x^(1/2)])^2 Pi I x^(1/2)/2, {x, Infinity, 4}]]]

does

$e^{2 i \sqrt{x}} \left(1-\frac{1}{4} i

  \sqrt{\frac{1}{x}}-\frac{5}{32 x}+\frac{21}{128} i
  \left(\frac{1}{x}\right)^{3/2}+\frac{507}{2048
  x^2}-\frac{4035 i
  \left(\frac{1}{x}\right)^{5/2}}{8192}-\frac{163395}{20
  97152 x^3}+\frac{50715 i
  \left(\frac{1}{x}\right)^{7/2}}{1048576}+\frac{4922662
  5}{1073741824 x^4}-\frac{218791125 i
  \left(\frac{1}{x}\right)^{9/2}}{4294967296}+O\left(\left(\frac{1}{x}\right)^5\right)\right)$

This phenomenon can be interpreted as bug in Mathematica Series routine.

References

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

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