47-460: Not to be confused with a J followed by a zero, for which see J0 (disambiguation) . [REDACTED] Look up jo , Jo , or JO in Wiktionary, the free dictionary. Jo , jo , JO , or J.O. may refer to: Arts and entertainment [ edit ] Jo (film) , a 1972 French comedy Jo (TV series) , a French TV series "Jo",
94-470: A common Korean surname which can be romanized as Jo Codes [ edit ] JO, ISO 3166 country code for Jordan .jo , the Internet country code top-level domain for Jordan JO, IATA code for JALways , a subsidiary of Japan Airlines Other uses [ edit ] jō ( 杖 ), a wooden staff used in some Japanese martial arts jō ( 丈 ), a Japanese unit of length equivalent to
141-524: A letter of the Cyrillic alphabet JO , symbol for rapid services on the Sōbu Line and Yokosuka Line . See also [ edit ] Joe (disambiguation) Jojo (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Jo . If an internal link led you here, you may wish to change the link to point directly to
188-723: A nonnegative integer, the above relations imply directly that J − ( m + 1 2 ) ( x ) = ( − 1 ) m + 1 Y m + 1 2 ( x ) , Y − ( m + 1 2 ) ( x ) = ( − 1 ) m J m + 1 2 ( x ) . {\displaystyle {\begin{aligned}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{aligned}}} These are useful in developing
235-467: A song by Goldfrapp from Tales of Us "Jo", a song by Mr. Oizo from Lambs Anger Jo a fictional character in the Star Wars franchise People [ edit ] Jo (given name) Jô , Brazilian footballer João Alves de Assis Silva (born 1987) Josiel Alves de Oliveira (born 1988), Brazilian footballer also known as Jô Jō (surname) , a Japanese surname Cho (Korean name) ,
282-507: Is a linear differential equation, solutions can be scaled to any amplitude. The amplitudes chosen for the functions originate from the early work in which the functions appeared as solutions to definite integrals rather than solutions to differential equations. Because the differential equation is second-order, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in
329-1006: Is also called Hansen-Bessel formula. 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 one of Schläfli's integrals, for Re( x ) > 0 : J α ( x ) = 1 π ∫ 0 π cos ( α τ − x sin τ ) d τ − sin ( α π ) π ∫ 0 ∞ e − x sinh t − α t d t . {\displaystyle 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.} The Bessel functions can be expressed in terms of
376-502: Is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to x − 1 / 2 {\displaystyle x^{-{1}/{2}}} (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x . (The series indicates that − J 1 ( x )
423-420: Is an integer or half-integer . Bessel functions for integer α {\displaystyle \alpha } are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates . Spherical Bessel functions with half-integer α {\displaystyle \alpha } are obtained when solving
470-823: Is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not: H − α ( 1 ) ( x ) = e α π i H α ( 1 ) ( x ) , H − α ( 2 ) ( x ) = e − α π i H α ( 2 ) ( x ) . {\displaystyle {\begin{aligned}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{aligned}}} In particular, if α = m + 1 / 2 with m
517-734: 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 ) . {\displaystyle Y_{-n}(x)=(-1)^{n}Y_{n}(x).} Both J α ( x ) and Y α ( x ) are holomorphic functions of x on
SECTION 10
#1732772743444564-1270: Is not an integer; when α is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x . The series expansion for I α ( x ) is thus similar to that for J α ( x ) , but without the alternating (−1) factor. K α {\displaystyle K_{\alpha }} can be expressed in terms of Hankel functions: K α ( x ) = { π 2 i α + 1 H α ( 1 ) ( i x ) − π < arg x ≤ π 2 π 2 ( − i ) α + 1 H α ( 2 ) ( − i x ) − π 2 < arg x ≤ π {\displaystyle K_{\alpha }(x)={\begin{cases}{\frac {\pi }{2}}i^{\alpha +1}H_{\alpha }^{(1)}(ix)&-\pi <\arg x\leq {\frac {\pi }{2}}\\{\frac {\pi }{2}}(-i)^{\alpha +1}H_{\alpha }^{(2)}(-ix)&-{\frac {\pi }{2}}<\arg x\leq \pi \end{cases}}} Using these two formulae
611-455: Is related to J α ( x ) by Y α ( x ) = J α ( x ) cos ( α π ) − J − α ( x ) sin ( α π ) . {\displaystyle Y_{\alpha }(x)={\frac {J_{\alpha }(x)\cos(\alpha \pi )-J_{-\alpha }(x)}{\sin(\alpha \pi )}}.} In
658-1089: Is related to the development of Bessel functions in terms of the Bessel–Clifford function . In terms of the Laguerre polynomials L k and arbitrarily chosen parameter t , the Bessel function can be expressed as J α ( x ) ( x 2 ) α = e − t Γ ( α + 1 ) ∑ k = 0 ∞ L k ( α ) ( x 2 4 t ) ( k + α k ) t k k ! . {\displaystyle {\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}}{4t}}\right)}{\binom {k+\alpha }{k}}}{\frac {t^{k}}{k!}}.} The Bessel functions of
705-874: Is the digamma function , the logarithmic derivative of the gamma function . There is also a corresponding integral formula (for Re( x ) > 0 ): Y n ( x ) = 1 π ∫ 0 π sin ( x sin θ − n θ ) d θ − 1 π ∫ 0 ∞ ( e n t + ( − 1 ) n e − n t ) e − x sinh t d t . {\displaystyle 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^{nt}+(-1)^{n}e^{-nt}\right)e^{-x\sinh t}\,dt.} In
752-411: Is the gamma function , a shifted generalization of the factorial function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by 2 {\displaystyle 2} in x / 2 {\displaystyle x/2} ; this definition is not used in this article. The Bessel function of the first kind
799-408: 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
846-429: Is the derivative of J 0 ( x ) , much like −sin x is the derivative of cos x ; more generally, the derivative of J n ( x ) can be expressed in terms of J n ± 1 ( x ) by the identities below .) For non-integer α , the functions J α ( x ) and J − α ( x ) are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order n ,
893-876: Is then found to be the Bessel function of the second kind, as discussed below. Another definition of the Bessel function, for integer values of n , is possible using an integral representation: J n ( x ) = 1 π ∫ 0 π cos ( n τ − x sin τ ) d τ = 1 π Re ( ∫ 0 π e i ( n τ − x sin τ ) d τ ) , {\displaystyle J_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(n\tau -x\sin \tau )\,d\tau ={\frac {1}{\pi }}\operatorname {Re} \left(\int _{0}^{\pi }e^{i(n\tau -x\sin \tau )}\,d\tau \right),} which
940-578: The Frobenius method to Bessel's equation: J α ( x ) = ∑ m = 0 ∞ ( − 1 ) m m ! Γ ( m + α + 1 ) ( x 2 ) 2 m + α , {\displaystyle J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },} where Γ( z )
987-738: The Hankel functions of the first and second kind , H α ( x ) and H α ( x ) , defined as H α ( 1 ) ( x ) = J α ( x ) + i Y α ( x ) , H α ( 2 ) ( x ) = J α ( x ) − i Y α ( x ) , {\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&=J_{\alpha }(x)+iY_{\alpha }(x),\\[5pt]H_{\alpha }^{(2)}(x)&=J_{\alpha }(x)-iY_{\alpha }(x),\end{aligned}}} where i
SECTION 20
#17327727434441034-770: The Helmholtz equation in spherical coordinates . 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: Bessel functions also appear in other problems, such as signal processing (e.g., see FM audio synthesis , Kaiser window , or Bessel filter ). Because this
1081-1244: The 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 H α ( 1 ) ( x ) = J − α ( x ) − e − α π i J α ( x ) i sin α π , H α ( 2 ) ( x ) = J − α ( x ) − e α π i J α ( x ) − i sin α π . {\displaystyle {\begin{aligned}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{aligned}}} If α
1128-512: 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 at a non-zero value, then the Bessel functions are entire functions of α . The Bessel functions of the second kind when α is an integer is an example of the second kind of solution in Fuchs's theorem . Another important formulation of the two linearly independent solutions to Bessel's equation are
1175-521: The generalized hypergeometric series as J α ( x ) = ( x 2 ) α Γ ( α + 1 ) 0 F 1 ( α + 1 ; − x 2 4 ) . {\displaystyle J_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\;_{0}F_{1}\left(\alpha +1;-{\frac {x^{2}}{4}}\right).} This expression
1222-521: The order of the Bessel function. Although α {\displaystyle \alpha } and − α {\displaystyle -\alpha } produce the same differential equation, 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 {\displaystyle \alpha } . The most important cases are when α {\displaystyle \alpha }
1269-561: The Chinese zhang jō ( 畳 ), a Japanese unit of area corresponding to the area of a standard tatami mat (1×½ ken or 18 square Japanese feet) JO, U.S. Navy rating-abbreviation for "journalist" Journal Officiel de la République Française , the official gazette of the Government of France Jo, a Vodun (deity) in the Fon pantheon. Jo language , a Duala language Ё ,
1316-1614: 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 α → n Y α ( x ) . {\displaystyle Y_{n}(x)=\lim _{\alpha \to n}Y_{\alpha }(x).} If n is a nonnegative integer, we have the series Y n ( z ) = − ( z 2 ) − n π ∑ k = 0 n − 1 ( n − k − 1 ) ! k ! ( z 2 4 ) k + 2 π J n ( z ) ln z 2 − ( z 2 ) n π ∑ k = 0 ∞ ( ψ ( k + 1 ) + ψ ( n + k + 1 ) ) ( − z 2 4 ) k k ! ( n + k ) ! {\displaystyle Y_{n}(z)=-{\frac {\left({\frac {z}{2}}\right)^{-n}}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {z^{2}}{4}}\right)^{k}+{\frac {2}{\pi }}J_{n}(z)\ln {\frac {z}{2}}-{\frac {\left({\frac {z}{2}}\right)^{n}}{\pi }}\sum _{k=0}^{\infty }(\psi (k+1)+\psi (n+k+1)){\frac {\left(-{\frac {z^{2}}{4}}\right)^{k}}{k!(n+k)!}}} where ψ ( z ) {\displaystyle \psi (z)}
1363-714: The case where n = 0 : (with γ {\displaystyle \gamma } being Euler's constant ) Y 0 ( x ) = 4 π 2 ∫ 0 1 2 π cos ( x cos θ ) ( γ + ln ( 2 x sin 2 θ ) ) d θ . {\displaystyle Y_{0}\left(x\right)={\frac {4}{\pi ^{2}}}\int _{0}^{{\frac {1}{2}}\pi }\cos \left(x\cos \theta \right)\left(\gamma +\ln \left(2x\sin ^{2}\theta \right)\right)\,d\theta .} Y α ( x )
1410-693: The condition Re( x ) > 0 is met. It can also be shown that J α 2 ( x ) + Y α 2 ( x ) = 8 cos ( α π ) π 2 ∫ 0 ∞ K 2 α ( 2 x sinh t ) d t , {\displaystyle J_{\alpha }^{2}(x)+Y_{\alpha }^{2}(x)={\frac {8\cos(\alpha \pi )}{\pi ^{2}}}\int _{0}^{\infty }K_{2\alpha }(2x\sinh t)\,dt,} only when | Re(α) | < 1 / 2 and Re(x) ≥ 0 but not when x = 0 . We can express
1457-693: 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 H α ( x ) , H α ( x ) for e ± i x {\displaystyle e^{\pm ix}} and J α ( x ) {\displaystyle J_{\alpha }(x)} , Y α ( x ) {\displaystyle Y_{\alpha }(x)} for cos ( x ) {\displaystyle \cos(x)} , sin ( x ) {\displaystyle \sin(x)} , as explicitly shown in
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1504-992: The first and second Bessel functions in terms of the modified Bessel functions (these are valid if − π < arg z ≤ π / 2 ): J α ( i z ) = e α π i 2 I α ( z ) , Y α ( i z ) = e ( α + 1 ) π i 2 I α ( z ) − 2 π e − α π i 2 K α ( z ) . {\displaystyle {\begin{aligned}J_{\alpha }(iz)&=e^{\frac {\alpha \pi i}{2}}I_{\alpha }(z),\\[1ex]Y_{\alpha }(iz)&=e^{\frac {(\alpha +1)\pi i}{2}}I_{\alpha }(z)-{\tfrac {2}{\pi }}e^{-{\frac {\alpha \pi i}{2}}}K_{\alpha }(z).\end{aligned}}} I α ( x ) and K α ( x ) are
1551-476: The first kind Yo , often written as j0 in Leet J00 (disambiguation) See also [ edit ] JO (disambiguation) 0J (disambiguation) [REDACTED] Topics referred to by the same term This disambiguation page lists articles associated with the same title formed as a letter–number combination. If an internal link led you here, you may wish to change the link to point directly to
1598-437: The first kind are finite at the origin ( x = 0 ); while for negative non-integer α , Bessel functions of the first kind diverge as x approaches zero. It is possible to define the function by x α {\displaystyle x^{\alpha }} times a Maclaurin series (note that α need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying
1645-435: The following relationship is valid (the gamma function has simple poles at each of the non-positive integers): J − n ( x ) = ( − 1 ) n J n ( x ) . {\displaystyle 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
1692-404: The form e . For real x > 0 {\displaystyle x>0} where J α ( x ) {\displaystyle J_{\alpha }(x)} , Y α ( x ) {\displaystyle Y_{\alpha }(x)} are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of
1739-414: The integration limits indicate integration along a contour that can be chosen as follows: from −∞ to 0 along the negative real axis, from 0 to ± π i along the imaginary axis, and from ± π i to +∞ ± π i along a contour parallel to the real axis. 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,
1786-995: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=J0&oldid=1236487539 " Category : Letter–number combination disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Bessel function Bessel functions , first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel , are canonical solutions y ( x ) of Bessel's differential equation x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 0 {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0} for an arbitrary complex number α {\displaystyle \alpha } , which represents
1833-585: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Jo&oldid=1258309859 " Category : Disambiguation pages Hidden categories: Articles containing Japanese-language text Short description is different from Wikidata All article disambiguation pages All disambiguation pages J0 (disambiguation) (Redirected from J0 (disambiguation) ) J0 may refer to: j 0 {\displaystyle j_{0}} , Zeroth order Bessel function of
1880-1595: The modified Bessel functions are (for Re( x ) > 0 ): I α ( x ) = 1 π ∫ 0 π e x cos θ cos α θ d θ − sin α π π ∫ 0 ∞ e − x cosh t − α t d t , K α ( x ) = ∫ 0 ∞ e − x cosh t cosh α t d t . {\displaystyle {\begin{aligned}I_{\alpha }(x)&={\frac {1}{\pi }}\int _{0}^{\pi }e^{x\cos \theta }\cos \alpha \theta \,d\theta -{\frac {\sin \alpha \pi }{\pi }}\int _{0}^{\infty }e^{-x\cosh t-\alpha t}\,dt,\\[5pt]K_{\alpha }(x)&=\int _{0}^{\infty }e^{-x\cosh t}\cosh \alpha t\,dt.\end{aligned}}} Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Re(ω) > 0 ): 2 K 0 ( ω ) = ∫ − ∞ ∞ e i ω t t 2 + 1 d t . {\displaystyle 2\,K_{0}(\omega )=\int _{-\infty }^{\infty }{\frac {e^{i\omega t}}{\sqrt {t^{2}+1}}}\,dt.} It can be proven by showing equality to
1927-532: 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 > 0 and is finite at x = 0 for α = 0 . Analogously, K α diverges at x = 0 with the singularity being of logarithmic type for K 0 , and 1 / 2 Γ(| α |)(2/ x ) otherwise. Two integral formulas for
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1974-825: The result to J α 2 ( z ) {\displaystyle J_{\alpha }^{2}(z)} + Y α 2 ( z ) {\displaystyle Y_{\alpha }^{2}(z)} , commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following J α 2 ( x ) + Y α 2 ( x ) = 8 π 2 ∫ 0 ∞ cosh ( 2 α t ) K 0 ( 2 x sinh t ) d t , {\displaystyle J_{\alpha }^{2}(x)+Y_{\alpha }^{2}(x)={\frac {8}{\pi ^{2}}}\int _{0}^{\infty }\cosh(2\alpha t)K_{0}(2x\sinh t)\,dt,} given that
2021-412: The second kind, denoted by Y α ( x ) , occasionally denoted instead by N α ( x ) , are solutions of the Bessel differential equation that have a singularity at the origin ( x = 0 ) and are multivalued . These are sometimes called Weber functions , as they were introduced by H. M. Weber ( 1873 ), and also Neumann functions after Carl Neumann . For non-integer α , Y α ( x )
2068-1143: 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 as I α ( x ) = i − α J α ( i x ) = ∑ m = 0 ∞ 1 m ! Γ ( m + α + 1 ) ( x 2 ) 2 m + α , K α ( x ) = π 2 I − α ( x ) − I α ( x ) sin α π , {\displaystyle {\begin{aligned}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 },\\[5pt]K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}} when α
2115-1052: The spherical Bessel functions (see below). The Hankel functions admit the following integral representations for Re( x ) > 0 : H α ( 1 ) ( x ) = 1 π i ∫ − ∞ + ∞ + π i e x sinh t − α t d t , H α ( 2 ) ( x ) = − 1 π i ∫ − ∞ + ∞ − π i e x sinh t − α t d t , {\displaystyle {\begin{aligned}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{aligned}}} where
2162-416: The table below and described in the following sections. Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by N n and n n , respectively, rather than Y n and y n . Bessel functions of the first kind, denoted as J α ( x ) , are solutions of Bessel's differential equation. For integer or positive α , Bessel functions of
2209-428: The two linearly independent solutions to the modified Bessel's equation : x 2 d 2 y d x 2 + x d y d x − ( x 2 + α 2 ) y = 0. {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y=0.} Unlike
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