Chapter 13

Partial Differential Equations and Special Functions

13.1 Partial Differential Equations

Many complex physical, geometrical, and stochastic (probabilistic) phenomena are described by partial differential equations (PDEs), involving two or more independent variables. They are obviously much more difficult to solve than ordinary differential equations (ODEs). Some applications, including weather prediction, econometric models, fluid dynamics, and nuclear engineering, might involve simultaneous PDEs with large numbers of variables. Such problems are best tackled by powerful supercomputers. We will be content to consider a few representative second-order PDEs for which analytic solutions are possible.

Scalar and vector fields which depend on more than one independent variable, which we write in the notation image, etc., are very often obtained as solutions to PDEs. Some classic equations of mathematical physics which we will consider are the wave equation, the heat equation, Laplace’s equation, Poisson’s equation, and the Schrödinger equation for some exactly solvable quantum-mechanical problems.

Maxwell’s equations lead to the wave equation for components of the electric and magnetic fields, with the general form

image (13.1)

Analogous equations apply to sound and other wave phenomena. For waves in one spatial dimension, the wave equation reduces to

image (13.2)

The heat equation or diffusion equation is similar to the wave equation but with a first derivative in the time variable:

image (13.3)

where image is a constant determined by the thermal conductivity or diffusion coefficient.

Introducing the electrostatic potential image into the first of Maxwell’s equation, image, we obtain Poisson’s equation

image (13.4)

In a region free of electric charge, this reduces to Laplace’s equation

image (13.5)

In the calculus of complex variables, we encounter the two-dimensional Laplace equation for image

image (13.6)

For the case of monochromatic time dependence, the solution of the wave equation has the form

image (13.7)

where

image (13.8)

or some linear combination of sine and cosine or complex exponentials. Substituting (13.7) into the wave equation (13.1) we find

image (13.9)

where we have noted that image acts only on the factor image while image acts only on image. Since either of the functions (13.8) is a solution of the ordinary differential equation

image (13.10)

Equation (13.9) reduces to Helmholtz’s equation

image (13.11)

after canceling out image and defining image. The heat equation also reduces to Helmholtz’s equation for separable time dependence in the form

image (13.12)

Recall that ordinary differential equations generally yield solutions containing one or more arbitrary constants, which can be determined from boundary conditions. In contrast, solutions of partial differential equations often contain arbitrary functions. An extreme case is the second-order equation

image (13.13)

This has the solutions

image (13.14)

where image and image are arbitrary differentiable functions. More detailed boundary conditions must be specified in order to find more specific solutions.

13.2 Separation of Variables

The simplest way to solve PDEs is to reduce a PDE with n independent variables to n independent ODEs, each depending on just one variable. This isn’t always possible but we will limit our consideration to such cases. In the preceding section, we were able to reduce the wave equation and the heat equation to the Helmholtz equation when the time dependence of image was separable. Consider the Helmholtz equation in Cartesian coordinates

image (13.15)

If the boundary conditions allow, the solution image might be reducible to a separable function of image:

image (13.16)

Substituting this into (13.15), we obtain

image (13.17)

Division by image gives the simplified equation:

image (13.18)

Now, if we solve for the term image, we find this function of x alone must be equal to a function of y and z, for arbitrary values of image. The only way this is possible is for the function to equal a constant, say

image (13.19)

A negative constant is consistent with a periodic function image, otherwise it would have exponential dependence. Analogously,

image (13.20)

so that

image (13.21)

We have thereby reduced Eq. (13.15) to three ordinary differential equations:

image (13.22)

We solved an equation of this form in Section 8.5, for two alternative sets of boundary conditions. The preceding solution of the Helmholtz equation in Cartesian coordinates is applicable to the Schrödinger equation for the quantum-mechanical particle-in-a-box problem.

13.3 Special Functions

The designation elementary functions is usually applied to the exponential, logarithmic, and trigonometric functions and combinations formed by algebraic operations. Certain other special functions occur so frequently in applied mathematics that they acquire standardized symbols and names—often in honor of a famous mathematician. We will discuss a few examples of interest in physics, chemistry, and engineering, in particular, the special functions named after Bessel, Legendre, Hankel, Laguerre, and Hermite. We already encountered in Chapter 6 the gamma function, error function, and exponential integral. Special functions are most often solutions of second-order ODEs with variable coefficients, obtained after separation of variables in PDEs. So that you will have seen the names, here is a list of some other special functions you are likely to encounter (although not in this book): Airy functions, Beta functions, Chebyshev polynomials, Elliptic functions, Gegenbauer polynomials, Jacobi polynomials, Mathieu functions, Meijer G-functions, Parabolic cylinder functions, Theta functions, Whittaker functions.

For a comprehensive reference on special functions, see M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (National Bureau of Standards Applied Mathematics Series, Vol. 55,Washington, DC, 1964). This is now available online at http://dlmf.nist.gov and features graphics and hypertext links.

Following is a sampling of some “amazing” formulas involving gamma and zeta functions, two of the special functions we have already introduced. A reflection formula for gamma functions:

image (13.23)

An infinite product representation of the gamma function:

image (13.24)

where image, the Euler-Mascheroni constant. An integral for the Riemann zeta function:

image (13.25)

Euler’s relation connecting the zeta function with prime numbers, which we proved in Section 1.12:

image (13.26)

For a long time, I felt intimidated by these amazing formulas involving special functions. Some resembled near miracles like “pulling rabbits out of a hat.” I could never, in a thousand years, have come up with anything as brilliant as one of these myself. I would not have been surprised to see in some derivation “and then a miracle occurs” between the next-to-last and last equations (see Sidney Harris cartoon):

image

Over the years, I learned to be less intimidated, assuring myself these amazing results were the product of hundreds of brilliant mathematicians using inspired guesswork, reverse engineering, and all kinds of other nefarious practices for over 300 years. Ideally, one should be fully comfortable in using these results, resisting any thoughts that we are not entitled to exploit such brilliance. Something like the habitual comfort most of us have developed in using our cars, computers, and washing machines. Without a doubt, we all “stand on the shoulders of giants.”

13.4 Leibniz’s Formula

We will later need a formula for the nth derivative of the product of two functions. This can be derived stepwise as follows:

image (13.27)

Clearly, we are generating a series containing the binomial coefficients

image (13.28)

and the general result is Leibniz’s formula

image (13.29)

13.5 Vibration of a Circular Membrane

As our first example of a special function, we consider a two-dimensional problem: the vibration of a circular membrane such as a drumhead. The amplitude of vibration is determined by solution of the Helmholtz equation in two dimensions, most appropriately chosen as the polar coordinates image. Using the scale factors image and image for the two-dimensional Laplacian, the Helmholtz equation can be written

image (13.30)

This is subject to the boundary condition that the rim of the membrane at image is fixed, so that image. Also it is necessary that the amplitude be finite everywhere on the membrane, in particular, at image. Assuming a separable solution image, Helmholtz’s equation reduces to

image (13.31)

or

image (13.32)

We could complete a separation of variables by dividing by image. This would imply that the function of image in square brackets equals a constant, which we can write

image (13.33)

This is a familiar equation, with linearly independent solutions

image (13.34)

Since image is an angular variable, the periodicity image is required. This restricts the parameter m to integer values. Thus we obtain an ordinary differential equation for image:

image (13.35)

Changing the variable to image, with image, we can write

image (13.36)

This is recognized as Bessel’s equation (8.135). Only the Bessel functions image are finite at image, the Neumann functions image being singular there. Thus the solutions to Helmholtz’s equation are

image (13.37)

The boundary condition at image requires that

image (13.38)

The zeros of Bessel functions are extensively tabulated. Let image represent the nth root of image. The first three zeros are given by:

Image

The eigenvalues of k are given by

image (13.39)

Some modes of vibration of a circular membrane, as labeled by the quantum numbers m and n, are sketched in Figure 13.1. To simplify the figure, only the sign of the wavefunction is indicated: gray for positive, white for negative. Note that modes for image are twofold degenerate, corresponding to the factors image and image.

image

Figure 13.1 Modes of vibration image of a circular membrane. Wavefunctions are positive in gray areas, negative in white.

13.6 Bessel Functions

In Section 8.7, a series solution of Bessel’s differential equation was derived, leading to the definition of Bessel functions of the first kind:

image (13.40)

This applies as well for noninteger n with the replacements image. The Bessel functions are normalized in the sense that

image (13.41)

Bessel functions of integer order can be determined from expansion of a generating function:

image (13.42)

To see how this works, expand the product of the two exponentials:

image (13.43)

The terms which contribute to image are

image (13.44)

which gives the expansion for image. The expansion for image is found from the terms proportional to image and so forth.

One of the many integral representations of Bessel functions is

image (13.45)

An identity involving the derivative is

image (13.46)

In particular, image. An addition theorem for Bessel functions states that

image (13.47)

Asymptotic forms for the Bessel function for small and large values of the argument are given by

image (13.48)

and

image (13.49)

Bessel functions of the second kind show the following limiting behavior:

image (13.50)

and

image (13.51)

The two kinds of Bessel functions thus have the asymptotic dependence of slowly damped cosines and sines. In analogy with Euler’s formula image, we define Hankel functions of the first and second kind:

image (13.52)

The asymptotic forms of the Hankel functions are given by

image (13.53)

having the character of waves propagating to the right and left, respectively.

Trigonometric functions with imaginary arguments suggested the introduction of hyperbolic functions: image and image. Analogously, we can define modified Bessel functions (or hyperbolic Bessel functions) of the first and second kind as follows:

image (13.54)

and

image (13.55)

Their asymptotic forms for large x are

image (13.56)

We have given just a meager sampling of formulas involving Bessel functions. Many more can be found in Abramowitz & Stegun and other references. G.N. Whittaker’s classic A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1952) is a ponderous volume devoted entirely to the subject.

13.7 Laplace’s Equation in Spherical Coordinates

Laplace’s equation in spherical polar coordinates can be written

image (13.57)

We consider separable solutions

image (13.58)

where the functions image are known as a spherical harmonics. Substitution of (13.58) into (13.57), followed by division by image and multiplication by image, separates the radial and angular variables:

image (13.59)

This can hold true for all values of image only if each of the two parts of this partial differential equation equals a constant. Let the first part equal image and the second equal image. The ordinary differential equation for r becomes

image (13.60)

It is easily verified that the general solution is

image (13.61)

Returning to the spherical harmonics, these evidently satisfy a partial differential equation in two variables:

image (13.62)

Again we assume a separable solution with

image (13.63)

Substituting (13.63) into (13.62), dividing by image, and multiplying by image, we achieve separation of variables:

image (13.64)

Setting the term containing image to a constant image, we obtain the familiar differential equation

image (13.65)

Solutions periodic in image can be chosen as

image (13.66)

Alternative solutions are image and image.

13.8 Legendre Polynomials

Separation of variables in Laplace’s equation leads to an ODE for the function image:

image (13.67)

Let us first consider the case image and define a new independent variable

image (13.68)

This transforms Eq. (13.67) to

image (13.69)

which is known as Legendre’s differential equation. We can construct a solution to this linear second-order equation by exploiting Leibniz’s formula (13.29). Begin with the function

image (13.70)

which is a solution of the first-order equation

image (13.71)

Differentiating image times, we obtain

image (13.72)

where

image (13.73)

This is a solution of Eq. (13.69) for

image (13.74)

With a choice of constant such that image, the Legendre polynomials can be defined by Rodrigues’ formula:

image (13.75)

Reverting to the original variable image, the first few Legendre polynomials are

image (13.76)

The Legendre polynomials obey the orthonormalization relations

image (13.77)

A generating function for Legendre polynomials is given by

image (13.78)

An alternative generating function involves Bessel function of order zero:

image (13.79)

Another remarkable connection between Legendre polynomials and Bessel functions is the relation

image (13.80)

Returning to Eq. (13.67) for arbitrary values of m, the analog of Eq. (13.69) can be written

image (13.81)

The solutions are readily found to be

image (13.82)

known as associated Legendre functions. These reduce to the Legendre polynomials (13.75) when image. Since image is a polynomial of degree image, image is limited to the values image. The associated Legendre functions obey the orthonormalization relations

image (13.83)

The orthonormalized solutions to Eq. (13.67) are thus given by

image (13.84)

13.9 Spherical Harmonics

Combining the above functions of image and image, we obtain the spherical harmonics

image (13.85)

where

image (13.86)

The factor image is appended in applications to quantum mechanics, in accordance with the Condon and Shortley phase convention. Two important special cases are

image (13.87)

and

image (13.88)

Following is a listing of spherical harmonics image through image:

image

A graphical representation of these functions is given in Figure 13.2. Surfaces of constant absolute value are drawn, with intermediate shadings representing differing complex values of the functions. In quantum mechanics, spherical harmonics are eigenfunctions of orbital angular momentum operators such that

image (13.89)

and

image (13.90)

image

Figure 13.2 Contours of spherical harmonics as three-dimensional polar plots.

The spherical harmonics are an orthonormal set with respect to integration over solid angle:

image (13.91)

Linear combinations of image and image contain the real functions:

image (13.92)

These are called tesseral harmonics since they divide the surface of a sphere into tesserae—four-sided figures bounded by nodal lines of latitude and longitude. (Nodes are places where the wavefunction equals zero.) When image, the spherical harmonics are real functions:

image (13.93)

These correspond to zonal harmonics since their nodes are circles of latitude which divide the surface of the sphere into zones. When image, the spherical harmonics reduce to

image (13.94)

The corresponding real functions

image (13.95)

are called sectoral harmonics. The three types of surface harmonics are shown in Figure 13.3.

image

Figure 13.3 Three types of spherical harmonics plotted on surface of a sphere. Boundaries between shaded (positive) and white (negative) regions are nodes where wavefunction equals zero.

13.10 Spherical Bessel Functions

Helmholtz’s equation in spherical polar coordinates can be obtained by adding a constant to Laplace’s equation (13.57):

image (13.96)

As in the case of Laplace’s equation, Eq. (13.96) has separable solutions

image (13.97)

Since we now know all about spherical harmonics, we can proceed directly to the radial equation:

image (13.98)

Were it not for the factor 2 in the second term, this would have the form of Bessel’s equation (13.35). As in Section 13.5, we redefine the variables to image and image. We also rewrite image as n to conform to conventional notation. The differential equation now reads

image (13.99)

With the substitution image, the equation reduces to

image (13.100)

which is recognized as Bessel’s equation of odd-half order image (if n is an integer). The linearly independent solutions are image and image, the latter being proportional to image since the order is not an integer. The solutions to Eq. (13.99) are known as spherical Bessel functions and conventionally defined by

image (13.101)

and

image (13.102)

You can verify that Eq. (13.99) with image has the simple solutions image and image. In fact, spherical Bessel functions have closed-form expressions in terms of trigonometric functions and powers of x, given by

image (13.103)

Explicit formulas for the first few spherical Bessel functions follow:

image (13.104)

There are also spherical analogs of the Hankel functions:

image (13.105)

The first few are

image (13.106)

13.11 Hermite Polynomials

The quantum-mechanical harmonic oscillator satisfies the Schrödinger equation:

image (13.107)

To reduce the problem to its essentials, simplify the constants with image, or alternatively, replace imagex by x. Correspondingly, image. We must now solve a second-order differential equation with nonconstant coefficients:

image (13.108)

A useful first step is to determine the asymptotic solution to this equation, giving the form of image as image. For sufficiently large values of image, image, so that the differential equation can be approximated by

image (13.109)

This suggests the following manipulation:

image (13.110)

Now, the first-order differential equation

image (13.111)

can be solved exactly to give

image (13.112)

To build in this asymptotic behavior, let

image (13.113)

This reduces Eq. (13.108) to a differential equation for image:

image (13.114)

To construct a solution to Eq. (13.114), we begin with the function

image (13.115)

which is clearly the solution of the first-order differential equation

image (13.116)

Differentiating this equation image times using Leibniz’s formula (13.29), we obtain

image (13.117)

where

image (13.118)

We find that image satisfies

image (13.119)

which is known as Hermite’s differential equation. The solutions in the form

image (13.120)

are known as Hermite polynomials, the first few of which are enumerated below:

image (13.121)

Comparing Eq. (13.119) with Eq. (13.114), we can relate the parameters

image (13.122)

Referring back to the original harmonic-oscillator equation (13.107), this leads to the general formula for energy eigenvalues

image (13.123)

A generating function for Hermite polynomials is given by

image (13.124)

Using the generating function, we can evaluate integrals over products of Hermite polynomials, such as

image (13.125)

Thus, the functions

image (13.126)

form an orthonormal set with

image (13.127)

13.12 Laguerre Polynomials

The quantum-mechanical problem of a particle moving in a central field is represented by a three-dimensional Schrödinger equation with a spherically symmetric potential image:

image (13.128)

As in the case of Helmholtz’s equation, we have separability in spherical polar coordinates: image. In convenient units with image, the ODE for the radial function can be written

image (13.129)

We consider the electron in a hydrogen atom or hydrogenlike ion (image) “orbiting” around a nucleus of atomic number Z. The attractive Coulomb potential in atomic units (image) can be written

image (13.130)

It is again useful to find asymptotic solutions to the differential equation. When image the equation is approximated by

image (13.131)

noting that the energy E will be negative for bound states of the hydrogenlike system. We find the asymptotic solution

image (13.132)

As image, Eq. (13.129) is approximated by

image (13.133)

which is just Laplace’s equation in spherical coordinates. The solution finite at image suggests the limiting dependence

image (13.134)

We can incorporate both limiting forms by writing

image (13.135)

in terms of a new variable

image (13.136)

where n is a constant to be determined. The differential equation for image then works out to

image (13.137)

Following the strategy used to solve the Hermite and Legendre differential equations, we begin with a function

image (13.138)

where image is a positive integer. This satisfies the first-order differential equation

image (13.139)

Differentiating this equation image times using Leibniz’s formula (13.29), we obtain

image (13.140)

where

image (13.141)

Laguerre polynomials are defined by Rodrigues’ formula:

image (13.142)

We require a generalization known as associated Laguerre polynomials, defined by

image (13.143)

These are solutions of the differential equation

image (13.144)

Comparing Eqs. (13.137) and (13.144), we can identify

image (13.145)

where n must be a positive integer. The bound-state energy hydrogenlike eigenvalues are therefore determined:

image (13.146)

with the normalized radial functions

image (13.147)

The conventional definition of the constant is

image (13.148)

such that

image (13.149)

Laguerre and associated Laguerre polynomials can be found from the following generating functions:

image (13.150)

image (13.151)

13.13 Hypergeometric Functions

A geometric series is a function of whose terms constitute a geometric progression, in which the ratio of successive terms is equal. Consider, for example, with z and a, in general, being complex quantities,

image (13.152)

This series converges to the value image, provided that image. A certain type of generalization of a geometric series is known as a hypergeometric series or hypergeometric function. This has the form of a power series in z in which the coefficients image are replaced by ratios of rational functions of constants.

A rudimentary example of a hypergeometric function can be written

image (13.153)

Here image, known as the Pochhammer symbol (also called a rising or ascending factorial), is defined by

image (13.154)

or

image (13.155)

The last form is valid even if a is not an integer. For image reduces to the geometric series for z. When a is a negative integer, the series terminates, for example,

image (13.156)

Problem 13.13.1

Show that the Pochhammer symbol image.

Problem 13.13.2

Show that image is a solution of the first-order differential equation

image (13.157)

and that, in fact, image.

Next consider the hypergeometric function

image (13.158)

The series is convergent for finite z, provided that image It is known as the confluent hypergeometric function (explanation later) or Kummer function. It is a solution of the differential equation:

image (13.159)

Two important properties of the confluent geometric function are Kummer’s first formula, again for image or a negative integer,

image (13.160)

and Kummer’s second formula:

image (13.161)

For large image, the function has the asymptotic forms:

image (13.162)

For certain combinations of a, c, and z, the confluent hypergeometric function reduces to some of the special functions considered earlier in this chapter. First, the rather trivial case:

image (13.163)

(Even more trivial is image.) Some other elementary functions are obtained from

image (13.164)

and

image (13.165)

The error function can be represented by

image (13.166)

When image, we obtain a relation for Bessel functions:

image (13.167)

Another formula for a Bessel function:

image (13.168)

For image, the Laguerre polynomial is given simply by

image (13.169)

Hermite polynomials reduce slightly to different forms for even and odd n. For image,

image (13.170)

while, for image,

image (13.171)

Gauss’ famous hypergeometric differential equation is given by

image (13.172)

where c does not equal image There exist 24 possible solutions involving transformed versions of image. We will consider only the simplest one, which is what was originally called the hypergeometric function:

image (13.173)

There is a huge amount of mathematical technology involving the hypergeometric function. The interested reader can consult numerous references which cover these in great detail. We will content ourselves with some simple results:

image (13.174)

image (13.175)

image (13.176)

The Legendre polynomials can be represented by:

image (13.177)

The concept of confluence involves the substitution image, followed by taking the limit image. This reduces the hypergeometric differential equation and the hypergeometric function to their analogs for the confluent hypergeometric function.

It is also possible to define generalized hypergeometric functions of higher order, involving more than three constants:

image (13.178)

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