Chapter 6 Bessel functions Bessel functions appear in a wide variety of physical problems. For example, separation of the Helmholtz or wave equation in circular cylindrical coordinates leads to Bessel’s equation. Bessel’s eq. The solutions of the Bessel’s eq. are called Bessel functions. In Chapter 3, we get the series solution of the above eq.

6.1 Bessel Functions of the First Kind, * Generating function, integer order , J n (x) Although Bessel functions are of interest primarily as solutions of differential equations, it is instructive and convenient to develop them from a completely different approach, that of the generating function. Let us introduce a function of two variables, (6.1)

Expanding it in a Laurent series, we obtain (6.2) The coefficient of t n , J n , is defined to be Bessel function of the first kind of integer order n. Expanding the exponentials, we have (6.3) Setting n =r- s , yields (6.4) Since for a positive integer m) ( note:

The coefficient of t n is then we have The coefficient of t n is then (6.5) This series form exhibits the behavior of the Bessel function J n for small x. The results for J 0 , J 1 , and J 2 are shown in Fig.6.1. The Bessel function s oscillate but are not periodic. . Figure 6.1 Bessel function, , and

Eq.(6.5) actually holds for n < 0 , also giving (6.6) Since the terms for s<n (corresponding to the negative integer (s-n) ) vanish, the series can be considered to start with s=n . Replacing s by s + n , we obtain (6.7 ) These series expressions may be used with n replaced by v to define J v and J -v for non-integer v . * Recurrence relations Differentiating Eq.(6.1) partially with respect to t , we find that (6.9)

and substituting Eq.(6.2) for the exponential and equating the coefficients of t n- 1 , we obtain (6.10) This is a three-term recurrence relation. On the other hand, differentiating Eq.(6.1) partially with respect to x , we have (6.11) Again, substituting in Eq.(6.2) and equating the coefficients of t n , we obtain the result (6.12) As a special case, (6.13)

Adding Eqs.(6.10) and (6.12) and dividing by 2, we have (6.14) Multiplying by x n and rearranging terms produces (6.15) Subtracting Eq.(6.12) from (6.10) and dividing by 2 yields (6.16) Multiplying by x -n and rearranging terms, we obtain (6.17)

*Bessel’s differential equation Please verify the follow result in class. In particular, we have shown that the functions Jn defined by our generating functions, satisfy Bessel’s eq., and thus are indeed Bessel functions

• Integral representation A particular useful and powerful way of treating Bessel functions employs integral representations. If we return to the generating function (Eq. (6.2)), and substitute t = e iθ , (6.23) in which we have used th e relations (6.24) and so on.

By employing the orthogonality properties of cousine and sine, In summation notation (6.25) equating real and imaginary parts, respectively. It might be noted that angleθ (in radius) has no dimensions. Likewise sinθ has no dimensions and function cos(xsinθ) is perfectly proper from a dimensional point of view. By employing the orthogonality properties of cousine and sine, (6.26a) (6.26b) in which n and m are positive integers (zero is excluded), we obtain

If these two equations are added together (6.27) (6.28) If these two equations are added together (6.29) As a special case, (6.30)

repeats itself in all four quadrants ( ), we may write Eq. (6.30) as Nothing that repeats itself in all four quadrants ( , ), we may write Eq. (6.30) as (6.30a) On the other hand, reverses its sign in the third and fourth quadrants so that Adding Eq. (6.30a) and i times Eq. (6.30b), we obtain the complex exponential representation (6.30b) (6.30c) This integral representation (Eq. (6.30c)) may be obtained somewhat more directly by employing contour integration.

• Example 6.11 Fraunhofer Diffraction, Circular Aperture In the theory of diffraction through a circular aperture we encounter the integral (6.31) for , the amplitude of the diffracted wave. Here is an azimuth angle in the plane of the circular aperture of radius a, and , is the angle defined by a point on a screen below the circular aperture relative to the normal through the center point. The parameter b is given by (6.32) with the wavelength of the incident wave. The other symbols are defined by Fig. 6.2 From Eq. (6.30c) , we get (6.33)

Figure 6.2 Fraunhofer diffraction –circular aperture

Equation (6. 15) enables us to integrate Eq. (6 Equation (6.15) enables us to integrate Eq. (6.33) immediately to obtain (6.34) The intensity of the light in the diffraction pattern is proportional to and (6.35) 6.2 Orthogonality

For v > 0, J v (0)=0. Thus, for a finite interval [0, a ], when is the m th zero of J v (i.e. ), we are able to have if m ≠ n , (6.49) This gives us orthogonality over the interval [0, a ]. * Normalization The normalization result may be written as (6.50) * Bessel series If we assume that the set of Bessel functions ( v fixed, m =1,2,… ) is complete, then any well-behaved function may be expanded in a Bessel series

The coefficients c vm are determined by using Eq.(6.50), (6.51) The coefficients c vm are determined by using Eq.(6.50), (6.52) * Continuum form If a → ∞, then the series forms may be expected to go over into integrals. The discrete roots become a continuous variable . A key relation is the Bessel function closure equation (6.59)

Figure 6.3 Neumann functions , , , and

6.3 Neumann function, Bessel function of the second kind, From the theory of the differential equations it is known that Bessel’s equation has two independent solutions, Indeed, for non-integral order v we have already found two solutions and labeled them and ,using the infinite series (Eq. (6.5)). The trouble is that when v is integral Eq.(6.8) holds and we have but one independent solution. A second solution may be developed by the method of Section 3.6. This yields a perfectly good solution of Bessel’s equation but is not the usual standard form. Definition and series form As an alternate approach, we have the particular linear combination of ' and (6.60)

This is Neumann function (Fig. 6.3). For nonintegral v , clearly satisfies Bessel’s equation, for i t is a linear combination of known solutions, and To verify that , our Neumann function or Bessel function of the second kind, actually does satisfy Bessel’s equation for integral n , we may process as follows. L’Hospital’s rule applied to Eq. (6.60) yields (6.65)

Differentiating Bessel’f equation for with repect to v , we have (6.66) Multiplying the equation for by (-1) v , substracting from the equation for (as suggested by Eq. (6.65)), and taking the limit , we obtain (6.67) For , an integer, the right-hand side vanishes by Eq. (6.8) and is seen to be a solution of Bessel’s equation. The most general solution for any v can be written as (6.68) Example Coaxial Wave Guides We are interested in an electromagnetic wave confined between the conducting cylindrical surfaces concentric , and . Most of the mathematics is worked out in Section 3.3. From EM knowledge,

( : electrical field along z axis) Let , we have This is the Bessel equation. If ,the solution is with . But, for the coaxial wave guide one generalization is needed. The origin is now excluded ( ). Hence the Neumann function may not be excluded. becomes (6.79) With the condition (6.80) we have the basic equatios for a TM (transverse magnetic ) wave.

The (tangential) electric field must vanish at the conducting surfaces (Direchlet boundary condition) or (6.81) (6.82) these transcendental equations may be solved for and the ratio . From the relation (6.83) and since must be positive for a real wave, the minimum frequency that will be propagated (in this TM mode) is (6.84) (6.82). This is with fixed by the boundary conditions, Eqs. (6.81) and the cutoff frequency of the wave guide.

6.4 Hankel function Many authors perfer to introduce the Hankel functions by means of integral representations and then use them to define the Neumann function, . We here introduce them a simple way as follows. As we have already obtained the Neumann function by more elementary (and less powerful) techniques, we may use it to define the Hankel functions, and : (6.85) (6.86) This is exactly analogous to taking (6.87)

6.5 Modified Bessel function , and For real arguments and are complex conjugates. The extent of the analogy will be seen better when the asymptotic forms are considered . Indeed, it is their asymptotic behavior that makes the Hankel functuions useful! 6.5 Modified Bessel function , and The Helmholtz equation, separated in circular cylindrical coordinates, leads to Eq. (6.22a), the Bessel equation. Equation (6.22a) is satisfied by the Bessel and Neumann functions and and any linear combination such as the Hankel functions and .Now the Helmholtz equation describes the space part of wave phenomena. If instesd we have a diffusion problem, then the Helmholtz equation is replaced by . (6.88)

The analog to Eq. (6.22a) is (6.89) The Helmholtz equation may be transformed into the diffusion equation by the transformation . Similarly, changes Eq. (6.22a) into Eq. (6.89) and shows that The solution of Eq. (6.89) are Bessel function of imaginary argument. To obtain a solution that is regular at the origin, we take as the regular Bessel function . It is customary (and convenient) to choose the normalization so that (6.90) (Here the variable is being replaced by x for simplicity.) Often this is written as (6.91)

In the terms of infinite series this is equivalent to removing the Series form In the terms of infinite series this is equivalent to removing the sign in Eq. (6.5) and writing (6.92) The extra normalization cancels the from each term and leaves real. For integral v this yields (6.93) Recurrence relations The recurrence relations satisfied by may be developed from the series expansions, but it is easier to work from the existing recurrence relations for . Let us replace x by –ix and rewrite Eq. (6.90) as (6.94)

Then Eq. (6.10) becomes Repalcing x by ix , we have a recurrence relation for , Equation (6.12) transforms to (6.95) (6.96) From Eq. (6.93) it is seen that we have but one independent solution when v is an integer, exactly as in the Bessel function solution of Eq. (6.108) is essentially a matter od convenience. We choose to define a second solution in terms of the Hankel function by (6.97)

The factor makes real when x is real. Using Eqs. (6. 60) and (6 The factor makes real when x is real. Using Eqs. (6.60) and (6.90), we may transform Eq. (6.97) to (6.98) analogous to Eq. (6.60) for The choice of Eq. (6.97) as a definition is somewhat unfortunate in that the function does not satisfy the same recurrence relations as . To avoid this annoyance other authors have included an additional factor of cos . This permits satisfy the same recurrence relations as , but it has the disadvantage of making for . To put the modified Bessel functions and in proper perspective, we introduce them here because: 1. These functions are solutions of the frequently encountered modified Bessel equation. 2. They are needed for specific physical problems such as diffusion problems.

6.6 Asmptotic behaviors Figure 6.4 Modified Bessel functions Frequently in physical problems there is a need to know how a given Bessel or modified Bessel functions for large values of argument, that is, the asymptotic behavior. Using the method of stepest descent studied in Chapter 2, we are able to derive the asymptotic behaviors of Hankel functions ( see page 450 in the text book for details) and related functions:

(6.99) 1. 2. The second kind Hankel function is just the complex conjugate of the first (for real argument), (6.100) 3. Since is the real part of (6.101) 4. The Neumann function is the imaginary part of , or (6.102) 5. Finally, the regular hyperbolic or modified Bessel function is given by (6.1 0 3) or (6.104)