In mathematics, '''Bessel functions''', first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions ''y''(''x'') of Bessel's differential equation:  : x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0  for an arbitrary real or complex number α (the ''order'' of the Bessel function); the most common and important cases are for α an integer or half-integer.  Although α and &amp;minus;α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α). Bessel functions are also known as '''cylinder functions''' or '''cylindrical harmonics''' because they are found in the solution to Laplace's equation in cylindrical coordinates.  ==Applications of Bessel function== Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = ''n''); in spherical problems, one obtains half-integer orders (α = ''n''&amp;nbsp;+&amp;nbsp;½). For example: * Electromagnetic waves in a cylindrical waveguide * Heat conduction in a cylindrical object * Modes of vibration of a thin circular (or annular) artificial membrane (such as a drum or other membranophone) * Diffusion problems on a lattice * Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle * Solving for patterns of acoustical radiation  Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).  ==Definitions== Since this is a second-order differential equation, there must be two linearly independent solutions.  Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below.  ===Bessel functions of the first kind : ''J''''&amp;alpha;''=== Bessel functions of the first kind, denoted as ''J''&amp;alpha;(''x''), are solutions of Bessel's differential equation that are finite at the origin (''x'' = 0) for integer &amp;alpha;, and diverge as ''x'' approaches zero for negative non-integer &amp;alpha;. The solution type (e.g., integer or non-integer) and normalization of ''J''&amp;alpha;(''x'') are defined by its properties below. It is possible to define the function by its Taylor series expansion around ''x'' = 0:Abramowitz and Stegun, [ p. 360, 9.1.10].  : J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m+\alpha+1)} {\left(\tfrac{1}{2}x\right)}^{2m+\alpha}   where Γ(''z'') is the gamma function, a generalization of the factorial function to non-integer values. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√''x'' (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large ''x''. (The Taylor series indicates that &amp;minus;''J''1(''x'') is the derivative of ''J''0(''x''), much like &amp;minus;sin&amp;nbsp;''x'' is the derivative of cos&amp;nbsp;''x''; more generally, the derivative of ''J''''n''(''x'') can be expressed in terms of ''J''''n''&amp;plusmn;1(''x'') by the identities below.)    For non-integer α, the functions ''J''&amp;alpha;(''x'') and ''J''&amp;minus;&amp;alpha;(''x'') are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order α, the following relationship is valid (note that the Gamma function becomes infinite for negative integer arguments):Abramowitz and Stegun, [ p. 358, 9.1.5].  :J_{-n}(x) = (-1)^n J_{n}(x).\,  This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.  ====Bessel's integrals==== Another definition of the Bessel function, for integer values of n, is possible using an integral representation:  :J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin \tau) \,\mathrm{d}\tau.  Another integral representation is:  :J_n (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{-\mathrm{i}\,(n \tau - x \sin \tau)} \,\mathrm{d}\tau.  This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by the addition of another term  :J_\alpha(x) =    \frac{1}{\pi} \int_0^\pi \cos(\alpha\tau- x \sin\tau)\,d\tau   - \frac{\sin(\alpha\pi)}{\pi} \int_0^\infty           e^{-x \sinh(t) - \alpha t} \, dt.   or for \alpha &gt; -\frac{1}{2} by :   J_\alpha(x)= \frac{1}{2^{\alpha-1}\Gamma(\alpha + \frac{1}{2}) \sqrt{\pi}\, x^\alpha} \int_0^x (x^2-\tau^2)^{\alpha-1/2}\cos \tau \, d\tau.   ====Relation to hypergeometric series==== The Bessel functions can be expressed in terms of the generalized hypergeometric series as :J_\alpha(x)=\frac{(x/2)^\alpha}{\Gamma(\alpha+1)}  \;_0F_1 (\alpha+1; -\tfrac{1}{4}x^2). This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.  ====Relation to Laguerre polynomials==== In terms of the Laguerre polynomials L_k and arbitrarily chosen parameter t, the Bessel function can be expressed asSzegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975. :\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha}= \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{k=0} \frac{L_k^{(\alpha)}\left( \frac{x^2}{4 t}\right)} \frac{t^k}{k!}.  ===Bessel functions of the second kind : ''Y''''&amp;alpha;=== The Bessel functions of the second kind, denoted by ''Y''''α''(''x''), are solutions of the Bessel differential equation. They have a singularity at the origin (''x'' = 0).    ''Y''''α''(''x'') is sometimes also called the '''Neumann function''', and is occasionally denoted instead by ''N''''α''(''x'').  For non-integer α, it is related to ''J''''α''(''x'') by:  :Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}.  In the case of integer order ''n'', the function is defined by taking the limit as a non-integer α tends to 'n':  :Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x),  which has the result (in integral form)  :Y_n(x) =    \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - n\theta) \, d\theta   - \frac{1}{\pi} \int_0^\infty           \left[ e^{n t} + (-1)^n e^{-n t} \right]           e^{-x \sinh t} \, dt.   ''Y''''α''(''x'') is necessary as the second linearly independent solution of the Bessel's equation when ''α'' is an integer. But ''Y''''α''(''x'') has more meaning than that. It can be considered as a 'natural' partner of ''J''''α''(''x''). See also the subsection on Hankel functions below.  When α is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:  :Y_{-n}(x) = (-1)^n Y_n(x).\,  Both ''J''α(''x'') and ''Y''α(''x'') are holomorphic functions of ''x'' on  the complex plane cut along the negative real axis.  When α is an integer, the Bessel functions ''J'' are entire functions of ''x''.  If ''x'' is held fixed, then the Bessel functions are entire functions of α.  ===Hankel functions: ''H''''α''(1), ''H''''α''(2)=== Another important formulation of the two linearly independent solutions to Bessel's equation are the '''Hankel functions''' ''H''''α''(1)(''x'') and ''H''''α''(2)(''x''), defined by:Abramowitz and Stegun, [ p. 358, 9.1.3, 9.1.4].  :H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)  :H_\alpha^{(2)}(x) = J_\alpha(x) - i Y_\alpha(x)  where ''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.  The importance of Hankel functions of the first and second kind lies more in theoretical development rather than in application. These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, 'simple' means  an appearance of the factor of the form e^{if(x)}. The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.  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:  :H_\alpha^{(1)} (x) = \frac{J_{-\alpha} (x) - e^{-\alpha \pi i} J_\alpha (x)}{i \sin (\alpha \pi)}  :H_\alpha^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}  if ''α'' is an integer, the limit has to be calculated. The following relationships are valid, whether ''α'' is an integer or not:Abramowitz and Stegun, [ p. 358, 9.1.6].  :H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_\alpha^{(1)} (x)   :H_{-\alpha}^{(2)} (x)= e^{-\alpha \pi i} H_\alpha^{(2)} (x).   The Hankel functions admit the following integral representations (useful in the calculus of the propagator of the Klein–Gordon field):Abramowitz and Stegun, [ p. 360, 9.1.25].  :H_\alpha^{(1)} (x)= \frac{e^{-\frac{1}{2} \alpha\pi i}}{\pi i}\int_{-\infty}^{+\infty} e^{ix\cosh t - \alpha t} \, dt.    :H_\alpha^{(2)} (x)= -\frac{e^{-\frac{1}{2} \alpha\pi i}}{\pi i}\int_{-\infty}^{+\infty} e^{-ix\cosh t - \alpha t} \, dt.   === Modified Bessel functions : ''I''''α'', ''K''''α''=== The Bessel functions are valid even for complex arguments ''x'', and an important special case is that of a purely imaginary argument.  In this case, the solutions to the Bessel equation are called the '''modified Bessel functions''' (or occasionally the '''hyperbolic Bessel functions''') of the first and second kind, and are defined by any of these equivalent alternatives:Abramowitz and Stegun, [ p. 375, 9.6.2, 9.6.10, 9.6.11].  :I_\alpha(x) = i^{-\alpha} J_\alpha(ix) =\sum_{m=0}^\infty \frac{1}{m! \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}  :K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)} = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) = -\frac{\pi}{2} i^{\alpha+1} e^{-i \pi \alpha} H_\alpha^{(2)}(-ix).  There exist many integral representations of these functions. The following for ''K''''α''(''x''), is useful for the calculus of the Feynman propagator in field theory:  :K_\alpha(x) = \frac{1}{2} e^{-\frac{1}{2}\alpha\pi i} \int_{-\infty}^{+\infty} e^{-ix\sinh t -\alpha t} \, dt   These are chosen to be real-valued for real and positive arguments ''x''. The series expansion for ''Iα''(''x'') is thus similar to that for ''Jα''(''x''), but without the alternating (&amp;minus;1)''m'' factor.  ''Iα''(''x'') and ''Kα''(''x'') are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, [ p. 374, 9.6.1].  :x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0.  Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, ''I''α and ''K''α are exponentially growing and decaying functions, respectively.  Like the ordinary Bessel function ''J''α, the function ''I''α goes to zero at ''x'' = 0 for α &gt; 0 and is finite at ''x'' = 0 for α = 0.  Analogously, ''K''α diverges at ''x''&amp;nbsp;=&amp;nbsp;0.  {| align=&quot;center&quot; |- |  |  |}  Plot of some modified Bessel functionsPlot of six modified Bessel functions. In solid line ''K''0, ''K''1, and ''K''2. In dashed line : ''I''0, ''I''1, and ''I''2. --&gt;  Modified Bessel functions  K_{1/3} and  K_{2/3} can be represented in terms of rapidly converged integralsM.Kh.Khokonov. ''Cascade Processes of Energy Loss by Emission of Hard Photons'' //  JETP, V.99, No.4, pp. 690-707  (2004).  : K_{1/3} (\xi) = \sqrt{3}\, \int_0^\infty \, \exp \left[- \xi   \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right] \ dx    : K_{2/3} (\xi) = \frac{1}{ \sqrt{3}} \,  \int_0^\infty \, \frac{3+2x^2}{\sqrt{1+x^2/3}}  \exp  \left[- \xi  \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right] \ dx   The '''modified Bessel function of the second kind''' has also been called by the now-rare names: * Basset function * modified Bessel function of the third kind * modified Hankel functionReferred to as such in: Teichroew, D.  ''The Mixture of Normal Distributions with Different Variances'', The Annals of Mathematical Statistics. Vol. 28, No. 2 (Jun., 1957), pp. 510–512 * MacDonald function  ===Spherical Bessel functions: ''j''''n'', ''y''''n''===   When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form:  :x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.  The two linearly independent solutions to this equation are called the '''spherical Bessel functions''' ''j''''n'' and ''y''''n'', and are related to the ordinary Bessel functions  ''J''''n'' and ''Y''''n'' by:Abramowitz and Stegun, [ p. 437, 10.1.1].  :j_{n}(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x),  :y_{n}(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x).  y_n is also denoted n_n or ηn; some authors call these functions the '''spherical Neumann functions'''.  The spherical Bessel functions can also be written as (Rayleigh's Formulas):Abramowitz and Stegun, [ p. 439, 10.1.25, 10.1.26]; :j_n(x) = (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\sin x}{x} , :y_n(x) = -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\cos x}{x}.  The first spherical Bessel function j_0(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: :j_0(x)=\frac{\sin x} {x} :j_1(x)=\frac{\sin x} {x^2}- \frac{\cos x} {x} :j_2(x)=\left(\frac{3} {x^2} - 1 \right)\frac{\sin x}{x} - \frac{3\cos x} {x^2}Abramowitz and Stegun, [ p. 438, 10.1.11]. :j_3(x)=\left(\frac{15}{x^3} - \frac{6}{x} \right)\frac{\sin x}{x} -\left(\frac{15}{x^2} - 1\right) \frac{\cos x} {x}, and :y_0(x)=-j_{-1}(x)=-\,\frac{\cos x} {x} :y_1(x)=j_{-2}(x)=-\,\frac{\cos x} {x^2}- \frac{\sin x} {x} :y_2(x)=-j_{-3}(x)=\left(-\,\frac{3}{x^2}+1 \right)\frac{\cos x}{x}- \frac{3 \sin x} {x^2}Abramowitz and Stegun, [ p. 438, 10.1.12]; :y_{3}\left( x\right)=j_{-4}(x) =\left( -\frac{15}{x^{3}}+\frac{6}{x}\right) \frac{\cos x}{x}-\left( \frac{15}{x^{2}}-1\right) \frac{\sin x}{x}.  The general identity is : \begin{align} J_{n+\frac 1 2}(x)=\sqrt{\frac 2 {\pi x}}\sum_{i=0}^\frac {n+1} 2 (-1)^{n-i} &amp; \left[ \sin(x) \left(\frac 2 x\right)^{n-2i} \frac {(n-i)!}{i!} {-\frac 1 2 -i \choose n-2i} \right. \\ &amp; \left.{} - \cos(x) \left(\frac 2 x\right)^{n+1-2i} \frac {(n-i)!}{i!} i {-\frac 1 2 -i \choose n-2i+1}\right], \end{align}  where the upper limit of summation is understood to be the largest integer less than or equal to (n+1)/2.  ====Generating function==== The spherical Bessel functions have the generating functions Abramowitz and Stegun, [ p. 439, 10.1.39]. :\frac 1 {z} \cos \sqrt{z^2 - 2zt}= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z),  :\frac 1 {z} \sin \sqrt{z^2 + 2zt}= \sum_{n=0}^\infty \frac{(-t)^n}{n!} y_{n-1}(z) .  ====Differential relations==== In the following f_n is any of j_n, y_n, h_n^{(1)}, h_n^{(2)} for n=0,\pm 1,\pm 2,\dots :\left(\frac{1}{z}\frac{d}{dz}\right)^m\left(z^{n+1}f_n(z)\right)=z^{(n-m)+1}f_{(n-m)}(z).Abramowitz and Stegun, [ p. 439, 10.1.23].  ===Spherical Hankel functions : ''h''&amp;nbsp;''n''=== There are also spherical analogues of the Hankel functions:  :h_n^{(1)}(x) = j_n(x) + i y_n(x) \,   :h_n^{(2)}(x) = j_n(x) - i y_n(x). \,   In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions.  In particular, for non-negative integers ''n'':  :h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!(2x)^m} \frac{(n+m)!}{(n-m)!}  and h_n^{(2)} is the complex-conjugate of this (for real x). It follows, for example, that j_0(x) = \sin(x)/x and y_0(x) = -\cos(x)/x, and so on.  ===Riccati&amp;ndash;Bessel functions: ''S''''n'', ''C''''n'', ''&amp;xi;''''n'', ''&amp;zeta;''''n''=== Riccati&amp;ndash;Bessel functions only slightly differ from spherical Bessel functions:  :S_n(x)=x j_n(x)=\sqrt{\pi x/2} \, J_{n+1/2}(x)  :C_n(x)=-x y_n(x)=-\sqrt{\pi x/2} \, Y_{n+1/2}(x)  :\xi_n(x) = x h_n^{(1)}(x)=\sqrt{\pi x/2} \, H_{n+1/2}^{(1)}(x)=S_n(x)-iC_n(x)  :\zeta_n(x)=x h_n^{(2)}(x)=\sqrt{\pi x/2} \, H_{n+1/2}^{(2)}(x)=S_n(x)+iC_n(x).  They satisfy the differential equation:  :x^2 \frac{d^2 y}{dx^2} + [x^2 - n (n+1)] y = 0.  This differential equation, and the Riccati&amp;ndash;Bessel solutions, arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g., Du (2004)Hong Du, &quot;Mie-scattering calculation,&quot; ''Applied Optics'' '''43''' (9), 1951&amp;ndash;1956 (2004) for recent developments and references.  Following Debye (1909), the notation \psi_n,\chi_n is sometimes used instead of S_n,C_n.  ==Asymptotic forms== The Bessel functions have the following asymptotic forms for non-negative α.  For small arguments 0 , one obtains:Arfken &amp; Weber.  :J_\alpha(x) \approx \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha   :Y_\alpha(x) \approx \begin{cases}   \frac{2}{\pi} \left[ \ln (x/2) + \gamma \right]  &amp; \text{if } \alpha=0 \\ \\   -\frac{\Gamma(\alpha)}{\pi} \left( \frac{2}{x} \right) ^\alpha &amp; \text{if } \alpha &gt; 0 \end{cases}   where \gamma is the Euler–Mascheroni constant (0.5772...) and \Gamma denotes the gamma function. For large arguments x \gg |\alpha^2 - 1/4|, they become:  :J_\alpha(x)\approx \sqrt{\frac{2}{\pi x}}         \cos \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right)  :Y_\alpha(x) \approx \sqrt{\frac{2}{\pi x}}         \sin \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right).  (For α=1/2 these formulas are exact; see the spherical Bessel functions above.) Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations.  For example, for large x \gg |\alpha^2 - 1/4|, the modified Bessel functions become:  :I_\alpha(x) \approx \frac{e^x}{\sqrt{2\pi x}} \left(1+ \frac{(1-2 \alpha)(1+2\alpha)}{8x}+ \cdots \right) ,  :K_\alpha(x) \approx \sqrt{\frac{\pi}{2x}} e^{-x}. Similarly, the last expressions are exact when \alpha=1/2.  For small arguments 0 , they become:  :I_\alpha(x) \approx \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha   :K_\alpha(x) \approx \begin{cases}   - \ln (x/2) - \gamma   &amp; \text{if } \alpha=0 \\ \\   \frac{\Gamma(\alpha)}{2} \left( \frac{2}{x} \right) ^\alpha &amp; \text{if } \alpha &gt; 0. \end{cases}   ==Properties== For integer order α = ''n'', ''J''''n'' is often defined via a Laurent series for a generating function:  :e^{(x/2)(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,  an approach used by P. A. Hansen in 1843.  (This can be generalized to non-integer order by contour integration or other methods.)  Another important relation for integer orders is the ''Jacobi–Anger expansion'':  :e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi},  and  :e^{iz \sin \phi} = \sum_{n=-\infty}^\infty J_n(z) e^{in\phi},  which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.  More generally, a series :f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_{k=1} a_k^\nu J_{\nu+k}(z) is called Neumann expansion of ''ƒ''. The coefficients for \nu=0 have the explicit form  : a_k^0=\frac{1}{2 \pi i} \int_{|z|=c} f(z) O_k(z) \, \mathrm d z,  where O_k is Neumann's polynomial.Abramowitz and Stegun, [ p. 363, 9.1.82] ff.  Selected functions admit the special representation :f(z)=\sum_{k=0} a_k^\nu J_{\nu+2k}(z) with :a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \frac{J_{\nu+2k}(z)}z  \mathrm d z due to the orthogonality relation \int_0^\infty J_\alpha(z) J_\beta(z) \frac {\mathrm d z} z= \frac 2 \pi \frac{\sin\left(\frac \pi 2 (\alpha-\beta)  \right)}{\alpha^2 -\beta^2}.  More generally, if ''ƒ'' has a branch-point near the origin of such a nature that f(z)= \sum_{k=0} a_k J_{\nu+k}(z), then  :\mathcal L \left\{\sum_{k=0} a_k J_{\nu+k} \right\}(s)= \frac 1 \sqrt{1+s^2} \sum_{k=0} \frac{a_k}{(s+\sqrt{1+s^2})^{\nu+k}} or  :\sum_{k=0} a_k \xi^{\nu+k}= \frac{1+\xi^2}{2\xi} \mathcal L \{f \} \left( \frac{1-\xi^2}{2\xi} \right),  where \mathcal L \{f \} is ''ƒ'''s Laplace transform.[ E. T. Whittaker, G. N. Watson, A course in modern Analysis p. 536]  Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:  :\begin{align}J_\nu(z) &amp;= \frac{ (\frac{z}{2})^\nu }{ \Gamma(\nu + \frac{1}{2} ) \sqrt{\pi} } \int_{-1}^{1} e^{izs}(1 - s^2)^{\nu - \frac{1}{2} } ds, \\ &amp;=\frac 2 d u,\end{align}  where ''&amp;nu;''&amp;nbsp;&gt;&amp;nbsp;&amp;minus;1/2 and ''z'' is a complex number. I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. ''Table of Integrals, Series, and Products'', seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Equation 8.411.10 This formula is useful especially when working with Fourier transforms.  The functions ''J''α, ''Y''α, ''H''α(1), and ''H''α(2) all satisfy the recurrence relations:  :\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)  : 2\frac{dZ_\alpha}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x)  where ''Z'' denotes ''J'', ''Y'', ''H''(1), or ''H''(2).  (These two identities are often combined, e.g. added or subtracted, to yield various other relations.) In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives).  In particular, it follows that:  :\left( \frac{d}{x dx} \right)^m \left[ x^\alpha Z_{\alpha} (x) \right] = x^{\alpha - m} Z_{\alpha - m} (x)  :\left( \frac{d}{x dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] = (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}.  ''Modified'' Bessel functions follow similar relations :  :e^{(x/2)(t+1/t)} = \sum_{n=-\infty}^\infty I_n(x) t^n,  and  :e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty  I_n(z) \cos(n\theta),  The recurrence relation reads  :C_{\alpha-1}(x) - C_{\alpha+1}(x) = \frac{2\alpha}{x} C_\alpha(x)  :C_{\alpha-1}(x) + C_{\alpha+1}(x) = 2\frac{dC_\alpha}{dx}  where ''C''α denotes ''I''α or  ''e''απ''i''''K''α. These recurrence relations are useful for discrete diffusion problems.  Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by ''x'', the solutions must satisfy an orthogonality relationship for appropriate boundary conditions.  In particular, it follows that:  :\int_0^1 x J_\alpha(x u_{\alpha,m}) J_\alpha(x u_{\alpha,n}) dx = \frac{\delta_{m,n}}{2} [J_{\alpha+1}(u_{\alpha,m})]^2 = \frac{\delta_{m,n}}{2} [J_{\alpha}'(u_{\alpha,m})]^2,  where ''α''&amp;nbsp;&gt;&amp;nbsp;&amp;minus;1, δ''m'',''n'' is the Kronecker delta, and ''u''α,m is the ''m''-th zero of ''J''α(''x'').  This orthogonality relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function is expanded in the basis of the functions ''J''α(''x'' ''u''α,m) for fixed α and varying ''m''.  An analogous relationship for the spherical Bessel functions follows immediately: :\int_0^1 x^2 j_\alpha(x u_{\alpha,m}) j_\alpha(x u_{\alpha,n}) dx = \frac{\delta_{m,n}}{2} [j_{\alpha+1}(u_{\alpha,m})]^2  Another orthogonality relation is the ''closure equation'':  :\int_0^\infty x J_\alpha(ux) J_\alpha(vx) dx = \frac{1}{u} \delta(u - v)  for ''α''&amp;nbsp;&gt;&amp;nbsp;&amp;minus;1/2 and where δ is the Dirac delta function. This property is used to construct an arbitrary function from a series of Bessel functions by means of the Hankel transform. For the spherical Bessel functions the orthogonality relation is:  :\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) dx = \frac{\pi}{2u^2} \delta(u - v)  for ''α''&amp;nbsp;&gt; &amp;nbsp;&amp;minus;1.  Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:  :A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},  where ''A''α and ''B''α are any two solutions of Bessel's equation, and ''C''α is a constant independent of ''x'' (which depends on α and on the particular Bessel functions considered).  For example, if ''A''α = ''J''α and ''B''α = ''Y''α, then ''C''α is 2/π.  This also holds for the modified Bessel functions; for example, if ''A''α = ''I''α and ''B''α = ''K''α, then ''C''α is&amp;nbsp;&amp;minus;1.  (There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)  ==Multiplication theorem== The Bessel functions obey a multiplication theorem  :\lambda^{-\nu} J_\nu (\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{(1-\lambda^2)z}{2}\right)^n J_{\nu+n}(z)   where \lambda and \nu may be taken as arbitrary complex numbers. A similar form may be given for Y_\nu(z) and ''etc.''Abramowitz and Stegun, [ p. 363, 9.1.74]. C. Truesdell, &quot;[ On the Addition and Multiplication Theorems for the Special Functions]&quot;, ''Proceedings of the National Academy of Sciences, Mathematics'',  (1950) pp.752–757.  ==Bourget's hypothesis== Bessel himself originally proved that for non-negative integers ''n'', the equation ''J''''n''(''x'')&amp;nbsp;=&amp;nbsp;0 has an infinite number of solutions in ''x''.F. Bessel, ''Untersuchung des Theils der planetarischen Störungen'', Berlin Abhandlungen (1824), article 14.  When the functions ''J''''n''(''x'') are plotted on the same graph, though, none of the zeros seem to coincide for different values of ''n'' except for the zero at ''x''&amp;nbsp;=&amp;nbsp;0.  This phenomenon is known as '''Bourget's hypothesis''' after the nineteenth century French mathematician who studied Bessel functions.  Specifically it states that for any integers ''n''&amp;nbsp;≥&amp;nbsp;0 and ''m''&amp;nbsp;≥&amp;nbsp;1, the functions ''J''''n''(''x'') and ''J''''n''+''m''(''x'') have no common zeros other than the one at ''x''&amp;nbsp;=&amp;nbsp;0.  The theorem was proved by Siegel in 1929.Watson, pp. 484–5  ==Derivatives of ''J'', ''Y'', ''I'', ''H'', ''K''== These formulas may be found in this &quot;Advanced Calculus for Engineers&quot;, F. B. Hildebrand, 6th printing, pp. 163–164 (1956) reference.  ===''p'' &amp;minus; 1 dependency=== :\frac{d}{dx}y_p(\alpha x)=\alpha y_{p-1}(\alpha x) - \frac{p}{x} y_p(\alpha x) (note that the above equation is for y = J, Y, I, H^{(1)}, H^{(2)}) :\frac{d}{dx}y_p(\alpha x)=-\alpha y_{p-1}(\alpha x) - \frac{p}{x} y_p(\alpha x) (note that the above equation is for y = K)  ===''p'' + 1 dependency=== :\frac{d}{dx}y_p(\alpha x)=-\alpha y_{p+1}(\alpha x) + \frac{p}{x} y_p(\alpha x) (note that the above equation is for y = J, Y, K, H^{(1)}, H^{(2)}) :\frac{d}{dx}y_p(\alpha x)=\alpha y_{p+1}(\alpha x) + \frac{p}{x} y_p(\alpha x) (note that the above equation is for y = I)  ===Other relationships=== :\frac{d}{dx}y_p(\alpha x)=\frac{\alpha}{2}[y_{p-1}(\alpha x) - y_{p+1}(\alpha x)] (note that the above equation is for y = J, Y, H^{(1)}, H^{(2)} only) :y_{p-1}(\alpha x) + y_{p+1}(\alpha x)=\frac{2p}{\alpha x}y_p(\alpha x) (note that the above equation is for y = J, Y, H^{(1)}, H^{(2)} only)  ==Selected identitiesSee, for example, Lide DR. CRC handbook of chemistry and physics: a ready-reference book of chemical    CRC Press, 2004, ISBN 0849304857, p. A-95== * I_{-1/2} \left(z\right)= \sqrt{\frac{2}{\pi z}}\cosh(z) ; * I_{1/2} \left(z\right)= \sqrt{\frac{2}{\pi z}}\sinh(z) ; * I_\nu(z)=\sum_{k=0} \frac{z^k}{k!} J_{\nu+k}(z); * J_\nu(z)=\sum_{k=0} (-1)^k \frac{z^k}{k!} I_{\nu+k}(z); * I_\nu (\lambda z)= \lambda^\nu \sum_{k=0} \frac{\left(\tfrac{1}{2}(\lambda^2-1)z\right)^k}{k!} I_{\nu+k}(z); * I_\nu (z_1+z_2)= \sum_{k=-\infty}^\infty I_{\nu-k}(z_1)I_k(z_2),\quad J_\nu(z_1\pm z_2)= \sum_{k=-\infty}^\infty J_{\nu \mp k}(z_1)J_k(z_2); * J_\nu(z)=\frac z {2 \nu} (J_{\nu-1}(z)+J_{\nu+1}(z)), \quad I_\nu(z)=\frac z {2 \nu} (I_{\nu-1}(z)-I_{\nu+1}(z)); * J_\nu'(z)=\tfrac{1}{2} (J_{\nu-1}(z)-J_{\nu+1}(z)), \quad I_\nu'(z)=\tfrac{1}{2}(I_{\nu-1}(z)+I_{\nu+1}(z)); * \left(\tfrac{1}{2}z\right)^\nu= \Gamma(\nu)\cdot \sum_{k=0} I_{\nu+2k}(z)(\nu+2k){-\nu\choose k} = \Gamma(\nu)\cdot\sum_{k=0}(-1)^k J_{\nu+2k}(z)(\nu+2k){-\nu \choose k} = \Gamma(\nu+1)\cdot \sum_{k=0}\frac 1{k!}\left(\tfrac1 2z\right)^k J_{\nu+k}(z). * K_\frac{1}{2}(z)=\sqrt{\frac{\pi}{2}} \mathrm{e}^{-z}z^{-1/2},\, z&gt;0   