Editing Bessel function (section)

=== Hankel functions: ''H''{{su|b=''α''|p=(1)}}, ''H''{{su|b=''α''|p=(2)}} <span class="anchor" id="Hankel functions"></span> ===
[[File:Plot of the Hankel function of the first kind H n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the Hankel function of the first kind {{math|''H''{{su|b=''n''|p=(1)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
[[File:Plot of the Hankel function of the second kind H n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the Hankel function of the second kind {{math|''H''{{su|b=''n''|p=(2)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]

Another important formulation of the two linearly independent solutions to Bessel's equation are the '''Hankel functions of the first and second kind''', {{math|''H''{{su|b=''α''|p=(1)}}(''x'')}} and {{math|''H''{{su|b=''α''|p=(2)}}(''x'')}}, defined as<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.3, 9.1.4].</ref>
<math display="block">\begin{align}
H_\alpha^{(1)}(x) &= J_\alpha(x) + iY_\alpha(x), \\[5pt]
H_\alpha^{(2)}(x) &= J_\alpha(x) - iY_\alpha(x),
\end{align}</math>
where {{mvar|i}} is the [[imaginary unit]]. These linear combinations are also known as '''Bessel functions of the third kind'''; they are two linearly independent solutions of Bessel's differential equation. They are named after [[Hermann Hankel]].

These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form {{math|''e''<sup>''i'' ''f''(x)</sup>}}. For real <math>x>0</math> where <math>J_\alpha(x)</math>, <math>Y_\alpha(x)</math> are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of [[Euler's formula]], substituting {{math|''H''{{su|b=''α''|p=(1)}}(''x'')}}, {{math|''H''{{su|b=''α''|p=(2)}}(''x'')}} for <math>e^{\pm i x}</math> and <math>J_\alpha(x)</math>, <math>Y_\alpha(x)</math> for <math>\cos(x)</math>, <math>\sin(x)</math>, as explicitly shown in the [[#Asymptotic forms|asymptotic expansion]].

The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the [[sign convention]] for the [[frequency]]).

Using the previous relationships, they can be expressed as
<math display="block">\begin{align}
H_\alpha^{(1)}(x) &= \frac{J_{-\alpha}(x) - e^{-\alpha \pi i} J_\alpha(x)}{i \sin \alpha\pi}, \\[5pt]
H_\alpha^{(2)}(x) &= \frac{J_{-\alpha}(x) - e^{\alpha \pi i} J_\alpha(x)}{- i \sin \alpha\pi}.
\end{align}</math>

If {{mvar|α}} is an integer, the limit has to be calculated. The following relationships are valid, whether {{mvar|α}} is an integer or not:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.6].</ref>
<math display="block">\begin{align}
H_{-\alpha}^{(1)}(x) &= e^{\alpha \pi i} H_\alpha^{(1)} (x), \\[6mu]
H_{-\alpha}^{(2)}(x) &= e^{-\alpha \pi i} H_\alpha^{(2)} (x).
\end{align}</math>

In particular, if {{math|1=''α'' = ''m'' + {{sfrac|1|2}}}} with {{mvar|m}} a nonnegative integer, the above relations imply directly that
<math display="block">\begin{align}
J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\[5pt]
Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).
\end{align}</math>

These are useful in developing the spherical Bessel functions (see below).

The Hankel functions admit the following integral representations for {{math|Re(''x'') > 0}}:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_360.htm p. 360, 9.1.25].</ref>
<math display="block">\begin{align}
H_\alpha^{(1)}(x) &= \frac{1}{\pi i}\int_{-\infty}^{+\infty + \pi i} e^{x\sinh t - \alpha t} \, dt, \\[5pt]
H_\alpha^{(2)}(x) &= -\frac{1}{\pi i}\int_{-\infty}^{+\infty - \pi i} e^{x\sinh t - \alpha t} \, dt,
\end{align}</math>
where the integration limits indicate integration along a [[methods of contour integration|contour]] that can be chosen as follows: from {{math|−∞}} to 0 along the negative real axis, from 0 to {{math|±{{pi}}''i''}} along the imaginary axis, and from {{math|±{{pi}}''i''}} to {{math|+∞ ± {{pi}}''i''}} along a contour parallel to the real axis.<ref name="p. 178"/>