Editing Bessel function (section)

== Asymptotic forms ==
The Bessel functions have the following [[asymptotic analysis|asymptotic]] forms. For small arguments <math>0<z\ll\sqrt{\alpha+1}</math>, one obtains, when <math>\alpha</math> is not a negative integer:<ref name=p360/>
<math display="block">J_\alpha(z) \sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right)^\alpha.</math>

When {{mvar|α}} is a negative integer, we have
<math display="block">J_\alpha(z) \sim \frac{(-1)^{\alpha}}{(-\alpha)!} \left( \frac{2}{z} \right)^\alpha.</math>

For the Bessel function of the second kind we have three cases:
<math display="block">Y_\alpha(z) \sim \begin{cases}
\dfrac{2}{\pi} \left( \ln \left(\dfrac{z}{2} \right) + \gamma \right) & \text{if } \alpha = 0 \\[1ex]
-\dfrac{\Gamma(\alpha)}{\pi} \left( \dfrac{2}{z} \right)^\alpha + \dfrac{1}{\Gamma(\alpha+1)} \left(\dfrac{z}{2} \right)^\alpha \cot(\alpha \pi) & \text{if } \alpha \text{ is a positive integer (one term dominates unless } \alpha \text{ is imaginary)}, \\[1ex]
 -\dfrac{(-1)^\alpha\Gamma(-\alpha)}{\pi} \left( \dfrac{z}{2} \right)^\alpha & \text{if } \alpha\text{ is a negative integer,}
\end{cases}</math>
where {{mvar|γ}} is the [[Euler–Mascheroni constant]] (0.5772...).

For large real arguments {{math|''z'' ≫ {{abs|''α''<sup>2</sup> − {{sfrac|1|4}}}}}}, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless {{mvar|α}} is [[half-integer]]) because they have [[zero of a function|zeros]] all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of {{math|arg ''z''}} one can write an equation containing a term of order {{math|{{abs|''z''}}<sup>−1</sup>}}:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_364.htm p. 364, 9.2.1].</ref>
<math display="block">\begin{align}
J_\alpha(z) &= \sqrt{\frac{2}{\pi z}}\left(\cos \left(z-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + e^{\left|\operatorname{Im}(z)\right|}\mathcal{O}\left(|z|^{-1}\right)\right) && \text{for } \left|\arg z\right| < \pi, \\
Y_\alpha(z) &= \sqrt{\frac{2}{\pi z}}\left(\sin \left(z-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + e^{\left|\operatorname{Im}(z)\right|}\mathcal{O}\left(|z|^{-1}\right)\right) && \text{for } \left|\arg z\right| < \pi.
\end{align}</math>

(For {{math|1=''α'' = {{sfrac|1|2}}}}, the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

The asymptotic forms for the Hankel functions are:
<math display="block">\begin{align}
H_\alpha^{(1)}(z) &\sim \sqrt{\frac{2}{\pi z}}e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 2\pi, \\
H_\alpha^{(2)}(z) &\sim \sqrt{\frac{2}{\pi z}}e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -2\pi < \arg z < \pi.
\end{align}</math>

These can be extended to other values of {{math|arg ''z''}} using equations relating {{math|''H''{{su|b=''α''|p=(1)}}(''ze''<sup>''im''π</sup>)}} and {{math|''H''{{su|b=''α''|p=(2)}}(''ze''<sup>''im''π</sup>)}} to {{math|''H''{{su|b=''α''|p=(1)}}(''z'')}} and {{math|''H''{{su|b=''α''|p=(2)}}(''z'')}}.<ref>[[NIST]] [[Digital Library of Mathematical Functions]], Section [https://dlmf.nist.gov/10.11#E1 10.11].</ref>

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, {{math|''J<sub>α</sub>''(''z'')}} is not asymptotic to the average of these two asymptotic forms when {{mvar|z}} is negative (because one or the other will not be correct there, depending on the {{math|arg ''z''}} used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for ''complex'' (non-real) {{mvar|z}} so long as {{math|{{abs|''z''}}}} goes to infinity at a constant phase angle {{math|arg ''z''}} (using the square root having positive real part):
<math display="block">\begin{align}
J_\alpha(z) &\sim \frac{1}{\sqrt{2\pi z}} e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 0, \\[1ex]
J_\alpha(z) &\sim \frac{1}{\sqrt{2\pi z}} e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } 0 < \arg z < \pi, \\[1ex]
Y_\alpha(z) &\sim -i\frac{1}{\sqrt{2\pi z}} e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 0, \\[1ex]
Y_\alpha(z) &\sim i\frac{1}{\sqrt{2\pi z}} e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } 0 < \arg z < \pi.
\end{align}</math>

For the modified Bessel functions, [[Hermann Hankel|Hankel]] developed [[asymptotic expansion]]s as well:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_377.htm p. 377, 9.7.1].</ref><ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_378.htm p. 378, 9.7.2].</ref>
<math display="block">\begin{align}
I_\alpha(z) &\sim \frac{e^z}{\sqrt{2\pi z}} \left(1 - \frac{4 \alpha^2 - 1}{8z} + \frac{\left(4 \alpha^2 - 1\right) \left(4 \alpha^2 - 9\right)}{2! (8z)^2} - \frac{\left(4 \alpha^2 - 1\right) \left(4 \alpha^2 - 9\right) \left(4 \alpha^2 - 25\right)}{3! (8z)^3} + \cdots \right) &&\text{for }\left|\arg z\right|<\frac{\pi}{2}, \\
K_\alpha(z) &\sim \sqrt{\frac{\pi}{2z}} e^{-z} \left(1 + \frac{4 \alpha^2 - 1}{8z} + \frac{\left(4 \alpha^2 - 1\right) \left(4 \alpha^2 - 9\right)}{2! (8z)^2} + \frac{\left(4 \alpha^2 - 1\right) \left(4 \alpha^2 - 9\right) \left(4 \alpha^2 - 25\right)}{3! (8z)^3} + \cdots \right) &&\text{for }\left|\arg z\right|<\frac{3\pi}{2}.
\end{align}</math>

There is also the asymptotic form (for large real <math>z</math>)<ref>[https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-81/issue-4/The-Kosterlitz-Thouless-transition-in-two-dimensional-abelian-spin-systems/cmp/1103920388.full Fröhlich and Spencer 1981 Appendix B]</ref>
<math display="block">\begin{align}
I_\alpha(z) = \frac{1}{\sqrt{2\pi z}\sqrt[4]{1+\frac{\alpha^2}{z^2}}}\exp\left(-\alpha \operatorname{arcsinh}\left(\frac{\alpha}{z}\right) + z\sqrt{1+\frac{\alpha^2}{z^2}}\right)\left(1 + \mathcal{O}\left(\frac{1}{z \sqrt{1+\frac{\alpha^2}{z^2}}}\right)\right).
\end{align}</math>

When {{math|1=''α'' = {{sfrac|1|2}}}}, all the terms except the first vanish, and we have
<math display="block">\begin{align}
I_{{1}/{2}}(z) &= \sqrt{\frac{2}{\pi}} \frac{\sinh(z)}{\sqrt{z}} \sim \frac{e^z}{\sqrt{2\pi z}} && \text{for }\left|\arg z\right| < \tfrac{\pi}{2}, \\[1ex]
K_{{1}/{2}}(z) &= \sqrt{\frac{\pi}{2}} \frac{e^{-z}}{\sqrt{z}}.
\end{align}</math>

For small arguments <math>0<|z|\ll\sqrt{\alpha + 1}</math>, we have
<math display="block">\begin{align}
I_\alpha(z) &\sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right)^\alpha, \\[1ex]
K_\alpha(z) &\sim \begin{cases}
  -\ln \left (\dfrac{z}{2} \right ) - \gamma & \text{if } \alpha=0 \\[1ex]
  \frac{\Gamma(\alpha)}{2} \left( \dfrac{2}{z} \right)^\alpha & \text{if } \alpha > 0
\end{cases}
\end{align}</math>