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by: Dr. Simeon Wiza


Dr. Simeon Wiza
GPA 3.96

Stephen Ellis

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Stephen Ellis
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This 9 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 228 at University of Washington taught by Stephen Ellis in Fall. Since its upload, it has received 37 views. For similar materials see /class/192435/phys-228-university-of-washington in Physics 2 at University of Washington.


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Date Created: 09/09/15
Lecture 23 Frobenius and Bessel More of Chapter 12 in Boas In our previous discussions we have focused on the case where we solved a differential equation Via a Taylor series expansion about a regular point the origin of the equation Now we want to consider the case where we expand about the origin when it is a regular singular point The general technique due to Frobenius see 1211 in Boas essentially corresponds to de ning a Laurent expansion about the origin although the singularity in the resulting solution may be a branch point rather than a simple pole The technique involves replacing our Ansatz of a simple power series with a powers series times a power yxianxquot vianxw 231 110 110 The exponents is then chosen so that the n 0 term yields a solution of the equation Since we are considering second order equations there will typically be two values for s corresponding to different behavior at the origin As suggested in our earlier discussions typically one solution is well behaved at the origin while the other solution is not well behaved The former solution is the one of physical interest for problems that include the origin in the region where the solution is expected to be finite As an interesting example of this behavior we now focus on Bessel s equation see 1212 1217 in Boas The differential equation studied by Bessel arises in several contexts in physics and the solutions display a variety of useful properties of which we will make only a brief survey Consider first our friend Laplace s equation but now in cylindrical coordinates Recall that in this case we are assuming that the underlying physics has the symmetry structure of a cylinder We keep the zaxis but reexpress the x and y coordinates in terms of a radial distance perpendicular to the zaxis and azimuthal angle around the zaxis In contrast to the notation of Boas we will use a more standard notation for these variables which will serve to distinguish them from spherical coordinates in our notation the angle 0 S lt 27 is an azimuthal periodic O is the same as 27 angle in both spherical and cylindrical coordinates while 0 S 6 S 7 is a polar angle and appears only in spherical coordinates We define Physics 228 Lecture 23 1 Winter 2008 x2 y2 p2tan i y 232 x pcos y psin In this notation Laplace s equation is 2 2 L a Tj 1 a I a 2 PO 233 1 a v2 p M a pap p26 2 62 pp As in the spherical case we assume that separation of variables is possible P Rp Z 2 As before this is only possible if after the separation 2 iii pile ii2 d2 Zz0 234 R pdp dp QDp 61 Z dz the various terms properly normalized are all equal to constants In particular the third term which depends only on 2 and not the other variables must be a constant We can also infer the separation constants from the eXpected forms of the various functions For infinite range of the 2 variable 00 lt z lt 00 we think in terms of real eXponentials Z 2 em Laplace style behavior but not orthogonal functions and write 1 d2 E322 k2 235 where the parameter k has units of inverse distance but is otherwise unconstrained Similarly we eXpect to describe the periodic dependence in terms of the compleX eXponentials eh or sin 17 cos 17 orthogonal functions so that 1 d2 2 D 236 where the periodic boundary conditions require that p is an integer Thus after separating out the z and dependence the cylindrical radial equation looks like after multiplying through by p2 Physics 228 Lecture 23 2 Winter 2008 d d 2 2 2 p5p Rpk p p Rp0 237 We obtain the usual form of Bessel s equation if we define a variable x kp a dimensionless variable unlike p and replace R p gt yx to obtain xzynxyrx2p2y0 2 238 yquotly39l p 2y0 x x From the second canonical form of the equation we see as expected that the origin is a regular singular point Note that from the way we derived this equation using cylindrical coordinates we eXpect that the periodicity in will require that p is an integer Indeed the original study by Bessel defining the usual Bessel function does correspond to integer values of p However the equation is thoroughly studied for general values of p and we will show now that halfinteger values of p are also of immediate interest In particular before proceeding to solve Bessel s equation let us take a brief detour to see how the same equation arises in the case of spherical coordinates Consider again Laplace s equation in spherical coordinates but now allow a nonzero righthandside This typically arises from some nonzero time dependence in either a diffusion problem see eg 1333 in Boas Vz P 10t2d Pdt or the wave equation V 1112d2 1 dl 2 Again we assume that separation of variables applies P Rrqgt 9T t With the assumption that the time dependence is exponential e g T em in the wave equation case T e kz xz in the diffusion case we can write the righthandside of the equation as V k2 the Helmholtz equation where as above k has the units of inverse distance in the wave equation case k is the wave number k 012 with v the velocity of the wave Assuming that we treat the 6 and dependence as in the discussion of the Associated Legendre equation in the previous Lecture the radial part of the equation becomes 1 21Rrlll k2r2Rr 239 dr dr Physics 228 Lecture 23 3 Winter 2008 As above we de ne x kr and since we know the answer make the replacement R r gt yxs Now the radial equation becomes x2Dx2 ll 1l 0 Ir I x 2 yT 24 2 x J17 4 J x2 ll1l0 J 1 2 xzyquotxy39x2 IE y 0 We recognize this as Bessel s equation but with p gt 1 12 a halfinteger value The spherical radial behavior of the inhomogeneous Laplace equation is given by a Bessel function of halfinteger order divided by J This form is called the spherical Bessel function We will return to this point below Nlt A 3x y L3 5 2310 Returning to the original equation of Bessel Eq 238 we substitute the form suggested by Frobenius Eq 231 to find 2aquot n sn s lxquotH an n sxquot x2 p2anxquot 710 2311 O 3 Zan s2 p2 an72xquot 0 172 2 ail 0 710 where the second term an2 only contributes for n 2 2 as noted As with our study of the Legendre equation the solutions split into two forms one based on even values of n and one based on odd values Consider the special case n 0 ie the coefficient of x5 which since there is no second term requires that s2 p2 gtsip 2312 Physics 228 Lecture 23 4 Winter 2008 Since we are initially interested in solutions that are well behaved at the origin x 0 we focus on s p 2 0 Note from the discussion above that p is not determined by this radial equation but rather is determined by an eigenvalue problem based on the angular behavior Here we are simply looking for the solutions of the radial equation which are characterized by their behavior at the origin and at infinity The previous equation provides us with a recursion relation for the coefficients Choosing as noted s p we have aniz aniz an4 np2p2 nn2p nn 2n2pn2p 2 2313 3 a2 1m do 22mm rmp139 With the arbitrary but conventional choice a0 1 21 Fl 17 we obtain the Bessel function of the first kind of order p J x w LET 2314 rm1rmp1 2 o m Clearly the Bessel function of order p behaves as xP as x gt 0 Note that changing the order by one p gt p l corresponds to keeping the terms with odd powers of x in the original sum ie we have included both types of solutions in this expression Due to the alternating signs in this series the Bessel function is an oscillating function much like the sine and cosine However in this case the distance between successive zeroes is not precisely a constant but they are tabulated Here are plots of J0x and J1 J0x Physics 228 Lecture 23 5 Winter 2008 The two ends of these plots the asymptotic behavior can be characterized simply as lim 0 J x 4 f P 1ox2 H P Fl p 2 2315 lime JP x cos x 111 O 7rx x From this last result we see that at least asymptotically the spacing between the zeros of J p x approaches 7 The second solution of the Bessel equation can be constructed from the case s p For noninteger values of p Jip x is an independent function and as suggested above is singular at the origin Jp x gtx P For the case of integer p values xgt0 we have due to the magic of the Gamma function that LP x 1P JP x p integer 2316 and more care is needed to define a second independent solution The standard form found in the literature is named after Neumann or Weber N xY Wkw 2317 where the form for integer p values is obtained by a careful limiting process The expression for integer p values exhibits a logarithmic singularity at the origin Returning to the spherical coordinate case with halfinteger p values p 1 12 the standard expression for the spherical Bessel function that is finite at the origin is note the remarkable form in terms of sinx which we will not derive here 1 d sin Jx21Jlxx gj 2318 The second spherical function singular at the origin is given by Physics 228 Lecture 23 6 Winter 2008 1 d yx J21Yl x 34 3 C x 2319 As outlined in Chapter 12 in Boas many different forms for the Bessel function have been studied and are often useful in special cases in physics including various recursion relations for the Bessel functions much as we saw for the Legendre polynomials A sample relation is d xPJP prp1 2320 We do not have the time to fully pursue these relations here As described in Chapter 13 of Boas we can put together the various functions we have studied to define solutions of the various 2nd order differential equations we meet in physics and fit the relevant boundary conditions For example see Chapter 135 consider the temperature inside a semiinfinite cylinder of uniform material with boundary conditions such that the temperature is a constant T0 on the end of the cylinder at z 0 but the temperature is held to zero on the boundary at p R0 The general solution can be eXpressed in terms of sums of the functions we have studied as in the form for positive 2 with constant coefficients Tm T MLZ ZTWJP kpeiquot e kz 2321 lap Here p must be integer valued to ensure periodic behavior in Since the specific boundary conditions have no variation in we choose p 0 The eigenvalues for k are fixed by the requirement that J0 0 on the boundary of the cylinder If the mth zero of J0 is x07quot J 0 xoym 0 recall the plots of the Bessel functions we have the eigenvalues for k given by km 2 xOYmR0 The x07quot the xpym for JP x JP xpym 0 play the same role as the constants m7 do for the sine function Note that from the large argument form of the Bessel function in Eq 2315 it follows that for m gtgt1 x x 7r The combined functions J 1 km p 6 with pml pm m and p integer valued are complete and orthogonal on the 2D surface of a disk of Physics 228 Lecture 23 7 Winter 2008 radius R0 analogously the Y m on the surface of a sphere Thus we have T 9452 ZCmJo kmpe k z 2322 ml with the coefficients specified by the boundary condition 00 T0 ltR0 z 0 2T0 Zch0 kmp ml 2T 0 m kaOJl I 2323 30 See the discussion in 135 for details The essential point for this purpose is the orthogonality relation for the Bessel functions This appears not between Bessel function of different order but rather between Bessel functions of the same order but involving different zeros ie the sum above is over m not p Recall that p is determined by the g5 dependence while In is related to the p dependence In our previous language we are working with a SturmLiouville problem for fixed p with the eigenfunctions characterized by different m values This is the analogue of the Associated Legendre problem where we observed orthogonal polynomials labeled by different values of l 2 m for a xed value of m The relevant relation is R0 2 zwnh2ph2P JJ dn l B20 Note that the righthandside is not zero for m n because the zeros of J p x are not the zeros of J1W1 Using this orthogonality above we have fpdeoJo kn3 Zcm pol3J0 kn3V0 km3 wm mAFg mm 2325 Using the recursion relation in Eq 23 20 above we have Physics 228 Lecture 23 8 Winter 2008 TpdproJo km 43cm mo 1 dx xJo xx kp T X d T m T 2 0 dxgx 1x 11001 ngiknp llwnp f 2 To J1knR0 0quot R70 J12 knRo 2326 n 2T0 2 n T knROJl knRO 0 x0nJl x0n 30 a the result claimed in Eq 23 23 Pulling the pieces together the temperature distribution inside the cylinder is given by 00 J0 kn ikz zgiw emzRo 2327 quot1 x0nJ1 x0n For our purposes the essential point is that we can write down the general solutions to diffusion equations wave equations etc in terms of a linear combination of the appropriate orthogonal and complete basis functions Then we can solve for the coefficients using the boundary conditions and the known properties of these functions Summa For physical systems in greater than 1 dimension and described by linear differential equations the method of separation of variables almost always serves to split the problem into simpler pieces one for each dimension The separate pieces in tum almost always correspond to one of the special equations we have studied with solutions that are the standard special functions we have discussed powers real eXponentials sinh s and cosh s compleX eXponentials sines and cosines Legendre polynomials spherical harmonics and Bessel functions Employing linear superposition we can write the solution to the full problem as sums of products of these functions Matching the boundary conditions including time dependence will result in eigenvalue problems for some of the separated equations but typically not all which serves to define the discrete eigenvalues over which we sum Physics 228 Lecture 23 9 Winter 2008


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