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J1(P)
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See also:J1(P) = -dJoP(P), Yi(P) dYo(p) 22. See also:Bessel's Functions as Coefficients in an Expansion.—It is clear that e'P See also:COS 4' =e'x or e'P See also:sir' 0=e," satisfy the See also:differential See also:equation (31), hence if these exponentials be See also:expanded in See also:series of cosines and sines of multiples of ¢, the coefficients must be Bessel's functions, which it is easy to see are of the first See also:kind To expand e'P See also:sin 4', put e'4' =t, we have then to expand e_P('_,-i) in See also:powers of t. Multiplying together the two absolutely convergent series \ e °7-1ml(2)mtm,e LP` - ( m~m 12p1 tm, we obtain for the coefficient of tm in the product l\ J 4 2mm! 2.201+2+2.4.2171+2.2m+4- ... } or Jm(P), hence the Bessel's functions were defined by Schlomilch as the coefficients of the powers of t in the expansion of elp(7-i '), and many of the properties of the functions can be deduced from this expansion. By differentiating both sides of (32) with respect to t, and equating the coefficients, of t"'-' on both sides, we find the relation J.-1(p) +Jm+i(P)=2p Jm(P), which connects three consecutive functions. Again, by differentiating both sides of (32) with respect to p, and equating the coefficients of corresponding terms, we find 2dr (P)-m 1(P)-Jmi(P)- In (32), let t=e b, and equate the real and imaginary parts, we have then cos (p sin 0)=Jo(P)+2Jz(p) cos 24)+2J3(P) cos 30+... sin (p sin 0) =2Ji(p) sin ¢+2J3(p) sin 30+.. , wee obtain expansions of cos (p cos 0), sin (p cos 0), by changing 4) into 1-4. On comparing these expansions with See also:Fourier's series, we find expressions for J,,,(p) as definite integrals, thus Jo(P) = J u cos (psin ¢)d¢, J,,,(p) _; fcos (p sin 0) cosmrbdetz. (m even) Jm(P) =,r' j sin (p sin ¢) sin m¢d¢ (m See also:odd). It can easily be deduced that when m is any See also:positive integer i,, (P) = jo cos (m0-p sin 0)1¢. 23. Bessel's Functions as Limits of See also:Legendre's Functions.—The See also:system of orthogonal surfaces whose parameters are cylindrical co-ordinates may be obtained as a limiting See also:case of those whose parameters are polar co-ordinates, when the centre of the See also:spheres moves off to an indefinite distance from the portion of space which is contemplated. It would therefore be expected that the normal forms e ±(Jm(Xp)s nm4) would be derivable as limits of r nd'IPn (cos 8)s mip, and we shall show that this is actually the case. If 0 be the centre of the spheres, take as new origin a point C on the See also:axis of z, such that OC =a; let P be a point whose polar coordinates are r, 0, 4) referred to 0 as origin, and cylindrical co-ordinates p, z, 4) referred to C as origin; we have p = r sin B, z = r cos 0 -a, hence (a) "P"(cose) = see() (1+0 "P"(cos 0) . Now let 0 move off to an See also:infinite distance from C, so that a becomes ( m-1 (2) .L ( ( ./II(n ni)II(n) 2) z" "=0 (31) or thus we have e'P('-d-1) = Jo (P) +tJ1(P) + . - . +tmJ m (p) + .. . —t 1Ji(n)+. .+(—I)'"t-mJ,n(P) (32) = F'tmJm(P) infinite, and at the same See also:time let n become infinite in such a way that n/a has a finite value X. Then as n L sec" B=L (sec a) = 1, L (t +a!) =eM' and it remains to find the limiting value of P,(cos 0). From the series (15), it may be at once proved that Ps(cos 0) = - (n+!) n (sin 2) 2+.. . 1 (—1)m8(n+mlz.2(m2m+I) (sin 2) zm where S is some number numerically less than unity and m is a fixed finite quantity sufficiently large; on proceeding to the limit, we have LP', (COS np) =I ---+224.1-...+(-I)m8122 42 (2m)2 where Si is less than unity. Hence L Pn (cos?P) = Jo(7sP)• n-a0 n d (- h2) m (-2)mPm ( ) d(PL)m hence L n "'Pn (cos P) =J,(P)• n- so n It may be shown that Yo (p) is obtainable as the limit of Qs (cos n) the zonal See also:harmonic of the second kind ; and that Ym(p) =Ln "'Q: (cos ) . 24. Definite Integral Solutions of Bessel's Equation.--Bessel's equation of See also:order m, where m is unrestricted, is satisfied by the expression p'° f e"e'(12- I)m-Idt, where the path of integration is either a See also:curve which is closed on the See also:Riemann's See also:surface on which the integrand is represented, or is taken between limits, at each of which e,Pi(t2-1)TM+f is zero. The equation is also satisfied by the expres- See also:sion f ewP(t - I 1)t TM-1dt where the integral is taken along a closed path as before, or between limits at See also:earn of which e12P(t-t 1)t_m_I vanishes. The following definite integral expressions for Bessel's functions are derivable from these fundamental forms. I i P) m f ~e'P cos ci sin 2"' J m(P)=II(-i)II(m-a) 0 where the real See also:part of m+z is positive. Ym(p)+zai.emn'sec ma.J„(p) m—1—n0 n(1 ) (2) '"f o +k r. ecp cosh 4, sinh 2mOdd) where the real parts of m+i, p are positive; if p is purely imaginary and positive the upper limit may be replaced by co . Ym(p)— ar.en— sec m7r.Jm(p) 2mnil"I(--in) (P) rn /' _e II(- ) 2 J o ;pa sinh 2'4;4 Ln mm has been introduced for them. We denote the two solutions of the equation by Io(r), Ko(r) when Io(r) = Jo(1r) =1+5 + 22'-424- + .. . _! f o cosh (r cos 0)4, and Ko(r) =Yo(or)+212rJo(Ir) =f: e`r cos h+dsp = f 1 cos (r sinh 1/')d#. The particular integral Ko(r) is so chosen that it vanishes when r is real and infinite; it is also represented by a0 cos V J o (v2+See also:r2)dv, (U e-"" J.° I (u2-1)du. The solutions of the equation dr2+s dr (I +5) uo are denoted by See also:Im(r),,( Km(r), where ))} Im(r)=2mII(m) I+2.2m-{-2+2.4.2m+2.2m+4+... =(2r)" w„Io(r), )) when m is an integer, and Km(r) _ (2Y)md 2),nKo(r) =e-#lm+r. Ym(ir)+ZwrJm(4r) 26. The Asymptotic Series for Bessel's Functions.—It may be shown, by means of definite integral expressions for the Bessel's functions, that j( )} Jm(P) = P P cos (211+5—0 + Q sin (2~+I-P) Ym(P) = . / e "' sec ma P sin (2'r+ -P) -Q cos (2F+ -P) where P and Q denote the series pI _(4m2-12)(4mn2-32) I .2 . (8p)2 + (4m2 - 12) (4m2 - 32) (4m2 - 52) (4m2 - 72) I.2.3.4(8p)' 4m5-12_ (4m2-12)(4m2-32) (4m2-52) Q- I.Bp I.2.3.(8p)3 -~.. These series for P, Q are divergent unless m is See also:half an odd integer, but it can be shown that they may be used for calculating the values of the functions, as they have the See also:property that if in the calculation we stop at any See also:term, the See also:error in the value of the See also:function is less than the next term; thus in using the series for calculation, we must stop at a term which is small. In such series the See also:remainder after n terms has a minimum for some value of n, and for greater values of n increases beyond all limits; such series are called semi-convergent or asymptotic. We have as particular cases of such series: Jo(P) I a 12.(-8p32)2+ 11.22.332 ..52.72 = \1,~d2p cos (4 - p) ) I.24(8p)4 .. Again, since we have P ':(cos p) =sin"' Bd"P,(cos 0) d(cos 0)"m dTMPn LnmPn (cos') = (cos!) d2u I du dr2+r dr-=o and by We find also Im(r) -" 1 .3.5.. (2m— 1) ,Jo cosh (r cos 0) sinzm¢d¢ Km(r) ( - I)mrm f° . 3 .5 . . . (2m -I) o e rcosh4 sinh 2.4,4 I =(-1)"'3.5.. .2m-Ir"` au cos u o (u2+r2)n+f u. under the same restrictions as in the last case; if p is a negative imaginary number, we may put so for the upper limit. Additional information and CommentsThere are no comments yet for this article.
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