Lecture 4b. Bessel functions. Introduction. Generalized factorial function. 4b.1. Using integration by parts it is easy to show that

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1 4b. Lecture 4b Using integration by parts it is easy to show that Bessel functions Introduction In the previous lecture the separation of variables method led to Bessel's equation y' ' y ' 2 y= () 2 Here we use instead of m to emphasize the complete generality of the separation constant (arbitrary comple number). Our goal in this lecture is to obtain the general solution of this equation. The resulting Bessel functions are among the most commonly encountered so-called special functions. They naturally arise in problems in cylindrical coordinates so are sometimes called cylinder functions. However, we will see that they also come up in spherical coordinates other applications. Before attempting a rigorous solution, let's eamine the qualitative behavior of the solutions to this equation. When gets very large 2 / 2 so the functions will approimately satisfy y ' ' y ' y= (2) This is independent of the parameter. Moreover if we neglect the y ' / term (not rigorously valid, but we are just getting a feel ) we have y ' ' y =. The solutions to this are simply cos, sin. It appears that as the solutions for any value of will oscillate like some combination of cos,sin. On the other h, when gets very small 2 / 2 2 / 2 the functions will approimately satisfy y ' ' y ' 2 2 y = (3) The solutions to this are y=, when y=, ln when =. So when it appears that one solution will go to zero as while the other will blow up like. For the special case =, one solution will approach a constant value while the other will blow up like ln. Generalized factorial function In developing our solutions we will need the generalized factorial function. This is! = t e t dt (4) Figure : Eample solutions to Bessel's equation. Thick (red) curves have =. Thin (blue) curves have =.5. Solid curves display behavior at while dotted curves display behavior. Since we have!=! (5) e t dt ==! (6) m!= t m e t dt (7) for m=,,2, where m!=m m m 2 is the regular integer factorial. Alternately we can use the gamma function write where! = (8) = t e t dt (9) Most math packages will evaluate either the generalized factorial (e.g., Maima) or the gamma function (e.g., Scilab). The generalized factorial function is well-defined for all real numbers ecept the negative integers where it blows up. Indeed! =!= () implies! =. Then! = 2! etc. require that m!=± for m=,2,. A plot of the factorial function is given in the following figure. A useful value in some applications is / 2!=/2. An approimation valid for large values is the Stirling formula!~2 /2 e () EE58: Advanced Electromagnetic Theory I Scott Hudson

2 4b.2 For 8 this is good to within %. r=± (8) This will give us two solutions with the behavior y=, that we would epect from our initial analysis. If = 2 is not an integer, then these are guaranteed to be linearly independent. That is, should not be integer or half integer. (More on this later.) The coefficient of the r term is [ r 2 2 ] a = (9) Ecept for the special case r 2 = r 2 which requires r= / 2 (more on that later) we must have a = (2) The coefficient of the r n term is Figure 2: Generalized factorial function! Bessel functions of the st kind Now let's solve the Bessel equation (). The functions p =/ q = 2 / 2 are singular at = but p, 2 q are analytic, so we need to use the Frobenious method. We look for solutions of the form y = r a n n = a n rn (2) where by assumption a. It is convenient to multiply () through by 2 to obtain Substituting the series 2 y ' ' y' 2 2 y= (3) y ' = r na n r n n = (4) y' ' = rnrn a n r n 2 into the differential equation we obtain Since [rnrn a n r na n 2 a n ] r n a n rn 2 = (5) rn r n rn 2 = r n 2 2 (6) we can write this as The coefficient of the r This requires [r n 2 2 ]a n rn a n 2 rn = (7) n =2 term is r 2 2 a = This requires [ n± 2 2 ] a n a n 2 = (2) a n 2 a n = n± 2 2 (22) We see that a =a 3 = a 5 = = so all odd-numbered coefficients are zero. Since n± 2 2 =n 2 ± 2n =n n± 2 (23) if we call n=2 k we can write a 2k 2 a 2k = 4 kk ± (24) for our coefficient recursion. Of the two solutions r=± let's start with r=. The first coefficient is arbitrary. Let's take then, since we will have Our solution is therefore a = 2! a 2 = 2! 4 a 4 = 2! = =2!! 2= 2! (25) (26) (27) a 2k = k (28) k! k! 2 2 2k EE58: Advanced Electromagnetic Theory I Scott Hudson

3 J = 2 k= k k!k! 2 (29) We call this the Bessel function of the first kind of order. The process with r= results in J = 2 k = k k!k! 2 Provided is not an integer, a general solution of () is (3) y =a J b J (3) (We'll see below that half-integer values present no problem.) Since [ k!k! 2 2 k ][ k!k! 2 ] (32) /2 2 = k k as k for any fied, we have that by the ratio test our series solution converges for all values of. Bessel functions of the second kind In the Frobenius method, if the two r values differ by an integer then our second solution may not be linearly independent. If r==m for non-negative integer m then one solution is J m = 2m k = Our other solution is (formally) J m = 2 m k = k!km! k 2 k!k m! k 2 (33) (34) However k m will take on negative integer values when k m. Since k m! = in those cases the corresponding terms will vanish. We are left with J m = 2 m k=m k k! k m! 2 Calling k =nm we can write this as J m = 2 m nm!n!2 2n2m n!nm! n 2 = m 2m = m J m 2n (35) (36) Therefore J m is clearly not linearly independent of J m we need to find another solution for the integerorder case. Following the Frobenius method, the second solution will have the form 4b.3 Y m =ln J m m b k k (37) k = Writing this as Y m =ln J m u plugging it into the differential equation results in 2 Y m ' ' Y m ' 2 m 2 Y m = (38) 2 u' ' u' 2 m 2 u= 2 J m ' (39) for the inhomogeneous differential equation satisfied by u= m b k k (4) k = We can put this series into the LODE solve for the coefficients b k obtain the second solution for the integerorder case. However, it is awkward to use J, J as solutions for not an integer then have to switch to J m,y m for an integer. A way around this is to use the Bessel function of the first kind, J, as one solution then to define the Bessel function of the second kind as Y J cos J sin (4) for any value of. If is not an integer then this is simply a linear combination of J, J, so it solves the Bessel equation is linearly independent of J. For an integer it becomes Y m = J m m J m = sinm (42) so we need to take the limit. We can do using L'Hospital's rule. We obtain [ [ J cos J ] ] Y m =lim m sin (43) =lim m J m J Since J contains a factor of / = ln we see that Y m will contain the ln behavior that we would get by using the Frobenious method. Taking derivatives with respect to of the J, J series is complicated, but doable. Following through with the calculation one obtains EE58: Advanced Electromagnetic Theory I Scott Hudson

4 4b.4 Y m = 2 ln 2 J m 2 m k = 2m k= m m k! where h =, h k = 2 3 k Euler's constant. k! 2 2k k k!km! h kh k m Fourier series integral forms 2 (44) =.5772 is Bessel functions also arise in certain Fourier series, this leads to useful integral forms for the functions. Consider the function cos sin which comes up in some antenna problems, among others places. Applying the Taylor series for cos to this function we have sin 2 sin 4 sin 6 cos sin = (45) 2! 4! 6! Now sin 2 = cos 2 2 sin 4 3 4cos 2 cos 4 = 8 sin 6 5 cos 2 6 cos 4 cos 6 = 32 (46) so on. Collecting all terms with the same dependence we obtain cos sin=[ 2 2 2! ! ! ] cos 2[ 2 2 2! ! cos 4[ 8 4 4! ! ] 6! ] (47) This is a Fourier series for cos sin. The coefficients are functions of which correspond to the power series of various Bessel functions as can be verified by comparing with (33) cos sin = J 2 J 2 cos 2 2 J 4 cos4 (48) Likewise, one finds sin sin =2 J sin 2 J 3 sin 3 (49) Applying the normal method of calculating Fourier coefficients we can therefore write 2 Finally, since cos sin cos m d ={ J m m even m odd 2 sin sin sinm d ={ J m m even m odd (5) (5) cos sin cos msin sin sin m=cos sin m (52) we have the integral form of J m Equivalent forms are J m = 2 cos sin m d (53) J m = cos sin m d (54) J m = e j sin m d (55) 2 These are valid only for non-negative integer values m. Using comple contour integration more general integral forms can be found (see the Carrier reference for details). For arbitrary one finds J = cos sin d sin e sinh d Y = sin sin d e e cos e sinh d (56) (57) Note that the first of these reduces to our result for =m an integer. Asymptotic behavior Asymptotic behavior in the limits, is important in many applications. The forms follow directly by taking the first terms of the series solutions. We have J ~ J ~! 2 with = Y ~ 2 2 ln Y ~! 2 (58) EE58: Advanced Electromagnetic Theory I Scott Hudson

5 4b.5 Now let's treat the limit. It's instructive to begin by considering the substitution y=u/ as it transforms () into u' ' 2 /4 u= (59) 2 When =/2 this becomes simply u ' 'u= which has solutions sin, cos. So for =/2 Bessel's equation is eactly solved by sin / cos /. Multiplying by constants to get the behavior of (58) we obtain J /2 = 2 sin Y /2 = 2 cos (6) Keep in mind that these are not asymptotic epressions, they are eact for all. We see that at least some of the Bessel functions are quite simple! Similarly one can show that J /2 = Y /2 Y /2 =J / 2. For arbitrary values of "asymptotic analysis" provides the asymptotic forms (see Carrier for details) J ~ 2 Y ~ 2 cos [2 ] 4 sin [2 ] 4 (6) Unlike the =/2 case, these are in general only valid for. Recursion derivative formulas An interesting result can be obtained by considering the sum J J Using (29) the 2k term of this function will be 2 k k!k! 2 2 k!k! 2 2 2k k [ k k ] k!k! = 2 2 = k k k!k! (62) This is just 2 / times a term from the J series. We therefore have J J = 2 J (63) An almost identical process shows that J J =2 J ' (64) Using these the definition of Y one can show that Y satisfies the same relations. So, with B representing either J or Y, or indeed any linear combination of these, we have B = 2 B B (65) B ' = 2 [ B B ] (66) The recursion (65) is very useful. For eample, if are able to calculate just J J then we can use (65) to calculate J 2, J 3, directly from J J without having to use series solutions for all of them. The derivative relation (66) is quite useful because Bessel functions derivatives arise in many applications. This allows us to calculate those directly from the Bessel functions. Half-integer-order Bessel functions Starting with J /2 = 2 cos J / 2 = 2 sin (67) we can use (65) to obtain J m /2 for any integer m. For eample J 3/2 = 2/2 J /2 J / 2 = 2 sin cos [ ] so on. From the definition of Y Y m /2 = J m /2 cosm/2 J m /2 sinm/2 = m J m /2 (68) (69) We see that the half-integer Bessel functions are actually just finite sums of elementary functions. These functions naturally arise in spherical coordinates, we will call something closely related to them spherical Bessel functions. Wronskian In Bessel's equation p =/. Therefore the formula for the Wronskian, equation (23) of Lecture c, is W =W e dt t =W (7) EE58: Advanced Electromagnetic Theory I Scott Hudson

6 4b.6 So the Wronskian behaves as a constant times the inverse of. Since this is true for all, we can consider use the small-argument asymptotic forms Therefore J Y ' J ' Y ~! 2! = ! 2! J Y ' J ' Y = 2 for all. This is useful in some applications. Hankel functions 2 (7) (72) The asymptotic forms of the Bessel functions Euler's formula cos j sin=e j suggest that combinations of the form J ± j Y might be useful. We are led to define the Hankel functions of the st 2 nd kind as H =J jy H 2 =J j Y (73) so the st 2 nd kind of Bessel functions can be written as linear combinations of the Hankel functions. A general solution to Bessel's equation can be formed from a linear combination of any two of the four functions J,Y, H, H 2. Note that for real non-negative, of these four functions only J is finite at =. The other three functions have a singularity there. So if we seek a solution that is finite everywhere, it must be use only J. References. Lebedev, N. N. Special Functions Their Applications. Dover Publications 972. ISBN Carrier, G. F., M. Krook C. E. Pearson. Functions of a Comple Variable. Hod Books 983. ISBN Bowman, F., Introduction to Bessel Functions. Dover Publications 958. ISBN Their asymptotic forms follow from (6) H ~ j 2 2 H 2 ~ j 2 2 Here we've used e j e j (74) e j m 2 2 = j m 2 (75) These are useful, particularly in scattering radiation problems, because they correspond to cylindrical waves. For eample H 2 ~ j 2 2 e j (76) represents a wave that travels radially outward from the z ais toward =. Since the Hankel function are linear combinations of solutions to Bessel's equation they are solutions to Bessel's equation. If are real then J,Y are real. In this case H, H 2 are comple conjugates of each other. Also, we have H H 2 = 2 J H H 2 = j 2Y (77) EE58: Advanced Electromagnetic Theory I Scott Hudson