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| '''Bessel functions''', first defined by the mathematician [[Daniel Bernoulli]] and generalized by [[Friedrich Bessel]], are the [[Canonical form|canonical]] solutions ''y''(''x'') of '''Bessel's [[differential equation]]''' | | More than 60% of the planet struggles with fat reduction. They need a technique to greater their lives, nevertheless it's tough to find a real weight reduction system that can deliver results. Therefore, the BMI (Body Mass Index) calculator was founded to help individuals maintain a healthy lifetime. So what is a BMI Calculator?<br><br>In conjunction to the BMI, its a advantageous idea to go look at the calculator page over at freedieting and run the Ideal Body Weight calculator and the Body Fat Calculator. These are superb tools inside helping you track your fat lose. The Goal Estimate Calculator is additionally greatly beneficial.<br><br>When calculating how much weight to lose the many important number to understand is your Body Fat Percentage. This amount informs you how much of your body is fat, and from there you can calculate how much is muscle, water, etc. For example, when you are a 586" female weighing 180 pounds and have a BP of 43%, then 77.4 pounds are fat, plus 102.6 pounds are muscle, skeletal mass, and water. You have a BMI of 28.2 - a little over the norm. Then you are able to figure out that if you would like your BF to be 23% - very perfect for a female - we have to lose 36 pounds of fat. The result: a fat of 144 pounds with a BP of 23%, that in turn offers we a BMI of 22.6, well in the regular range.<br><br>A great deal of folks have benefited from the [http://safedietplans.com/bmi-calculator bmi calculator] over the past limited years yet, there are more accurate alternatives available nowadays. If you are not certain what method is best for your condition, then you need to consult a health professional for right guidance.<br><br>Additionally, people differ inside how muscular they are. A very muscular person will have low body fat and nonetheless have a especially high BMI. Professional body builders are fat, by this criterion. Arnold Schwarzenegger, inside his prime, was 6'2 plus 235 pounds http://www.bodybuildinguniverse.com/arnold.htm; that provides him a BMI of 30.2! There are running backs inside American sports that are 5'8" or 5'9" and over 200 pounds. Running backs are not fat. Heck, Michael Jordan was 6'6" and 216 pounds - borderline obese per the above mentioned!<br><br>To watch both videos below takes regarding 90 minutes. Here are the features of it. It shows 6 persons inside their home towns with diabetes going to Dr. Cousen's center. One elder man had his blood sugar brought down to normal yet left following 2 week since the diet was too difficult to follow. You can eat as much because you wish however it is still hard for individuals to follow.<br><br>Then I merely go running about twice a week to keep what I've worked especially hard to achieve. I no longer feel randomly depressed, nor do I binge eat simply because I don't feel superior. I've found it funny that having lost fat I do feel more confident. I even have more vitality and am able to focus a lot better during the day. |
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| : <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0</math>
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| for an arbitrary [[complex number]] α (the '''order''' of the Bessel function). The most important cases are for α an [[integer]] or [[half-integer]].
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| Although α and −α produce the same differential equation for real α, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α. Bessel functions are also known as '''cylinder functions''' or the '''[[cylindrical harmonics]]''' because they appear in the solution to [[Laplace's equation]] in [[cylindrical coordinates]].
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| ==Applications of Bessel functions==
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| 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''+1/2). For example:
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| * [[Electromagnetic radiation|Electromagnetic waves]] in a cylindrical [[waveguide]]
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| * Pressure amplitudes of inviscid rotational flows
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| * [[Conduction (heat)|Heat conduction]] in a cylindrical object
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| * Modes of vibration of a thin circular (or annular) [[Acoustic membrane|artificial membrane]] (such as a [[drum]] or other [[membranophone]])
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| * Diffusion problems on a lattice
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| * Solutions to the radial [[Schrödinger equation]] (in spherical and cylindrical coordinates) for a free particle
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| * Solving for patterns of acoustical radiation
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| * Frequency-dependent friction in circular pipelines
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| Bessel functions also appear in other problems, such as signal processing (e.g., see [[FM synthesis]], [[Kaiser window]], or [[Bessel filter]]).
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| ==Definitions==
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| Because 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. Different variations are described below.
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| ===Bessel functions of the first kind : ''J''<sub>α</sub>===
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| Bessel functions of the first kind, denoted as ''J''<sub>α</sub>(''x''), are solutions of Bessel's differential equation that are finite at the origin (''x'' = 0) for integer or positive α, and diverge as ''x'' approaches zero for negative non-integer α. It is possible to define the function by its [[Taylor series]] expansion around ''x'' = 0:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_360.htm p. 360, 9.1.10].</ref>
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| :<math> J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m+\alpha} </math>
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| where Γ(''z'') is the [[gamma function]], a shifted generalization of the [[factorial]] function to non-integer values. The Bessel function of the first kind is an [[entire function]] if α is an integer. 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 −''J''<sub>1</sub>(''x'') is the derivative of ''J''<sub>0</sub>(''x''), much like −sin(''x'') is the derivative of cos(''x''); more generally, the derivative of ''J<sub>n</sub>''(''x'') can be expressed in terms of ''J''<sub>''n''±1</sub>(''x'') by the identities [[Bessel function#Properties|below]].)
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| [[Image:Bessel Functions (1st Kind, n=0,1,2).svg|thumb|300px|right|Plot of Bessel function of the first kind, ''J''<sub>α</sub>(''x''), for integer orders α = 0, 1, 2]]
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| For non-integer α, the functions ''J''<sub>α</sub>(''x'') and ''J''<sub>−α</sub>(''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 has simple poles at each of the non-positive integers):<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_358.htm p. 358, 9.1.5].</ref>
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| :<math>J_{-n}(x) = (-1)^n J_{n}(x).\,</math>
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| 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.
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| ====Bessel's integrals====
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| Another definition of the Bessel function, for integer values of ''n'', is possible using an integral representation:<ref name=Temme>{{cite book|last=Temme|first=Nico M.|title=Special functions : an introduction to the classical functions of mathematical physics|year=1996|publisher=Wiley|location=New York [u.a.]|isbn=0471113131|pages=228-231|edition=2. print.}}</ref>
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| :<math>J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin(\tau)) \,d\tau.</math>
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| Another integral representation is:<ref name=Temme />
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| :<math>J_n (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(n \tau - x \sin(\tau))} \,d\tau.</math>
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| 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 (for Re(''x'') > 0), one of Schäfli's integrals:<ref name=Temme />
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| :<math>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. </math> <ref>Watson, [http://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA176 p. 176]</ref><ref>http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html</ref><ref>http://www.nbi.dk/~polesen/borel/node15.html</ref><ref>Arfken & Weber, exercise 11.1.17.</ref>
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| ====Relation to hypergeometric series====
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| The Bessel functions can be expressed in terms of the [[generalized hypergeometric series]] as<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_362.htm p. 362, 9.1.69].</ref>
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| :<math>J_\alpha(x)=\frac{(\frac{x}{2})^\alpha}{\Gamma(\alpha+1)} \;_0F_1 (\alpha+1; -\tfrac{x^2}{4}).</math>
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| This expression is related to the development of Bessel functions in terms of the [[Bessel–Clifford function]].
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| ====Relation to Laguerre polynomials====
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| In terms of the [[Laguerre polynomials]] ''L<sub>k</sub>'' and arbitrarily chosen parameter ''t'', the Bessel function can be expressed as<ref>[[Gábor Szegő|Szegö, G.]] Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.</ref>
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| :<math>\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha}= \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{k=0}^\infty \frac{L_k^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{{k+ \alpha \choose k}} \frac{t^k}{k!}.</math>
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| ===Bessel functions of the second kind : ''Y''<sub>α</sub>===
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| The Bessel functions of the second kind, denoted by ''Y''<sub>α</sub>(''x''), occasionally denoted instead by ''N''<sub>α</sub>(''x''), are solutions of the Bessel differential equation that have a singularity at the origin (''x'' = 0). These are sometimes called '''Weber functions''' after [[Heinrich Martin Weber]], and also '''Neumann functions''' after [[Carl Neumann]].
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| [[Image:Bessel Functions (2nd Kind, n=0,1,2).svg|thumb|300px|right|Plot of Bessel function of the second kind, ''Y''<sub>α</sub>(''x''), for integer orders α = 0, 1, 2.]]
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| For non-integer α, ''Y''<sub>α</sub>(''x'') is related to ''J''<sub>α</sub>(''x'') by:
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| :<math>Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}.</math>
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| In the case of integer order ''n'', the function is defined by taking the limit as a non-integer α tends to ''n'':
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| :<math>Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x).</math>
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| There is also a corresponding integral formula (for Re(''x'') > 0),<ref>Watson, [http://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA178 p. 178].</ref>
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| :<math>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. </math>
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| ''Y''<sub>α</sub>(''x'') is necessary as the second linearly independent solution of the Bessel's equation when α is an integer. But ''Y''<sub>α</sub>(''x'') has more meaning than that. It can be considered as a 'natural' partner of ''J''<sub>α</sub>(''x''). See also the subsection on Hankel functions below.
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| When α is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
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| :<math>Y_{-n}(x) = (-1)^n Y_n(x).\,</math>
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| Both ''J''<sub>α</sub>(''x'') and ''Y''<sub>α</sub>(''x'') are [[holomorphic function]]s of ''x'' on the [[complex plane]] cut along the negative real axis. When α is an integer, the Bessel functions ''J'' are [[entire function]]s of ''x''. If ''x'' is held fixed, then the Bessel functions are entire functions of α.
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| Other choices are possible for the Bessel functions of the second kind, as any linearly independent solution may be used. This is the second alternative for the second solution in [[Fuchs's theorem]] when α is an integer.
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| ===Hankel functions: ''H''<sub>α</sub><sup>(1)</sup>, ''H''<sub>α</sub><sup>(2)</sup>===
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| Another important formulation of the two linearly independent solutions to Bessel's equation are the '''Hankel functions''' ''H''<sub>α</sub><sup>(1)</sup>(''x'') and ''H''<sub>α</sub><sup>(2)</sup>(''x''), defined by:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_358.htm p. 358, 9.1.3, 9.1.4].</ref>
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| :<math>H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)</math>
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| :<math>H_\alpha^{(2)}(x) = J_\alpha(x) - i Y_\alpha(x)</math>
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| 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]].
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| 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<sup>if(x)</sup>''. The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.
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| 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]]).
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| Using the previous relationships they can be expressed as:
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| :<math>H_\alpha^{(1)} (x) = \frac{J_{-\alpha} (x) - e^{-\alpha \pi i} J_\alpha (x)}{i \sin (\alpha \pi)}</math>
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| :<math>H_\alpha^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}.</math>
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| If α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_358.htm p. 358, 9.1.6].</ref>
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| :<math>H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_\alpha^{(1)} (x) </math>
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| :<math>H_{-\alpha}^{(2)} (x)= e^{-\alpha \pi i} H_\alpha^{(2)} (x). </math>
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| In particular, if α = ''m'' + 1/2 with ''m'' a nonnegative integer, the above relations imply directly that
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| :<math>J_{-(m+\frac{1}{2})}(x) = (-1)^{m+1} Y_{m+\frac{1}{2}}(x) </math>
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| :<math>Y_{-(m+\frac{1}{2})}(x) = (-1)^m J_{m+\frac{1}{2}}(x). </math>
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| These are useful in developing the spherical Bessel functions (below).
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| The Hankel functions admit the following integral representations for Re(''x'') > 0:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_360.htm p. 360, 9.1.25].</ref>
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| :<math>H_\alpha^{(1)} (x)= \frac{1}{\pi i}\int_{-\infty}^{+\infty+i\pi} e^{x\sinh t - \alpha t} \, dt, </math>
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| :<math>H_\alpha^{(2)} (x)= -\frac{1}{\pi i}\int_{-\infty}^{+\infty-i\pi} e^{x\sinh t - \alpha t} \, dt, </math>
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| where the integration limits indicate integration along a [[Methods of contour integration|contour]] that can be chosen as follows: from −∞ to 0 along the negative real axis, from 0 to ±iπ along the imaginary axis, and from ±iπ to +∞±iπ along a contour parallel to the real axis.<ref>Watson, [http://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA178 p. 178]</ref>
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| ===Modified Bessel functions : ''I''<sub>α</sub>, ''K''<sub>α</sub>===<!-- [[K-distribution]] links to this section. -->
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| The Bessel functions are valid even for [[complex number|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:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_375.htm p. 375, 9.6.2, 9.6.10, 9.6.11].</ref>
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| :<math>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}</math>
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| :<math>K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)}.</math>
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| These are chosen to be real-valued for real and positive arguments ''x''. The series expansion for ''I''<sub>α</sub>(''x'') is thus similar to that for ''J''<sub>α</sub>(''x''), but without the alternating (−1)<sup>''m''</sup> factor.
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| If −π < arg(''x'') ≤ π/2, ''K''<sub>α</sub>(''x'') can be expressed as a Hankel function of the first kind:
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| :<math>K_\alpha(x) = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix),</math>
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| and if −π/2 < arg(''x'') ≤ π, it can be expressed as a Hankel function of the second kind:
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| :<math>K_\alpha(x) = \frac{\pi}{2} (-i)^{\alpha+1} H_\alpha^{(2)}(-ix).</math>
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| We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if −π < arg(''z'') ≤ π/2):
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| :<math>\begin{align}
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| J_\alpha(iz) &=e^{\frac{\alpha i\pi}{2}} I_\alpha(z)\\
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| Y_\alpha(iz) &=e^{\frac{(\alpha+1)i\pi}{2}}I_\alpha(z)-\frac{2}{\pi}e^{-\frac{\alpha i\pi}{2}}K_\alpha(z).
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| \end{align}</math>
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| ''I''<sub>α</sub>(''x'') and ''K''<sub>α</sub>(''x'') are the two linearly independent solutions to the '''modified Bessel's equation''':<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_374.htm p. 374, 9.6.1].</ref>
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| :<math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0.</math>
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| Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, ''I''<sub>α</sub> and ''K''<sub>α</sub> are [[exponential growth|exponentially growing]] and [[exponential decay|decaying]] functions, respectively. Like the ordinary Bessel function ''J''<sub>α</sub>, the function ''I''<sub>α</sub> goes to zero at ''x'' = 0 for α > 0 and is finite at ''x'' = 0 for α = 0. Analogously, ''K''<sub>α</sub> diverges at ''x'' = 0.
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| {| align="center"
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| | [[Image:BesselI Functions (1st Kind, n=0,1,2,3).svg|none|thumb|300px|Modified Bessel functions of 1st kind, ''I''<sub>α</sub>(''x''), for α = 0, 1, 2, 3]]
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| | [[Image:BesselK Functions (n=0,1,2,3).svg|none|thumb|300px|Modified Bessel functions of 2nd kind, ''K''<sub>α</sub>(''x''), for α = 0, 1, 2, 3]]
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| |}
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| <!-- <center>[[File:ModifiedBessel.png|Plot of some modified Bessel functions]]<br />Plot of six modified Bessel functions. In solid line ''K''<sub>0</sub>, ''K''<sub>1</sub>, and ''K''<sub>2</sub>. In dashed line : ''I''<sub>0</sub>, ''I''<sub>1</sub>, and ''I''<sub>2</sub>.</center> -->
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| Two integral formulas for the modified Bessel functions are (for Re(''x'') > 0):<ref>Watson, [http://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA181 p. 181].</ref>
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| :<math>I_\alpha(x) = \frac{1}{\pi}\int_0^\pi \exp(x\cos(\theta)) \cos(\alpha\theta) \,d\theta - \frac{\sin(\alpha\pi)}{\pi}\int_0^\infty \exp(-x\cosh t - \alpha t) \,dt ,</math>
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| :<math>K_\alpha(x) = \int_0^\infty \exp(-x\cosh t) \cosh(\alpha t) \,dt.</math>
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| Modified Bessel functions ''K''<sub>1/3</sub> and ''K''<sub>2/3</sub> can be represented in terms of rapidly converged integrals<ref>M.Kh.Khokonov. ''Cascade Processes of Energy Loss by Emission of Hard Photons'', JETP, V.99, No.4, pp. 690-707 (2004). Derived from formulas sourced to I. S. Gradshteĭn and I. M. Ryzhik, ''Table of Integrals, Series, and Products'' (Fizmatgiz, Moscow, 1963; Academic, New York, 1980).</ref>
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| :<math> \begin{align}
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| K_{\frac{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 \\
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| K_{\frac{2}{3}} (\xi) &= \frac{1}{ \sqrt{3}} \, \int_0^\infty \, \frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}} \exp \left[- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \,\right] \,dx.\end{align}</math>
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| The '''modified Bessel function of the second kind''' has also been called by the now-rare names:
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| * '''Basset function''' after [[Alfred Barnard Basset]]
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| * '''Modified Bessel function of the third kind'''
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| * '''Modified Hankel function'''<ref>Referred 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</ref>
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| * '''Macdonald function''' after [[Hector Munro Macdonald]]
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| * '''Weber function'''<ref name="mhtlab.uwaterloo.ca">http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf</ref>
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| * '''Neumann function'''<ref name="mhtlab.uwaterloo.ca"/>
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| ===Spherical Bessel functions: ''j<sub>n</sub>'', ''y<sub>n</sub>''===
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| [[Image:Spherical Bessel j Functions (n=0,1,2).svg|thumb|300px|right|Spherical Bessel functions of 1st kind, ''j<sub>n</sub>''(''x''), for ''n'' = 0, 1, 2]]
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| [[Image:Spherical Bessel y Functions (n=0,1,2).svg|thumb|300px|right|Spherical Bessel functions of 2nd kind, ''y<sub>n</sub>''(''x''), for ''n'' = 0, 1, 2]]
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| When solving the [[Helmholtz equation]] in spherical coordinates by separation of variables, the radial equation has the form:
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| :<math>x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.</math>
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| The two linearly independent solutions to this equation are called the '''spherical Bessel functions''' ''j''<sub>''n''</sub> and ''y''<sub>''n''</sub>, and are related to the ordinary Bessel functions ''J''<sub>''n''</sub> and ''Y''<sub>''n''</sub> by:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_437.htm p. 437, 10.1.1].</ref>
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| :<math>j_{n}(x) = \sqrt{\frac{\pi}{2x}} J_{n+\frac{1}{2}}(x),</math>
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| :<math>y_{n}(x) = \sqrt{\frac{\pi}{2x}} Y_{n+\frac{1}{2}}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-\frac{1}{2}}(x).</math>
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| ''y<sub>n</sub>'' is also denoted ''n<sub>n</sub>'' or [[Eta (letter)|η]]<sub>n</sub>; some authors call these functions the '''spherical Neumann functions'''.
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| The spherical Bessel functions can also be written as ('''Rayleigh's formulas'''):<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_439.htm p. 439, 10.1.25, 10.1.26];</ref>
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| :<math>j_n(x) = (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\sin(x)}{x} ,</math>
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| :<math>y_n(x) = -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\cos(x)}{x}.</math>
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| The first spherical Bessel function ''j''<sub>0</sub>(''x'') is also known as the (unnormalized) [[sinc function]]. The first few spherical Bessel functions are:
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| :<math>j_0(x)=\frac{\sin(x)} {x}</math>
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| :<math>j_1(x)=\frac{\sin(x)} {x^2}- \frac{\cos(x)} {x}</math>
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| :<math>j_2(x)=\left(\frac{3} {x^2} - 1 \right)\frac{\sin(x)}{x} - \frac{3\cos(x)} {x^2}</math><ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_438.htm p. 438, 10.1.11].</ref>
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| :<math>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},</math>
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| and
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| :<math>y_0(x)=-j_{-1}(x)=-\,\frac{\cos(x)} {x}</math>
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| :<math>y_1(x)=j_{-2}(x)=-\,\frac{\cos(x)} {x^2}- \frac{\sin(x)} {x}</math>
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| :<math>y_2(x)=-j_{-3}(x)=\left(-\,\frac{3}{x^2}+1 \right)\frac{\cos(x)}{x}- \frac{3\sin(x)} {x^2}</math><ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_438.htm p. 438, 10.1.12];</ref>
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| :<math>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}.</math>
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| ====Generating function====
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| The spherical Bessel functions have the generating functions <ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_439.htm p. 439, 10.1.39].</ref>
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| :<math>\frac 1 {z} \cos \left (\sqrt{z^2 - 2zt} \right )= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z), </math>
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| :<math>\frac 1 {z} \sin \left ( \sqrt{z^2 + 2zt} \right )= \sum_{n=0}^\infty \frac{(-t)^n}{n!} y_{n-1}(z) .</math>
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| ====Differential relations====
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| In the following ''f<sub>n</sub>'' is any of <math>j_n, y_n, h_n^{(1)}, h_n^{(2)}</math> for <math>n=0,\pm 1,\pm 2,\dots</math>
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| :<math>\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).</math><ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_439.htm p. 439, 10.1.23].</ref>
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| ===Spherical Hankel functions: ''h''<sub>''n''</sub><sup>(1)</sup>, ''h''<sub>''n''</sub><sup>(2)</sup>===
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| There are also spherical analogues of the Hankel functions:
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| :<math>h_n^{(1)}(x) = j_n(x) + i y_n(x) \, </math>
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| :<math>h_n^{(2)}(x) = j_n(x) - i y_n(x). \, </math>
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| In fact, there are simple closed-form expressions for the Bessel functions of [[half-integer]] order in terms of the standard [[trigonometric function]]s, and therefore for the spherical Bessel functions. In particular, for non-negative integers ''n'':
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| :<math>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)!}</math>
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| and <math>h_n^{(2)}</math> is the complex-conjugate of this (for real ''x''). It follows, for example, that <math>j_0(x) = \sin(x)/x</math> and <math>y_0(x) = -\cos(x)/x</math>, and so on.
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| The spherical Hankel functions appear in problems involving spherical wave propagation, for example in [[Electromagnetic_wave_equation#Multipole_expansion|the multipole expansion of the electromagnetic field]].
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| ===Riccati–Bessel functions: ''S''<sub>''n''</sub>, ''C''<sub>''n''</sub>, ξ<sub>''n''</sub>, ζ<sub>''n''</sub>===
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| [[Jacopo Riccati|Riccati]]–Bessel functions only slightly differ from spherical Bessel functions:
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| :<math>S_n(x)=x j_n(x)=\sqrt{\frac{\pi x}{2}} \, J_{n+\frac{1}{2}}(x)</math>
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| :<math>C_n(x)=-x y_n(x)=-\sqrt{\frac{\pi x}{2}} \, Y_{n+\frac{1}{2}}(x)</math>
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| :<math>\xi_n(x) = x h_n^{(1)}(x)=\sqrt{\frac{\pi x}{2}} \, H_{n+\frac{1}{2}}^{(1)}(x)=S_n(x)-iC_n(x)</math>
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| :<math>\zeta_n(x)=x h_n^{(2)}(x)=\sqrt{\frac{\pi x}{2}} \, H_{n+\frac{1}{2}}^{(2)}(x)=S_n(x)+iC_n(x).</math>
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| They satisfy the differential equation:
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| :<math>x^2 \frac{d^2 y}{dx^2} + [x^2 - n (n+1)] y = 0.</math>
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| This differential equation, and the Riccati–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)<ref>Hong Du, "Mie-scattering calculation," ''Applied Optics'' '''43''' (9), 1951–1956 (2004)</ref> for recent developments and references.
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| Following [[Peter Debye|Debye]] (1909), the notation <math>\psi_n,\chi_n</math> is sometimes used instead of <math>S_n,C_n</math>.
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| ==Asymptotic forms==
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| The Bessel functions have the following [[Asymptotic analysis|asymptotic]] forms for non-negative α. For small arguments <math>0 < z \ll \sqrt{\alpha + 1}</math>, one obtains:<ref name="Arfken & Weber">Arfken & Weber.</ref>
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| :<math>J_\alpha(z) \sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right) ^\alpha </math>
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| :<math>Y_\alpha(z) \sim \begin{cases} \frac{2}{\pi} \left[ \ln (z/2) + \gamma \right] & \text{if } \alpha=0 \\ \\ -\frac{\Gamma(\alpha)}{\pi} \left( \frac{2}{z} \right) ^\alpha & \text{if } \alpha > 0 \end{cases} </math>
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| where γ is the [[Euler–Mascheroni constant]] (0.5772...) and Γ denotes the [[gamma function]]. For large real arguments <math>x \gg |\alpha^2 - 1/4|</math>, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless 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 arg(''z'') one can write an equation containing a term of order |''z''|<sup>−1</sup>:<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_364.htm p. 364, 9.2.1];</ref>
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| :<math>J_\alpha(z)=\sqrt{\frac{2}{\pi z}}\left(\cos \left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)+e^{|\operatorname{Im}(z)|}O(|z|^{-1})\right)\text{ for }|\arg z|<\pi</math>
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| :<math>Y_\alpha(z)=\sqrt{\frac{2}{\pi z}}\left(\sin \left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)+e^{|\operatorname{Im}(z)|}O(|z|^{-1})\right)\text{ for }|\arg z|<\pi.</math>
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| (For α = 1/2 the last terms in these formulas drop out completely; see the spherical Bessel functions above.) Even though these equations are true, better approximations may be available for complex ''z''. For example, ''J''<sub>0</sub>(''z'') when ''z'' is near the negative real line is approximated better by <math>J_0(z)\approx\sqrt{\frac{-2}{\pi z}}\cos \left(z+\frac{\pi}{4}\right)</math> than by <math>J_0(z)\approx\sqrt{\frac{2}{\pi z}}\cos \left(z-\frac{\pi}{4}\right)</math>.
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| The asymptotic forms for the Hankel functions are:
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| :<math>H_\alpha^{(1)}(z)\sim\sqrt{\frac{2}{\pi z}}\exp\left(i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)\right)\text{ for }-\pi<\arg z<2\pi</math>
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| :<math>H_\alpha^{(2)}(z)\sim\sqrt{\frac{2}{\pi z}}\exp\left(-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)\right)\text{ for }-2\pi<\arg z<\pi</math>
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| These can be extended to other values of arg(''z'') using equations relating <math>H_\alpha^{(1)}(ze^{im\pi})</math> and <math>H_\alpha^{(2)}(ze^{im\pi})</math> to ''H''<sub>α</sub><sup>(1)</sup>(''z'') and ''H''<sub>α</sub><sup>(2)</sup>(''z'').<ref>[[NIST]] [[Digital Library of Mathematical Functions]], Section [http://dlmf.nist.gov/10.11#E1 10.11].</ref>
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| It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, ''J''<sub>α</sub>(''z'') is not asymptotic to the average of these two asymptotic forms when ''z'' is negative (because one or the other will not be correct there, depending on the 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) ''z'' so long as |''z''| goes to infinity at a constant phase angle arg ''z'' (using the square root having positive real part):
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| :<math>J_\alpha(z)\sim\frac{\exp\left( i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)\right)}{\sqrt{2\pi z}}\text{ for }-\pi<\arg z<0</math>
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| :<math>J_\alpha(z)\sim\frac{\exp\left(-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)\right)}{\sqrt{2\pi z}}\text{ for }0<\arg z<\pi</math>
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| :<math>Y_\alpha(z)\sim-i\frac{\exp\left( i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)\right)}{\sqrt{2\pi z}}\text{ for }-\pi<\arg z<0</math>
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| :<math>Y_\alpha(z)\sim-i\frac{\exp\left(-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)\right)}{\sqrt{2\pi z}}\text{ for }0<\arg z<\pi</math>
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| For the modified Bessel functions, [[Hermann Hankel|Hankel]] developed [[asymptotic expansion]]s as well:
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| :<math>I_\alpha(z) \sim \frac{e^z}{\sqrt{2\pi z}} \left(1 - \frac{4 \alpha^{2} - 1}{8 z} + \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9)}{2! (8 z)^{2}} - \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9) (4 \alpha^{2} - 25)}{3! (8 z)^{3}} + \cdots \right)\text{ for }|\arg z|<\pi/2 ,</math><ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_377.htm p. 377, 9.7.1];</ref>
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| :<math>K_\alpha(z) \sim \sqrt{\frac{\pi}{2z}} e^{-z} \left(1 + \frac{4 \alpha^{2} - 1}{8 z} + \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9)}{2! (8 z)^{2}} + \frac{(4 \alpha^{2} - 1) (4 \alpha^{2} - 9) (4 \alpha^{2} - 25)}{3! (8 z)^{3}} + \cdots \right)\text{ for }|\arg z|<3\pi/2.</math><ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_378.htm p. 378, 9.7.2];</ref>
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| When α = 1/2 all the terms except the first vanish and we have
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| :<math>I_{1/2}(z)=\sqrt{\frac{2}{\pi z}}\sinh(z) \sim \frac{e^z}{\sqrt{2\pi z}}\text{ for }|\arg z|<\pi/2 ,</math>
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| :<math>K_{1/2}(z)=\sqrt{\frac{\pi}{2z}} e^{-z}</math>
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| For small arguments <math>0 < |z| \ll \sqrt{\alpha + 1}</math>, we have:
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| :<math>I_\alpha(z) \sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right) ^\alpha </math>
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| :<math>K_\alpha(z) \sim \begin{cases} - \ln (z/2) - \gamma & \text{if } \alpha=0 \\ \\ \frac{\Gamma(\alpha)}{2} \left( \frac{2}{z} \right) ^\alpha & \text{if } \alpha > 0. \end{cases} </math>
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| ==Properties==<!-- This section is linked from [[Bessel function]] -->
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| For integer order α = ''n'', ''J''<sub>''n''</sub> is often defined via a [[Laurent series]] for a generating function:
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| :<math>e^{(\frac{x}{2})(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,\!</math>
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| an approach used by [[P. A. Hansen]] in 1843. (This can be generalized to non-integer order by [[Methods of contour integration|contour integration]] or other methods.) Another important relation for integer orders is the ''[[Jacobi–Anger expansion]]'':
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| :<math>e^{iz \cos(\phi)} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi},\!</math>
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| and
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| :<math>e^{iz \sin(\phi)} = \sum_{n=-\infty}^\infty J_n(z) e^{in\phi},\!</math>
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| which is used to expand a [[plane wave]] as a sum of cylindrical waves, or to find the [[Fourier series]] of a tone-modulated [[Frequency modulation|FM]] signal.
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| More generally, a series
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| :<math>f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_{k=1} a_k^\nu J_{\nu+k}(z)\!</math>
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| is called Neumann expansion of ''ƒ''. The coefficients for ν = 0 have the explicit form
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| : <math>a_k^0=\frac{1}{2 \pi i} \int_{|z|=c} f(z) O_k(z) \,dz,\!</math>
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| where ''O<sub>k</sub>'' is [[Neumann polynomial|Neumann's polynomial]].<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_363.htm p. 363, 9.1.82] ff.</ref>
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| Selected functions admit the special representation
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| :<math>f(z)=\sum_{k=0} a_k^\nu J_{\nu+2k}(z)\!</math>
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| with
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| :<math>a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \frac{J_{\nu+2k}(z)}z \,dz\!</math>
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| due to the orthogonality relation
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| :<math>\int_0^\infty J_\alpha(z) J_\beta(z) \frac {dz} z= \frac 2 \pi \frac{\sin\left(\frac \pi 2 (\alpha-\beta) \right)}{\alpha^2 -\beta^2}.</math>
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| More generally, if ''ƒ'' has a branch-point near the origin of such a nature that
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| :<math>f(z)= \sum_{k=0} a_k J_{\nu+k}(z),</math>
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| then
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| :<math>\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}}</math>
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| or
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| :<math>\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),</math>
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| where <math>\mathcal L \{f \}</math> is ''f''<nowiki>'</nowiki>s [[Laplace transform]].<ref>[http://books.google.com/books?id=Mlk3FrNoEVoC&lpg=PA522&dq=bessel%20neumann%20series&pg=PA536#v=onepage&q=bessel%20neumann%20series&f=false E. T. Whittaker, G. N. Watson, A course in modern Analysis p. 536]</ref>
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| Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:
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| :<math>\begin{align}J_\nu(z) &= \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, \\ &=\frac 2{{\left(\frac z 2\right)}^\nu\cdot \sqrt{\pi} \cdot \Gamma\left(\frac 1 2-\nu\right)} \int_1^\infty \frac{\sin(z u)}{(u^2-1)^{\nu+\frac 1 2}} \,du,\end{align}</math>
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| where ν > −1/2 and ''z'' ∈ '''C'''.<ref>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</ref> This formula is useful especially when working with [[Fourier transforms]].
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| The functions ''J''<sub>α</sub>, ''Y''<sub>α</sub>, ''H''<sub>α</sub><sup>(1)</sup>, and ''H''<sub>α</sub><sup>(2)</sup> all satisfy the [[recurrence relation]]s:
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| :<math>\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)\!</math>
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| :<math> 2\frac{dZ_\alpha}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x)\!</math>
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| where ''Z'' denotes ''J'', ''Y'', ''H''<sup>(1)</sup>, or ''H''<sup>(2)</sup>. (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:
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| :<math>\left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ x^\alpha Z_{\alpha} (x) \right] = x^{\alpha - m} Z_{\alpha - m} (x)</math>
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| :<math>\left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] = (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}.</math>
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| ''Modified'' Bessel functions follow similar relations :
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| :<math>e^{(\frac{x}{2})(t+1/t)} = \sum_{n=-\infty}^\infty I_n(x) t^n,\!</math>
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| and
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| :<math>e^{z \cos( \theta)} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos(n\theta).\!</math>
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| The recurrence relation reads
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| :<math>C_{\alpha-1}(x) - C_{\alpha+1}(x) = \frac{2\alpha}{x} C_\alpha(x)\!</math>
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| :<math>C_{\alpha-1}(x) + C_{\alpha+1}(x) = 2\frac{dC_\alpha}{dx}\!</math>
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| where ''C''<sub>α</sub> denotes ''I''<sub>α</sub> or ''e''<sup>απ''i''</sup>''K''<sub>α</sub>. These recurrence relations are useful for discrete diffusion problems.
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| 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:
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| :<math>\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,\!</math>
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| where α > −1, δ<sub>''m'',''n''</sub> is the [[Kronecker delta]], and ''u''<sub>α, m</sub> is the ''m''-th [[root of a function|zero]] of ''J''<sub>α</sub>(''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''<sub>α</sub>(''x'' ''u''<sub>α, m</sub>) for fixed α and varying ''m''.
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| An analogous relationship for the spherical Bessel functions follows immediately:
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| :<math>\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.\!</math>
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| Another orthogonality relation is the ''closure equation'':<ref>Arfken & Weber, section 11.2</ref>
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| :<math>\int_0^\infty x J_\alpha(ux) J_\alpha(vx) \,dx = \frac{1}{u} \delta(u - v)\!</math>
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| for α > −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:
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| :<math>\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) \,dx = \frac{\pi}{2u^2} \delta(u - v)\!</math>
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| for α > −1.
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| Another important property of Bessel's equations, which follows from [[Abel's identity]], involves the [[Wronskian]] of the solutions:
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| :<math>A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},\!</math>
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| where ''A''<sub>α</sub> and ''B''<sub>α</sub> are any two solutions of Bessel's equation, and ''C''<sub>α</sub> is a constant independent of ''x'' (which depends on α and on the particular Bessel functions considered). For example, if ''A''<sub>α</sub> = ''J''<sub>α</sub> and ''B''<sub>α</sub> = ''Y''<sub>α</sub>, then ''C''<sub>α</sub> is 2/π. This also holds for the modified Bessel functions; for example, if ''A''<sub>α</sub> = ''I''<sub>α</sub> and ''B''<sub>α</sub> = ''K''<sub>α</sub>, then ''C''<sub>α</sub> is −1.
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| (There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)
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| ==Multiplication theorem==
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| The Bessel functions obey a [[multiplication theorem]]
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| :<math>\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) </math>
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| where λ and ν may be taken as arbitrary complex numbers, see <ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_363.htm p. 363, 9.1.74].</ref><ref>C. Truesdell, "[http://www.pnas.org/cgi/reprint/36/12/752.pdf On the Addition and Multiplication Theorems for the Special Functions]", ''Proceedings of the National Academy of Sciences, Mathematics'', (1950) pp.752–757.</ref> The above expression also holds if <math>J</math> is replaced by <math>Y</math>. The analogous identities for modified Bessel functions are
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| :<math>\lambda^{-\nu} I_\nu (\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{(\lambda^2-1)z}{2}\right)^n I_{\nu+n}(z) </math>
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| and
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| :<math>\lambda^{-\nu} K_\nu (\lambda z) = \sum_{n=0}^\infty \frac{(-1)^n}{n!} \left(\frac{(\lambda^2-1)z}{2}\right)^n K_{\nu+n}(z). </math>
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| ==Bourget's hypothesis==
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| Bessel himself originally proved that for non-negative integers ''n'', the equation ''J''<sub>''n''</sub>(''x'') = 0 has an infinite number of solutions in ''x''.<ref>F. Bessel, ''Untersuchung des Theils der planetarischen Störungen'', Berlin Abhandlungen (1824), article 14.</ref> When the functions ''J''<sub>''n''</sub>(''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'' = 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'' ≥ 0 and ''m'' ≥ 1, the functions ''J<sub>n</sub>''(''x'') and ''J''<sub>''n''+''m''</sub>(''x'') have no common zeros other than the one at ''x'' = 0. The hypothesis was proved by [[Carl Ludwig Siegel]] in 1929.<ref>Watson, pp. 484–5</ref>
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| ==Selected identities<ref>See, for example, Lide DR. CRC handbook of chemistry and physics: a ready-reference book of chemical CRC Press, 2004, ISBN 0-8493-0485-7, p. A-95</ref>==
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| :<math>K_\frac{1}{2}(z)=\sqrt{\frac{\pi}{2}} \mathrm{e}^{-z}z^{-\tfrac{1}{2}},\, z>0; </math>
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| :<math>I_{-\frac{1}{2}} (z)= \sqrt{\frac{2}{\pi z}}\cosh(z) ;</math>
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| :<math>I_{\frac{1}{2}} \left(z\right)= \sqrt{\frac{2}{\pi z}}\sinh(z) ;</math>
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| :<math>I_\nu(z)=\sum_{k=0} \frac{z^k}{k!} J_{\nu+k}(z);</math>
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| :<math>J_\nu(z)=\sum_{k=0} (-1)^k \frac{z^k}{k!} I_{\nu+k}(z);</math>
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| :<math>I_\nu (\lambda z)= \lambda^\nu \sum_{k=0} \frac{\left((\lambda^2-1)\frac z 2\right)^k}{k!} I_{\nu+k}(z);</math>
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| :<math>I_\nu (z_1+z_2)= \sum_{k=-\infty}^\infty I_{\nu-k}(z_1)I_k(z_2)</math>
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| :<math>J_\nu(z_1\pm z_2)= \sum_{k=-\infty}^\infty J_{\nu \mp k}(z_1)J_k(z_2);</math>
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| :<math>I_\nu(z)=\frac z {2 \nu} (I_{\nu-1}(z)-I_{\nu+1}(z));</math>
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| :<math>J_\nu(z)=\frac z {2 \nu} (J_{\nu-1}(z)+J_{\nu+1}(z));</math>
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| :<math>J_\nu'(z)=\frac{J_{\nu-1}(z)-J_{\nu+1}(z)}{2}\quad(\nu\neq 0), \quad J_0'(z)=-J_1(z);</math>
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| :<math>I_\nu'(z)=\frac{I_{\nu-1}(z)+I_{\nu+1}(z)}{2},\quad I_0'(z)=I_1(z);</math>
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| :<math>\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}</math>
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| :<math>= \Gamma(\nu+1)\cdot \sum_{k=0}\frac 1{k!}\left(\tfrac1 2z\right)^k J_{\nu+k}(z).</math>
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| ==See also==
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| * [[Anger function]]
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| * [[Bessel–Clifford function]]
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| * [[Bessel polynomials]]
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| * [[Fourier–Bessel series]]
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| * [[Frequency Modulation]]
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| * [[Hahn–Exton q-Bessel function]]
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| * [[Hankel transform]]
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| * [[Jackson q-Bessel function]]
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| * [[Kelvin functions]]
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| * [[Lerche–Newberger sum rule]]
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| * [[Lommel function]]
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| * [[Lommel polynomial]]
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| * [[Neumann polynomial]]
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| * [[Propagator]]
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| * [[Sonine formula]]
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| * [[Struve function]]
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| * [[Vibrations of a circular drum]]
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| * [[Anger function|Weber function]]
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| * [[Wright generalized Bessel function]]
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| ==Notes==
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| {{Reflist|colwidth=30em}}
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| ==References==
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| * {{Abramowitz_Stegun_ref2|9|355|10|435}}
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| * [[George B. Arfken|Arfken, George B.]] and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition (Harcourt: San Diego, 2005). ISBN 0-12-059876-0.
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| * Bayin, S.S. ''Mathematical Methods in Science and Engineering'', Wiley, 2006, Chapter 6.
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| * Bayin, S.S., ''Essentials of Mathematical Methods in Science and Engineering'', Wiley, 2008, Chapter 11.
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| * Bowman, Frank ''Introduction to Bessel Functions'' (Dover: New York, 1958). ISBN 0-486-60462-4.
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| * G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", ''Ann. Phys. Leipzig'' '''25''' (1908), p. 377.
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| *{{dlmf|first=F. W. J. |last=Olver|first2=L. C. |last2=Maximon|id=10}}
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| * {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.5. Bessel Functions of Integer Order | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=274}}
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| * B Spain, M.G. Smith, ''Functions of mathematical physics'', Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.
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| * N. M. Temme, ''Special Functions. An Introduction to the Classical Functions of Mathematical Physics'', John Wiley and Sons, Inc., New York, 1996. ISBN 0-471-11313-1. Chapter 9 deals with Bessel functions.
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| * [[G. N. Watson|Watson, G.N.]], ''A Treatise on the Theory of Bessel Functions, Second Edition'', (1995) Cambridge University Press. ISBN 0-521-48391-3.
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| ==External links==
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| * {{springer|first=P.I. |last=Lizorkin|id=b/b015840|title= Bessel functions}}
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| *{{springer|first=L.N. |last=Karmazina|first2=A.P. |last2=Prudnikov|id=c/c027610|title= Cylinder function}}
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| *{{springer|first==N.Kh.|last= Rozov|id=B/b015830|title=Bessel equation}}
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| * Wolfram function pages on Bessel [http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ J] and [http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ Y] functions, and modified Bessel [http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ I] and [http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ K] functions. Pages include formulas, function evaluators, and plotting calculators.
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| *[http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html Wolfram Mathworld – Bessel functions of the first kind]
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| *Bessel functions [http://www.librow.com/articles/article-11/appendix-a-34 J<sub>ν</sub>], [http://www.librow.com/articles/article-11/appendix-a-35 Y<sub>ν</sub>], [http://www.librow.com/articles/article-11/appendix-a-36 I<sub>ν</sub>] and [http://www.librow.com/articles/article-11/appendix-a-37 K<sub>ν</sub>] in Librow [http://www.librow.com/articles/article-11 Function handbook].
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| {{DEFAULTSORT:Bessel Function}}
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| [[Category:Special hypergeometric functions]]
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| [[Category:Fourier analysis]]
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