Special functions mathematics engineers pdf

When this is the case, it is useful to have tests of convergence as we did in studying infinite series. Although we could devise con- vergence tests based directly on the product, there are related infinite series whose convergence or divergence will settle the question in regard to the infinite product. If we denote the partia! A result more useful than considering the associated series of logarithms is contained in the following theorem.

The general theory of represent- ing functions by infinite products of the form 1. Remark: The notion of uniform convergence of infinite products plays an important role in the theory of in:finite products, much as it does in the theory of infinite series and improper integrais.

The usual way in which uniform convergence is established for in:finite products is by another Weierstrass M test. The interested reader should consult E. This is sometimes indeed the case. Use 1. This problem led Euler in to the now famous gamma function, a generalization of the factorial function that gives meaning to x! His result can be extended to certain negative numbers and even to complex numbers.

The notation r x that is now widely accepted for the gamma function is not due to Euler, however, but was introduced in by A. Legendre , who was also responsible for the duplication formula for the gamma function. Nearly years after Euler's discovery of it, the theory concerning the gamma function was greatly expanded by means of the theory of entire functions developed by K.

Weierstrass Because it is a generalization of n! Another function useful in various applications is the related beta function, often called the eulerian integral of the first kind. The logarithmic derivative of the gamma function leads to the digamma function, while further differentiations produce the family of polygamma functions, all of which are related to the zeta function of G. Riemann It appears occasionally by itself in physical applications mostly in the form of some integral , but much of its importance stems from its usefulness in developing other functions such as Bessel functions Chaps.

The gamma function has severa! Gauss As a consequence we have the following theorem. Theorem 2. While many of the special functions satisfy some linear differential equation, it has been shown that the gamma function does not satisfy any linear differential equation with rational coefficients.

It is sometimes considered a nuisance that n! Because of this, some authors adopt the notation x! We will not use the notation of Gauss, nor will we use the factorial notation except when dealing with nonnegative integer values. The gamma function rarely appears in the form 2. Since integrals are fairly easy to manipulate, 2. Equation 2. Last, we note that 2. Nonetheless, the integral 2. Let us first establish the equivalence of 2. To investigate the behavior of f x as x approaches the value zero from the right, we use the recurrence formula 2.

See Theorem 1. Values of f x are commonly tabulated for the interval 1:sx:s2 e. For example, f 2. The gamma function is defined for both positive and negative values of x by Eq. Also using 2. From 2. Solutlon: Repeated use of 2. U sing properties of even functions, and recalling Eqs. Simplification of this idE ntity leads to 2. An important special case of 2. An important identity involving the gamma function and sine function can now he derived hy using 2. Uecalling Eq.

Solution: Making use of 2. Weierstrass was the first to show that any entire function under appropriate restrictions with an infinite number of zeros, such as sinx and cosx, is essentially determined by its zeros.

This result led to the infinite product representations of such functions and, in particular, to the infinite product representa- tion of the gamma function.

A list of some of the most important properties of the gamma function can be found in the Appendix for easy reference. Use Eq. Hint: See problem Prove that e-ttx-i dt converges uniformly in 1s;xs;2. Using the recurrence formula 2. Show that Consider the following example.

The special case y 2 1 1 2 3 X Figure 2. Other important probability density functions are introduced in Exercises 2. Exemple 7: Calculate the moments of the gamma distribution defined by 2. Such generalizations are not new, however; mathe- maticians over the years have concerned themselves with this concept. Example 8. Exercises 2. The Gamma Function and Related Functlons 81 3. Compute the fractional-order derivatives. However, since it occurs so frequently in practice, a special designation for it is widely accepted.

Solutlon: By comparison with 2. Example 10 illustrates one of the basic approaches we use in the evaluation of nonelementary integrais. That is, we replace part of or all the integrand by its series representation or integral representa- tion and then interchange the order in which the operations are carried out.

Using the notation of problem 15 in Exercises 2. This function most com- monly arises in probability theory, particularly those applications involving the chi-square distribution. By substituting the series respresentation for e-t in 2. Formally derive the asymptotic series 2. It has characteristics quite distinct from those of the gamma function, however, since it is related to the derivative of f x. If n denotes a positive integer, it follows from 2. By repeated application of 2. Now, from Eqs.

Next, splitting the inside integral into a sum of integrais, an,d recalling the integral relation see problem 17 in Exercises 2.

We are simply using a mathematical gimmick here in order to formally derive 2. We then rewrite 2. To show this, we simply replace t by -t in 2. To facilitate the computations involving such expressions, it is helpful to have an accurate asymptotic formula from which to approximate f x.

To evaluate C, we would normally need to know the exact behavior of series 2. Of course, for larger values of n the formula is even more accurate. Although Stirling's series is valid for all positive x, it is used primarily for evaluating the gamma function for large arguments. Although 2. Instead, it may be preferable to have power series expansions for such calculations.

Termwise differentiation of 2. Continued differentiation of 2. Of course, the values of the digamma and polygamma functions must usually be obtained from tables.

We also found that the zeta function is closely related to the logarithm of the gamma function and to the polygamma functions. Although the zeta function was known to Euler, it was Riemann in who established most of its properties, which now are yery important in the field of number theory, among others.

Thus it bears his name. Therefore we deduce that 2. Abramowitz and 1. Stegun eds. The Gamma Function and Related Functlons eliminates ali terms from 2. Continuing in this fashion, it can eventually be shown that the infinite product over all prime numbers greater than 1 leads to 2.

Other relations involving this function, as well as some special values, are taken up in the exercises. The graph of C x - 1 is shown in Fig. For com- parison, the dashed line is the graph of 2-x. Whittaker and G. Note the logarithmic scale on the vertical ruds. Starting with Eq. Use problem 21 to establish that. Show that the mth derivative of Eq. Write the sum of the series in terms of the digamma and polygamma functions and evaluate.

Some of these functions were introduced in Chap. The error function derives its name from its importance in the theory of errors, but it also occurs in probability theory and in certain heat conduction problems on infinite domains. The closely related Fresnel integrais, which are fundamental in the theory of optics, can be derived directly from the error function.

A special case of the incomplete gamma function Sec. Elliptic integrais first arose in the problems associated with computing the arclength of an ellipse and a lemniscate a curve in the shape of a figure eight. Some early results conceming elliptic integrais were discovered by L. Euler and J. Landen, but virtually the whole theory of these integrais was developed by Legendre over a period spanning 40 years. The inverses of the elliptic integrais, called elliptic functions, were independently introduced in by C.

Jacobi and N. Abel Many of the properties of elliptic functions, however, had already been developed as early as by Gauss. Elliptic functions have the distinction of being doubly periodic, with one real period and one imaginary period. Among other areas of application, the elliptic functions are important in solving the pendulum problem Sec.

By representing the exponential function in 3. Other Functions Defined by Integrais Exercises 3. Establish the relations see Sec. Obtain the series representations. Among other applications, this density function appears in the analysis of random noise, such as that found at the input to radar and sonar detection devices.

The goveming equation is a partial differential equation called the one-dimensional heat equation. We assume the reader is familiar with the Laplace transform method for solving ordinary differential equations.

Because the functions involved in solving a partia! Other Functions Defined by Integrais We recognize 3. The remaining boundary condition in 3. Solution: The formal solution is that given by 3. We assume the beam is fixed at the end x - oo. Following the procedure used in Sec. The remain- ing boundary conditions in 3. If the boundary condition in 3. O Hint: Use the result of problem 18 in Exercises 3. Also many integrais of a more complicated nature can be expressed in terms of the exponential integrais.

Comparison of 3. For example, from the series for y a, x [see Eq. X -1 Figure 3. For large arguments, we can use Eq. Other Functions Defined by Integrais By using 3. The graphs of these func- tions are shown in Fig.

Exercises 3. Show that. L a t 3 t Hint: Use integration by parts. L2 1-t 2 t Hint: Start with partia! Beeause of its origin, it is ealled an elliptic integral. Some of the importance connected with these integrals lies in the following theorem, which we state without proof. If R x, y is a rational function in x and y and P x is a polynomial of degree 4 or less, then the integral JR x, VP x dx can always be expressed in terms of elliptic integrals.

Summing forces see Fig. Equation 3. The constant C is proportional to the energy of the system. Solving 3. Such a function exists and is called a Jacobian elliptic function. Much of the theory of elliptic functions is couched in the language of complex variables, and thus we do not pursue their general theory.

Some elementary properties, however, are taken up in the exercises. Verify the identities. Verify the addition formulas. Chapter Legendre Polynomials and 4 Related Functions 4. Other similar problems dealing with either gravita- tional potentials or electrostatic potentials also lead to Legendre polynomials, as do certain steady-state heat conduction problems in spherical solids, and so forth.

There exist a whole class of polynomial sets which have many properties in common and for which the Legendre polynomials represent the simplest example. Each polynomial set satisfies several recurrence formulas, is involved in numerous integral relationships, and forms the basis for series expansions resembling Fourier trigon- ometric series, where the sines and cosines are replaced by members of the polynomial set.

Because of all the similarities in these polynomial sets and because the Legendre polynomials are the simplest such set, our development of the properties associated with the Legendre polynomials will be more extensive than similar developments in Chap. The Legendre functions of the second kind arise as a second solution set of Legendre's equation independent of the Legendre polynomials , and the associated functions are related to derivatives of the Legendre polynomials.

Central to the discussion of problems of gravita- tional attraction is Newton 's law of universal gravitation: Every particle of matter in the universe attracts every other particle with a force whose direction is that of the line joining the two, and whose magnitude is directly as the product of their masses and inversely as the square of their distance from each other. The force field generated by a single particle is usually considered to be conservative. That is, there exists a potential function V such that the gravitational force F at a point of free space i.

Because of spherical symmetry of the gravitational field, the potential function V depends on only the radial distance r. Valuable information on the properties of potentials like 4. He found that the coefficients appearing in this expansion were polynomials that exhibited interesting properties.

Legendre Polynomials and Related Functlons p Figure 4. Let the point Q representa point of free space r units from P and b units from the origin O. Our task at this point is to develop w x, t in a power series in the variable t. The factor 2x - tr is simply a finite binomial series, and thus 4. Thus, recalling Eq. By recognizing that [see Eq. The first few Legendre polynomials are listed in Table 4. Making an observation, we note that when n is an even number, the polynomial Pn x is an even function; and when n is odd, the polynomial is an odd function.

Returning now to Eq. Special values and recurrence formulas The Legendre polynomials are rich in recurrence relations and identities. Remark: Actually, 4. We accomplish this by leaving the first sum in 4. Thus, 4. We refer to 4. One of the primary uses of 4. Substituting the series for w x, t directly into 4. Thus, by equating the coefficient of tn to zero in 4. Certain combinations of 4. For example, suppose we first differentiate 4.

We may well wonder if any relation exists between derivatives of the Legendre polynomials and Legendre polynomials of the sarne index. The answer is yes, but to derive this relation, we must consider second derivatives of the polynomials. By taking the derivative of both sides of 4. Expanding the product term in 4. Solving the partia! DE by the separation-of-variables technique leads to a system of ordinary DEs, and sometimes one of these is Legendre's DE.

This is precisely the case, for example, in solving for the steady-state temperature distribution independent of the azimuthal angle in a solid sphere. We delay any further discussion of such problems, however, until Sec. Remark: Any function fn x that satisfies Legendre's equation, i. Consequently, any further solutions of Legendre's equation can be selected in such a way that they automatically satisfy the whole set of recurrence relations already derived. The set of solutions Qn x introduced in Sec.

Exercises 4. Use the series 4. Legendre Polynomlals and Related Functions 4. Using the Cauchy product of two power series Sec. What shape does the string assume in the vertical plane in each case? Pn X from which 4. Thus, all odd terms in 4. Legendre Polynomials and Related Functlons 4. Of particular interest is the interval lx s 1, but since the integrand in 4.

The equality in 4. Using Rodrigues' formula 4. Representing Pn x by Rodrigues' formula 4. Then, by comparing your result with Eq. Use the Laplace integral formula 4. Because the general term in such series is a polynomial, we can interpret a Legendre series as some generalization of a power series for which the general term is also a polynomial, that is, x - a n.

However, to develop a given function f in a power series requires that the function f be at least continuous and differentiable in the interval of con- vergence. Legendre series are only one member of a fairly large and special class of series collectively referred to as generalized Fourier series, all of which have many properties in common. Besides their obvious mathematical interest, it turns out that the applica- tions of generalized Fourier series are very extensive-so much so, in fact, that they involve almost every facet of applied mathematics.

To prove 4. Hence, 4. To further simplify 4. Using the fact that and Eq. For example, if qm x denotes an arbitrary polynomial of degree m, then since P0 x , P1 x , Example 1: Express x 2 in a series of Legendre polynomials. Hence, When the polynomial qm x is of a high degree, solving a system of simultaneous equations for the c's as we did in Example 1 is very tedious.

A more systematic procedure can be developed by using the orthogonality property 4. We begin by writing 4. As a consequence of the fact that a polynomial of degree m can be expressed as Legendre series involving only Pm x and lower-order Legendre polynomials, we have the following theorem.

For now it suffices to say that for certain functions the series 4. Series of this type are called Legendre series, and because they belong to the larger class of generalized Fourier series, the coefficients 4.

However, if the function f is not too complicated, we can sometimes use various properties of the Legendre polynomials to evaluate such integrais in closed forro. The following example illustrates the point. Remark: Because the interval of convergence of 4. That is, even if f is defined for all X, the representation will not be vali d beyond the interval -1 :s x :s 1 unless f is a polynomial.

Hence, owing to the even-odd property of the Legendre polynomials depending on the index n, we note that f x Pn x is an odd function when n is even, and in this case it follows that see problem 25 in Exercises 4. Referring to Eq. Show that the orthogonality relation 4. Use Rodrigues' formula 4. Hint: Use problem What we mean is-if a value of x is selected in the chosen interval and each term of the series is evaluated for this value of x, will the sum of the series be f x?

If so, we say the series converges pointwise to f x. From a practical point of view, such conditions should be broad enough to cover most situations of concem and still simple enough to be easily checked for the given function. Definltion 4. A function f is said to be piecewise continuous in the interval a :5 x :5 b provided that a f x is defined and continuous at all but a finite number of points in the interval.

It is not essential that a piecewise continuous function f be defined at every point in the interval of interest. Also the interval of interest may be open or closed, or open at one end and closed at the other see Fig. Definition 4. A function f is said to be smooth in the interval a :s;x :s; b if it has a continuous derivative there. The function in b is also continuous, but because the derivative is discontinuous, i.

Lemma 4. If the function f is piecewise continuous in the closed interval -1 :5x l, then! Hence, the series on the left is a convergent series because its sumis finite , and therefore it follows that. The Legendre polynomials satisfy the identity Proof: We begin by multiplying the recurrence relation 4. Theorem 4. Interchanging the order of sumtnation and integration and recalling the Christoffel-Darboux formula Lemma 4.

Letting we can express the partia! Legendre Polynomlals and Related Functlons Because any linear combination of solutions is also a solution of a homogeneous DE, it has become customary to define the second solution of 4. We refer to Qn x as the Legendre function of the second kind of integral order. Hence, we select the Legendre functions Qn x so that necessarily 4. The first few Legendre functions of the second kind are sketched in Fig.

While Eq. For example, if! Other properties are taken up in the exercises. Legendre Polynomials and Related Functlons 5. Deduce the result of problem 10 by using the wronskian relation in problem 6 and appropriate recurrence relations. Use the result of Eq. Solve Legendre's equation 1-x 2 y" - 2.

The DE 4. By taking m derivatives of 4. The associated Legendre functions have many properties in com- mon with the simpler Legendre polynomials Pn x and Legendre functions of the second kind Qn x. Many of these properties can be developed directly from the corresponding relation involving either Pn x or Qn x by taking derivatives and applying the definitions 4. But because P';: x has two indices instead of just one, there exist a wider variety of possible relations than for Pn x.

To derive the three-term recurrence formula for P';: x , we start with the known relation [see Eq. The details of proving 4.

As a final comment we mention that although it is essentially only a mathematical curiosity, there is another orthogonality relation for the associated Legendre functions given by 1 J P: x P!

Directly from Eq. Applying the Leibniz formula 4. Derive the generating function relation 2m! Since that time the study of special functions has been closely linked with the study of DEs. For example, Legendre polynomials and the associated Legendre functions are prominent in applications featuring spherical geometry, such as finding the electric potential inside a spherical shell or the steady- state temperature inside a homogeneous solid sphere.

On the other hand, Bessel functions see Chaps. The most important partial DE or PDE in mathematical physics is Laplace's equation, also known as the potential equation. This sarne equation is satisfied by the gravitational potential in free space, the electrostatic potential in a uniform dielectric, the magnetic potential in free space, and the velocity potential of inviscid, irrotational flows.

Laplace's equa- tion 4. To solve 4. The direct substitution of 4. Thus we have "separated the variables. Equating each side of 4. Recalling Example 2 in Sec. To determine the potential in this region, we must again solve Laplace's equation 4.

Assuming the sphere is void of any heat sources, we wish to determine the steady-state temperature distribution everywhere within the sphere. The general form of Laplace's equation in spherical coor- dinates is given by see problem 12 in Exercises 4.

Although the theory associated with such series follows in a natural way from the theory of one variable, it goes beyond the intended scope of this text. As a final observation here, we note that for the special case where the prescribed temperatures are in- dependent of the angle 8, the temperatures inside the sphere will also be independent of 8.

Find the electric potential in the exterior of the unit sphere, assuming the potential on the surface is prescribed as given in problem 1. Use the result of problem 4 to verify that 4. Solve the electric-potential problem in Sec. Find the temperature inside the sphere. A spherical shell has an inner radius of 1 unit and an outer radius of 2 units. Show that the laplacian in problem 11 can also be expressed as 2 2 a 2 au 1 a.

The Legendre polynomials discussed in Chap. Other polynomial sets which commonly occur in applications are the Hermite, Laguerre, and Chebyshev polynomials. More general poly- nomial sets are defined by the Gegenbauer and Jacobi polynomials, which include the others as special cases. The study of general polynomial sets like the Jacobi polynomials facilitates the study of each polynomial set by focusing on those properties that are characteristic of all the individual sets.

Moreover, it can be shown that any orthogonal polynomial set satisfying these three conditions is necessarily a member of the Jacobi polynomial set, or a limiting case such as the Hermite and Laguerre polynomials. Thus, it follows that 5. This definition occurs most often in statistical applications.

J 2nk! Solution: From 5. Let us start with the generating-function relations oo tn L! We have the following theorem for them. Theorem 5. Other Orthogonal Polynomials then the Hermite series 5. The proof of Theorem 5. A fundamental problem in wave mechanics concems the motion of a particle bound in a potential well. A particular example of this important class of problems is the linear oscillator also called the simple harmonic oscillator , the solutions of which lead to Hermite polynomials.

Thus, asymptotically we expect the solution of 5. Based on this observation, we make the assumption that 5. The substitution of 5. The other solutions of Hermite's equation are not appropriate in this problem. Derive the Fourier transform relations. Derive the Hermite series relations. Hn-m X ln problems 16 to 19, derive the series relationship. Similarly, substituting 5. To obtain the governing DE for the Laguerre polynomials, we begin by differentiating 5.

We begin by multiplying the two series 5. Other Orthogonal Polynomlals 5. An important application involving the Laguerre poly- nomials is to find the wave function associated with the electron in a hydrogen atom. For physical reasons, we must require 0 8 to be a periodic function with period 2Jr. Derive the Rodrigues formula. Hint: Use the Leibniz formula 5. Derive the recurrence formulas. By repeated differentiation of the series 5.

Show that 6. Assume 5. This result gives the average displacement of the electron from the nucleus. Indeed, the Gegenbauer and Jacobi polynomials are two such generalizations. The Gegenbauer polynomials are closely connected with axially symmetric potentials in n dimensions and contain the Legendre, Hermite, and Chebyshev polynomials as special cases.

The Jacobi polynomials are more general yet, since they contain the Gegenbauer polynomials as a special case. One of the main advantages of developing properties of the Gegenbauer polynomials is that each recurrence formula, etc.

Hence, from 5. By following a procedure similar to that used to verify the relation 5. The Chebyshev polynomials have acquired great practical impor- tance in polynomial approximation methods. Specifically, it has been shown that a series of Chebyshev polynomials converges more rapidly than any other series of Gegenbauer polynomials, and it converges much more rapidly than a power series.

Fox and I. Next, replacing the left-hand side of 5. Also the orthogonality property and governing DE are given, respectively, by k -:f:.

Exercises 5. Use any of the results of problems 4 to 8 and the recurrence formula 5. By using Eq. Using the recurrence formula 5. Other Orthogonal Polynomials Bessel first achieved fame by computing the orbit of Halley's comet. Nonetheless, Bessel functions were first discovered in by D.

Bernoulli , who provided a series solution representing a Bessel function for the oscillatory displacements of a heavy hanging chain see Sec. Euler later developed a series similar to that of Bernoulli, which was also a Bessel function, and Bessel's equation appeared in a article by Euler dealing with the vibrations of a circular drumhead. Fourier also used Bessel functions in his classical treatise on heat in , but it was Bessel who first recognized their special properties.

Bessel functions are closely associated with problems possessing circular or cylindrical symmetry. For example, they arise in the study of free vibrations of a circular membrane and in finding the temperature distribution in a circular cylinder.

They also occur in electromagnetic theory and numerous other areas of physics and engineering. Because of their dose association with cylindrical domains, the solutions of Bessel's equation are also called cylinder functions. Consequently, Eq. Finally, replacing the inside series in 6. The Bessel functions that arise most frequently in practice are J 0 x and J 1 x , whose series representations are J.

Observe that these functions exhibit an oscillatory behavior somewhat like that of the sinusoidal functions, except that the amplitude maximum departure from the x axis of the Bessel functions diminishes as x increases and the infinitely many zeros of these functions are not evenly spaced. The location of these zeros is of great theoretical and practical importance, but the theory goes beyond the scope of this text. The replacement of p with -p in 6.

The cases where p is half-integral are of special interest because these Bessel functions are actually elementary functions.

Fortunately, many of the properties of JP x and J -p x can be developed directly from their series definition. For example, suppose we multiply the series for JP x by xP and then differentiate the result with respect to x.

Observe that it is not restricted to integer values ofp. To start, we rewrite Eqs. Therefore, under these conditions a general solution of Bessel's equation 6. Among other areas of application, Bessel's equation arises in the solution of various partia! Show that the generating-function relation 6. By using the series representation 6. The generating function 6. Use the series 6. Use problem 19 to derive Lommel's formula 2 sinp. X - 2t -pl2 oo tn Legendre polynomial oo tn There are several such representations, but foremost is one involving Bessel functions of integral order.

P to get 00 e-ix sin. To verify 6. O where we are using properties of even and odd functions and have expressed cosxt in a power series. The residual integral in 6. Finally, by substituting the result of 6. The direct verification of 6. Fourier presented basic papers to the Academy of Sciences in Paris conceming the representation of functions by trigonometric series.

This new concept intrigued many of the researchers of the era who subsequently tried to use Fourier's series representation in their own work. Bessel Functlons The problem solved by Bessel was actually one proposed by Johann Kepler The ellipse A'PA in Fig. A line drawn through P perpendicular to A'A defines the point Q on the circle. We assume that time t is measured from an instant when P is at A.

Bessel's approach to solving Kepler's problem was to use Fourier's method of trigonometric series. By comparing 6. Some of these, such as Example 1 following, are simple problems of a geometric nature. Additional problems of this type are taken up in the exercises. By comparing this last expression with 6.

A circle of unit radius rolls along a straight line, the original point of contact describing a curve called a cycloid. These may appear in the form of either indefinite or definite integrais.

We discuss each case separately. These integral formulas are valid for any p 2: O. Example 2: Reduce f x 2J 2 x dx to an integral involving only J 0 x. Solutlon: The given integral does not exactly match either 6. Thus, we have and a second integration by parts finally leads to The last integral involving J 0 x cannot be evaluated in closed form, and so our integration is complete.

Since it cannot be evaluated in closed form, the integral fi J 0 t dt has been tabulated. Some additional integral relations of this general type appear in the exercises. The usual procedure is to replace the Bessel function by its series representation or an integral representation and then in- terchange the order in which the operations are carred out. To start, let us replace the product tP12JP 2Vt with its series representation, i.

L:o k! This time the resulting series is more difficult to identify, but it is actually a binomial series. Recalling the Legendre duplication formula and Eq. Yet it is possible to justify a limiting procedure whereby the real part of a approaches zero. Thus, if we formally replace a in 6. Both 6. Exercises 6. Verify the identity! Hint: Differentiate problem 19 with respect to s. These usually appear in the form of addition formulas or orthogonal expansions.

We do derive 6. Addition theorems such as 6. To prove them, let us start with the identity see problem 7 in Exercises 6. Recognizing that the product of the series on the right-hand sides in 6. Such series belong to the class of generalized Fourier series. Jp kmx J Jp knx dx -f! JP kmx ]dx Performing integration by parts on the right-hand side and dividing by the factor k! To deduce its value, we take the limit of 6.

Because the right-hand side of 6. Such a series is called a Fourier-Bessel series or simply a Bessel series. This last integral is simply 6. If f and f' are piecewise continuous functions on Q;;x ;;b, then the Bessel series 6. From Eq. For purposes of constructing a general solution of 6. Because it is a linear combination of solutions of 6.

We conclude, therefore, that the general solution of Bessel's equation 6. Because the limit 6. From 6. Hence, combining this last expression with 6. However, it can be shown that see problem 9 in Exercises 6. X 2k-n Y,. Also note that these functions have oscillatory characteristics similar to those of Jn X. Bessel Functlons X -1 Figure 6. For example, it is easily established that see problems 10 and 11 in Exercises 6.

Furthermore, it can be shown that see problem 12 in Exercises 6. Use the result of problem 5 to finda general solution of Bessel's equation 6. From the results of problems 18 and 19 in Exercises 6. Show that a! The sarne can be said of many problems of a more complicated nature when their solutions can be expressed in terms of Bessel functions. Bessel Functlons To derive the solution formula 6. First, let us set 6.

For those cases whenp is not integral, we can express the general solution 6. This problem was first discussed in by D. Bernoulli and later in by Euler, both many years prior to Bessel's legendary paper in on the properties of Bessel functions.

Consider a uniform heavy fiexible chain of length L, fixed at the upper end and free at the lower end see Fig. When the chain is slightly disturbed from its position of equilibrium in a vertical plane, it undergoes "small" oscillations. Let p denote the constant mass per unit length and y the horizontal displacement of the chain at time t. Taking the origin at the bottom of the chain as shown in Fig.

Let us suppose the tension T is due entirely to the weight of the chain below a given point x. Although 6. We recognize 6. Hence, the general solution is 6. The complete solution of the oscillating-chain problem consists of the superposition of all vibration modes, i. Although similar in definition to the standard Bessel functions, the modified Bessel functions are most clearly distinguished by their nonoscillatory behavior.

For this reason, they often appear in ap- plications that are different in nature from those for the standard functions. The general family of cylinder functions also include spherical Bessel functions, Hankel functions, Kelvin's functions, Lommel functions, Struve functions, Airy functions, and Anger and Weber functions.

Of these, Hankel functions have special significance in that they enable us to obtain asymptotic formulas for large arguments for all the other types of Bessel functions. It is of the form 7. To avoid the imaginary ar- guments in 7. Next, observing that the above series itself is a real quantity multiplied by iP, we are motivated to define the real function p?

Except that the alternating factor -l k is missing, the series 7. Thus each term of the series 7. We conclude, therefore, that IP x cannot have a positive zero and so cannot exhibit an oscillatory behavior like that of JP x. Moreover, by using an argument similar to that in Sec. Hence for p not an integer, a general solution of 7. In the sarne manner, we find that see problem 10 in Exercises 7. Because it is a linear combination of solutions, the function KP x is also a solution of 7.

By using techniques K 11 x 3 X Figure 7. Similarly, by using the relation 7. Hence, comparing coefficients of tn on both sides of the equation, we arrive at the simple add. Exercises 7. Develop the asymptotic formulas. Use the result of problem 7 in Exercises 6. Replace a by ia and b by ib in Eq. Bessel Functlons of Other Klnds 7. We recognize 7. Since n takes on integral values, all Bessel functions in 7. We previously found that the particular half-integral order Bessel function J x is an elementary function given by [recall Eq.

To combine the multiplicative factor x- appearing in front of 7. Observe that the general behavior of these functions is that of the standard Bessel functions. This leads to see the exercises 7. Some of the properties associated with these modified spherical Bessel functions are tak.

Exerclses 7. From the series representation 7. By use of any of the recurrence relations, show that. Develop the generating-function relations 1 oo tn a -cosv'x 2 Email or Username Forgot your username? Password Forgot your password? Keep me signed in. Please wait No SPIE account? Create an account Institutional Access:. Author s : Larry C. Copublished with Oxford University Press. Softcover version of PM Buy this book on SPIE. This will count as one of your downloads.

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The Gamma Function and Related Functions. Other Functions Defined by Integrals. Legendre Polynomials and Related Functions. Other Orthogonal Polynomials. Bessel Functions. Bessel Functions of Other Kinds. Applications Involving Bessel Functions.

The Hypergeometric Function.