essential mathematics for the physical sciences7 bessel functions and friends 7-1 7.1 small-argument...

Essential Mathematics for thePhysical Sciences

Volume I: Homogeneous boundary value problems, Fourier methods, andspecial functions

Essential Mathematics for thePhysical Sciences

Volume I: Homogeneous boundary value problems, Fourier methods, andspecial functions

Brett BordenDepartment of Physics, Naval Postgraduate School, Monterey, USA

James LuscombeDepartment of Physics, Naval Postgraduate School, Monterey, USA

Morgan & Claypool Publishers

Copyright ª 2017 Morgan & Claypool Publishers

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ISBN 978-1-6817-4485-8 (ebook)ISBN 978-1-6817-4484-1 (print)ISBN 978-1-6817-4487-2 (mobi)

DOI 10.1088/978-1-6817-4485-8

Version: 20171001

IOP Concise PhysicsISSN 2053-2571 (online)ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise PhysicsPublished by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA,94901, USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

Contents

Preface viii

Author biography x

1 Partial differential equations 1-1

Exercise 1-7

2 Separation of variables 2-1

2.1 Helmholtz equation 2-1

2.2 Helmholtz equation in rectangular coordinates 2-3

2.3 Helmholtz equation in cylindrical coordinates 2-4

2.4 Helmholtz equation in spherical coordinates 2-6

2.5 Roadmap: where we are headed 2-8

Summary 2-9

Exercises 2-10

Reference 2-10

3 Power-series solutions of ODEs 3-1

3.1 Analytic functions and the Frobenius method 3-2

3.2 Ordinary points 3-5

3.3 Regular singular points 3-10

3.4 Wronskian method for obtaining a second solution 3-16

3.5 Bessel and Neumann functions 3-19

3.6 Legendre polynomials 3-23

Summary 3-26

Exercises 3-26

References 3-27

4 Sturm–Liouville theory 4-1

4.1 Differential equations as operators 4-1

4.2 Sturm–Liouville systems 4-3

4.3 The SL eigenvalue problem, L[y]=−λwy 4-5

4.4 Dirac delta function 4-10

4.5 Completeness 4-11

v

4.6 Hilbert space: a brief introduction 4-13

Summary 4-14

Exercises 4-14

References 4-16

5 Fourier series and integrals 5-1

5.1 Fourier series 5-1

5.2 Complex form of Fourier series 5-6

5.3 General intervals 5-8

5.4 Parseval’s theorem 5-13

5.5 Back to the delta function 5-14

5.6 Fourier transform 5-16

5.7 Convolution integral 5-21

Summary 5-25

Exercises 5-26

References 5-28

6 Spherical harmonics and friends 6-1

6.1 Properties of the Legendre polynomials, Pl(x) 6-2

6.2 Associated Legendre functions, Plm(x) 6-11

6.3 Spherical harmonic functions, Ylm(θ, ϕ) 6-12

6.4 Addition theorem for Ylm(θ, ϕ) 6-13

6.5 Laplace equation in spherical coordinates 6-20

Summary 6-22

Exercises 6-22

References 6-26

7 Bessel functions and friends 7-1

7.1 Small-argument and asymptotic forms 7-1

7.2 Properties of the Bessel functions, Jn(x) 7-3

7.3 Orthogonality 7-8

7.4 Bessel series 7-10

7.5 Fourier–Bessel transform 7-12

7.6 Spherical Bessel functions 7-13

7.7 Expansion of plane waves in spherical coordinates 7-18

Essential Mathematics for the Physical Sciences

vi

Summary 7-19

Exercises 7-20

Reference 7-22

Appendices

A Topics in linear algebra A-1

B Vector calculus B-1

C Power series C-1

D Gamma function, Γ(x) D-1


vii

Preface

Physics is expressed in the language of mathematics. Mathematics is deeplyingrained in how physics is taught and how it is practiced. Like any language, itshapes our cognition about the workings of the physical world. The concept ofeigenvalue, for example, shapes our understanding of the quantum realm.Mathematics is so much a part of physics, it is safe to say that physics would notbe possible without it. A study of the mathematics used in science is thus a soundintellectual investment (for training as scientists and engineers), and courses inmathematical physics have long been a component of physics curricula.

Why there is such an intimate connection between science and mathematics isdifficult to say. Perhaps it is because those who invent mathematics, humans, arepart of the same world that scientists (other humans) try to describe. In 1960 thephysicist Eugene Wigner published ‘The unreasonable effectiveness of mathematicsin the natural sciences’, an essay probing the connection between mathematics andphysics1. He states that:

…the mathematical formulation of the physicist’s often crude experience leadsin an uncanny number of cases to an amazingly accurate description of a largeclass of phenomena. This shows that the mathematical language has more tocommend it than being the only language which we can speak; it shows that itis, in a very real sense, the correct language.

Nowhere is this more true than in our conception of physical reality as described byfields governed by partial differential equations (PDEs).A fieldψ(r, t) has a value at eachpoint of space (labeled by the position vector r) and at each instant of time t. Physicalsystems are described by numerous fields: the velocity field of a fluid; the pressure field ofacoustical waves; and the electromagnetic field. To study advanced physics onemust, atthe very least, become proficient in the methods used to solve PDEs.

These considerations determine the structure of this book, which is in two parts, notjust with respect to the topics presented, but in terms of the role a course inmathematicalmethods plays in a physics curriculum, complete with its attendant time constraints. Thefirst volume is centered on methods of solving PDEs and the special functionsintroduced. Solving PDEs cannot be done, however, outside the context in whichthey apply to physical systems. The solutions to PDEs must conform to boundaryconditions, a set of additional constraints in space or time to be satisfied at theboundaries of the system, that small part of the universe under study. This first volumeis devoted to homogeneous boundary-value problems (BVPs), homogeneous implying asystem lacking a forcing function, or source function. The second volume takes up (inaddition to other topics) inhomogeneous problems where, in addition to the intrinsicPDE governing a physical field, source functions are an essential part of the system.

This text is based on a course offered by the Department of Physics at the NavalPostgraduate School (NPS) in Monterey, California. Our students are somewhat

1Wigner E P 1960 Commun. Pure Appl. Math. 13 1.

viii

unique for several reasons. They are, for the most part, mid-career military officerswho have been sent to obtain a graduate degree in physics as part of theirpreparation to assume technical responsibilities in a development or acquisitionprogram office. A few of the PhD students will go on to teach at one of the militaryacademies. And while our entering students usually have solid undergraduatepreparation in the hard sciences, they have not typically been ‘in practice’ of theseskills for most of the decade before reporting to NPS. They also have a limitedamount of time to refresh these skills, acquire new ones, and write a thesis thatjustifies the awarding of an advanced degree in physics: our MS students mustcomplete the process in two years. Given these time constraints, we have chosen toinclude only the material students will see elsewhere in some part of their two-yearcourse of study, although that material is still quite extensive in scope.

We have further addressed the two-year time constraint by constructing thestudents course matrix so that different classes can be ‘mutually constructive’—i.e.material required to support one course is concurrently offered by another course inthe matrix. Math refresher courses are essential to this strategy. Upon reporting toNPS, students start with a comprehensive ten-week review of lower-divisionmathematics: multi-dimensional calculus; ordinary differential equations (ODEs)with constant coefficients; and linear algebra. But unlike traditional reviews, thisfirst course is coordinated to complement topics as they appear in the other courses—ODEs, for example, are introduced in week 2 so as to coincide with motionequations in the course on elementary mechanics.

This book is based on the assumption that it follows a math review course, andwas designed to coincide with the second quarter of student study, which isdominated by BVPs but also requires an understanding of special functions andFourier analysis. In our experience, standard texts are not well suited to this‘interleaving’ approach and, because they typically present the material in encyclo-pedic fashion, serve better as references than texts. While produced in response tothe needs of NPS students, we believe that our approach to presenting mathematicalphysics would work well at other universities facing similar time constraints.

The second volume contains the material of an additional ten-week course inmathematical methods required of our PhD students, topics not typically required inthe coursework that MS students are required to complete but, nevertheless,essential for more advanced work (such as complex variable methods). The amountof mathematics one could be called upon to know in a scientific career is trulydaunting; clearly many more topics could be included in a text on mathematicalphysics, although at the expense of our goal of producing a text covering only theessentials. Notably, this text does not include a discussion of probability andstatistics (which is better taught as an integrated part of our statistical physicscourse). Nor do we discuss tensor analysis on manifolds (which is an integral part ofan elective course on general relativity).

Brett BordenJames LuscombeMonterey, CA


ix

Author biography

Brett Borden

Brett Borden received his undergraduate degree from the University of Wisconsin inMadison and the PhD from the University of Texas at Austin (both in Physics). Hejoined the Research Department at The Naval Weapons Center in China Lake, CAin 1980. In 2002 he joined the Faculty of The Naval Postgraduate School inMonterey, CA, where he is Professor of Physics (Emeritus). Dr Borden is a Fellow ofThe Institute of Physics.

James Luscombe

James H Luscombe received the PhD in Physics from the University of Chicago in1983. After post-doctoral positions at the University of Toronto and Iowa StateUniversity, he joined the Research Laboratory of Texas Instruments, where heworked on the development of nanoelectronic devices. In 1994, he joined the NavalPostgraduate School in Monterey, CA, where he is Professor of Physics. He wasChair of the Department of Physics from 2003–2009. He teaches a wide variety oftopics, including general relativity, statistical mechanics, mathematical methods,and quantum computation. He has published more than 60 research articles, and hasgiven more than 100 conference presentations; he holds two patents.

x

IOP Concise Physics

Essential Mathematics for the Physical SciencesVolume I: Homogeneous boundary value problems, Fourier methods, and special functions

Brett Borden and James Luscombe

Chapter 1

Partial differential equations

In this chapter we offer a quick survey of the different types of partial differentialequations (PDEs) encountered in physics. Solving such equations is the subjectmatter of the rest of the book. The goal here is to get a list of the generic types ofPDEs in front of you as soon as possible; peek ahead to table 1.1.

Continuity equationLet us start with the continuity equation, a PDE describing the behavior in time andspace of conserved quantities. Consider a quantityQ t( ) of some substance containedin a fixed volume V at time t (the precise nature of the substance is immaterial; itcould be charge, matter, or energy). To say that Q is conserved means that it cannotjust ‘disappear’ from V, the only way Q can decrease is if the substance ‘leaves thebuilding’, i.e. flows through the surface S bounding V. The conservation of Q ismodeled by the equation

∮ ·= − J r atQ t t

dd

( ) ( , ) d , (1.1)S

where J r t( , ) is the local current density vector (dimensions of substance per unitsurface area per unit time) and = ˆa n ad d is the infinitesimal element of surface area,considered as a vector quantity in the direction of the outwards unit normal n̂ of S atlocation r. Equation (1.1) is known as a balance equation:the rate of change of Q isaccounted for by the net flow through S. If the net flow is positive (negative),Q decreases (increases) in time. For fixed V, equation (1.1) is equivalent to

∫ ∫∮ ∫· ·

ρ ρ

∇

= = ∂∂

= − = −

rr

J r a J r

tQ t

tt r

tt

r

t t r

dd

( )dd

( , )d( , )

d

( , ) d ( , )d ,(1.2)V V

S V

3 3

3

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http://dx.doi.org/10.1088/978-1-6817-4485-8ch1

where ρ r t( , ) is the local density function (substance per volume), and we haveinvoked the divergence theorem.We can take the time derivative inside the integralin the second equality in equation (1.2) because V is fixed. Because equation (1.2)holds for any V, we have the differential form of equation (1.1) known as thecontinuity equation:

·ρ ∇∂∂

+ =Jt

0. (1.3)

The continuity equation shows up in any description of conserved quantities; forexample, in quantum mechanics for the probability of finding a particle in a volume.

Diffusion equationThe diffusion equation is a general type of PDEobtained from the continuity equation.It follows from the continuity equation by using a phenomenological model of thecurrent density vector, ρ∇= −J D . This relation (Fickʼs law) models flow as occurringagainst the direction of the gradient in ρ. Fickʼs law models what is observedmacroscopically—substances disperse in such a way as to remove density inhomoge-neities. The quantity D, the diffusion coefficient, is a proportionality factor betweenthe flux and the density gradient; it can be found experimentally. When Fickʼs lawis combined with the continuity equation (1.3), we obtain the diffusion equation:

ρ ρ∂∂

− ∇ =t

D 0. (1.4)2

Note the characteristic feature of the diffusion equation: first-order derivatives intime dependence, second-order derivatives in space dependence. You should be ableto see from equation (1.4) that D has dimensions of length squared per time.

Free-particle Schrödinger equationThe time-dependent Schrödinger equation for a free particle of mass m is1:

ψ ψℏ∂∂

+ ℏ ∇ =t m

i2

0, (1.5)2

2

where ψ (a complex-valued quantity) is such that *ψ ψr rt t r( , ) ( , )d3 is the probabilityof finding the particle within a volume rd3 at location r at time t. The Schrödingerequation has the form of the diffusion equation with a complex diffusion coefficient,

= ℏD mi (2 ). The mathematical method of solving equation (1.5) is the same asthat for equation (1.4); the same math has the same solution. It can be shown thatthe Schrödinger equation gives rise to a continuity equation with *ρ ψ ψ≡ and

* *ψ ψ ψ ψ∇ ∇≡ ℏ −J mi (2 )( ).

1 The Schrödinger equation cannot be derived from something more fundamental. It can be motivated, but notderived; it is something entirely new. We would not teach it, however, if it had not been successfully testedagainst the results of experimental findings. Nature is the arbiter of truth.


1-2

Heat equationThe heat conduction equation, or simply the heat equation, is a variant of thediffusion equation. Start from the continuity equation with the conserved quantitybeing thermal energy. From the first law of thermodynamics, =Q C Td dV , where

Qd is the heat transferred from an object under a change in temperature Td , and CV

is the heat capacity of the substance at constant volume. The specific heat cv is theheat capacity per mass. The change in thermal energy density ud associated with Tdis then ρ=u c Td dv , where ρ is the mass density. Fourierʼs law is the phenomeno-logical model of the heat current density, κ∇= −J T , where κ is a material-specificquantity called the thermal conductivity. Combining Fourierʼs law with the con-tinuity equation, we obtain the heat equation

α∂∂

− ∇ =Tt

T 0, (1.6)2

where α κ ρ≡ c( )v is the thermal diffusivity.

Inhomogeneous diffusion equationSuppose we have a quantity that is not conserved in the sense stated above. In thatcase, we can modify the balance equation (1.1) by introducing an additional sourceterm:

∮ ∫·= − +J r a rtQ t t S t r

dd

( ) ( , ) d ( , )d .S V

3

Here rS t( , ) is the local source density function representing the rate at which thesubstance comprising Q in V is either increasing or decreasing by a means otherthan flowing through the surface bounding V. Such a situation occurs in semi-conductors where charge carriers are locally produced by exposure to electro-magnetic radiation. Invoking Fickʼs law as above, we obtain the inhomogeneousdiffusion equation

ρ ρ∂∂

− ∇ =rr r

tt

D t S t( , )

( , ) ( , ). (1.7)2

The mathematics involved in solving inhomogeneous PDEs such as equation (1.7) issufficiently different from that involved in the solution of the correspondinghomogeneous PDE that we postpone the treatment of such equations until thesecond volume of this text.

Schrödinger equationThe Schrödinger equation for a particle in a potential energy environmentcharacterized by the function rV ( ) has the form of the inhomogeneous diffusionequation:


1-3

ψ ψ ψℏ∂∂

+ ℏ ∇ = r rt m

V ti2

( ) ( , ). (1.8)2

2

The inhomogeneous term in equation (1.8) involves the product of ψ (what we aretrying to solve for) and the potential energy function V.

Poisson equationIn electromagnetism, the static (time-independent) electric field vector E r( ) is relatedto the charge density function ρ r( ) through Gaussʼs law: · ρ ϵ∇ =E 0, where ϵ0 is acharacteristic quantity (the permittivity of free space). The static E field is obtainedfrom the gradient of a scalar field ϕ r( ), the electrostatic potential function, with

ϕ∇= −E . By combining ϕ∇= −E with Gaussʼs law, we obtain the Poissonequation, which is a second-order PDE describing the differential behavior of theelectrostatic potential:

ϕ ρ ϵ∇ = − r( ) . (1.9)20

Solving the Poisson equation for a given charge density function ρ r( ) is one of themajor tasks of the theory of electrostatics. PDEs in the form of the Poisson equationalso follow from the inhomogeneous diffusion equation under steady-state condi-tions, where all time derivatives are zero.

Laplace equationThe Laplace equation is the version of the Poisson equation when ρ =r( ) 0,

ϕ∇ = 0. (1.10)2

The Laplace equation holds a special place in the pantheon of PDEs. Under steady-state conditions, the temperature field satisfies the Laplace equation, ∇ =T 02 .

Wave equationLet us derive the wave equation. Consider a string of mass density ρ (mass/length),under tension T stretched along the x-axis. At time t, let the string have transversedisplacement ψ x t( , ) (figure 1.1). From Newtonʼs second law,

θ θ ρ ψ− = ∂∂+T T xt

( sin ) ( sin ) d .x x xd

2

2

For a displacement sufficiently small that ψ∂ ∂ ≪x 1, we can approximateθ θ ψ≈ ≈ ∂ ∂xsin tan so that θ ψ≈ ∂ ∂T T xsin , in which case the equation of

motion becomes in the limit →xd 0,

ψ ψ∂∂

− ∂∂

=x c t

10, (1.11)

2

2 2

2

2

where ρ≡c T2 . Equation (1.11) is the one-dimensional wave equation where c is thespeed of wave propagation on the string. By extending the derivation to a stretched


1-4

membrane and then to a three-dimensional elastic medium, the homogeneous waveequation in any number of dimensions is

ψ ψ∇ − ∂∂

=c t1

0. (1.12)22

2

2

Inhomogeneous wave equationIn electrodynamics it is shown that the time-dependent scalar potential ϕ r t( , )satisfies the inhomogeneous wave equation,which we cite without derivation:

ϕ ϕ ρ ϵ∇ − ∂∂

= −rr

rtc

tt

t( , )1 ( , )

( , )/ . (1.13)22

2

2 0

Summary of PDEsWe can stop now; we see the pattern. The most commonly encountered PDEs inphysics involve second-order spatial derivatives and up to second-order timederivatives, with or without source functions. In the first volume of this text weconcentrate on the homogeneous versions of these PDEs (no source terms), savinginhomogeneous differential equations for the second volume. Table 1.1 lists thesePDEs together with generic source functions, rS( ) or rS t( , ). The quantity rS( ) couldindicate, for example, ( ρ ϵ− r( ) 0) in equation (1.9), while rS t( , ) could indicate( ρ ϵ− r t( , ) 0) in equation (1.13).

Role of boundary conditionsIf different areas of physics involve the same PDEs, what distinguishes differentsubjects? Answer: the boundary conditions are different. As a simple example,consider the steady-state heat equation for a one-dimensional system2,

Figure 1.1. String under tension.

2We of course live in a three-dimensional world. There are systems, however, where the variations of quantitiesin two of the dimensions are negligible, leaving us with only variations in one dimension to consider.


1-5

=Tx

dd

0.2

2

The general solution of this equation is particularly simple: α β= +T x x( ) , where αand β are constants. To match the general solution with the temperature distributionof a particular system, we require more information. We require the boundary valuesof T at the ends of the system. Assuming the system occupies the interval ⩽ ⩽x L0 ,the values of T at x = 0 and at x = L need to be specified (although these are notthe only kinds of boundary conditions that could be specified).With that informa-tion supplied, we have the temperature distribution that satisfies the steady-stateheat equation and the boundary conditions:

= − +T x T L TxL

T( ) [ ( ) (0)] (0).

Moral of the story: PDEs are not completely specified until we have information aboutthe boundary conditions.

Looking aheadAs an aspiring student of physics, you must become proficient in solving boundary-value problems (BVPs) in each of the three major coordinate systems: rectangular(x y z, , ), cylindrical (ρ ϕ z, , ), and spherical ( θ ϕr, , ). Each coordinate system has itsown characteristic special functions—solutions to the Laplace equation—and youwill need to get comfortable with them. In this part of the text we treat homogeneousBVPs (those associated with the homogeneous PDEs), and in the second we turn tosolutions of inhomogeneous PDEs. For future reference we list the expression for theLaplacian operator in each of these coordinate systems.

Rectangular coordinates: (−∞ < < ∞x y z, , )

∇ = ∂∂

+ ∂∂

+ ∂∂x y z

.22

2

2

2

2

2

Cylindrical coordinates: ( ρ⩽ < ∞0 , ϕ π⩽ <0 2 , −∞ < < ∞z )

⎛⎝⎜

⎞⎠⎟ρ ρ

ρρ ρ ϕ

∇ = ∂∂

∂∂

+ ∂∂

+ ∂∂z

1 1.2

2

2

2

2

2

Table 1.1. Generic PDEs.

ψ∇ =r rt S( , ) ( )2 Poisson equation (Laplace if =rS( ) 0)

ψα

ψ∇ − ∂∂

=rr

rtt

tS t( , )

1 ( , )( , )2

Diffusion equation

ψ ψ∇ − ∂∂

=rr

rtc

tt

S t( , )1 ( , )

( , )22

2

2

Wave equation


1-6

Spherical coordinates: ( ⩽ < ∞r0 , θ π⩽ ⩽0 , ϕ π⩽ <0 2 )

⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠θ θ

θθ θ ϕ

∇ = ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂r r

rr r r

1 1sin

sin1

sin.2

22

2 2 2

2

2

Exercise1.1. The following equation represents a solution to the one-dimensional

homogeneous diffusion equation:

⎛⎝⎜

⎞⎠⎟∫ψ

πψ= ′ − − ′ ′

−∞

∞x t

Dtx

x xDt

x( , )1

4d exp

( )4

( , 0). (1.14)2

Show this. Verify that equation (1.14) solves the differential equation

ψ ψ∂∂

− ∂∂

=x D t

10. (1.15)

2

2

You should find that all parts of equation (1.14) are required to satisfy thediffusion equation, except apparently the factor of π4 . We will show inchapter 4 that the zero-time limit of equation (1.14) properly reproduces theinitial condition, i.e. ψ ψ=→ x t xlim ( , ) ( , 0)t 0 , which accounts for the factorof π4 .


1-7

IOP Concise Physics

Essential Mathematics for the Physical SciencesVolume I: Homogeneous boundary value problems, Fourier methods, and special functions

Brett Borden and James Luscombe

Chapter 2

Separation of variables

In this chapter we introduce a procedure for producing solutions of PDEs—themethod of separation of variables. We do not solve BVPs in this chapter; not yet, thatwill come later. We illustrate, in each of three coordinate systems, how PDEs arereduced to a set of ordinary differential equations (ODEs).

2.1 Helmholtz equationConsider the time-dependent heat equation (1.6), which we reproduce here:

ψα

ψ∇ − ∂∂

=t

10. (1.6)2

We use the symbol ψ to denote temperature because, as we will see, we are going touse T for another purpose. We are looking for a temperature distribution that isspace- and time-dependent, ψ ψ= r t( , ). The separation of variables method is toguess the solution of equation (1.6) as the product of two functions, each a functionof a single variable:

ψ =r rt R T t( , ) ( ) ( ). (2.1)

How do we know if our guess works? Substitute it into equation (1.6), whichyields

α∇ =T R R

Tt

1 dd

. (2.2)2

Note that the partial derivative with respect to time in equation (1.6) has beenreplaced with an ordinary derivative in equation (2.2) (because T depends only on

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http://dx.doi.org/10.1088/978-1-6817-4485-8ch2

t and so, as far as T t( ) is concerned, there is no difference between ∂ ∂t/ and td/d ).Now, divide equation (2.2) by ψ = RT to obtain1:

α∇ =

rr

RR

T tT t

t1( )

( )1 1

( )d ( )

d.

(2.3)

depends only on spatial variables depends only on time

2

� ��

We now come to a key step in this method, one that we will frequently invoke. Theterms on the left of equation (2.3) depend only on spatial variables, while the termson the right depend only on time. Such an equality is possible only if both sides areequal to a constant. (The term on the left is independent of t, the term on the right isindependent of r, yet they are equal for all values of r and t.) Let us rewriteequation (2.3) to make explicit the equality with a constant:

α∇ = − =

rr

RR k

T tT t

t1( )

( )1 1

( )d ( )

d, (2.4)2 2

where −k2 is the separation constant. We have written it as −k2 so that no matterwhat k turns out to be, the separation constant is manifestly negative. How did weknow to make it negative? Experience. If that makes you uneasy, try it with apositive separation constant and see how your answer turns out. If it does not makesense physically, reconsider your decision to try a positive constant.

The ODE on the right side of equation (2.4) is readily solved (check it!):

= α−T t A( ) e , (2.5)k t2

where A is a constant. A negative separation constant thus leads to a temperature thatdecays in time. Unless somethingwas forcing the temperature to indefinitely increase intime, an exponentially decaying solution in time conforms to our physical expectations.

That leaves the left side of equation (2.4):

∇ + =r rR k R( ) ( ) 0. (2.6)2 2

Equation (2.6) is known as the Helmholtz equation, a PDE involving the spatialvariables only. To make progress with equation (2.6), we must declare a coordinatesystem (what we will do shortly). Note what has happened, however. By guessing aproduct solution ψ =r rt R T t( , ) ( ) ( ), the original PDE, equation (1.6), a differentialequation in four variables, has been reduced to a PDE involving three variables plusan ODE, the solution of which is readily obtained to produce the time variation,equation (2.5). That has been accomplished at the expense of introducing anunknown quantity (k2) into the analysis. The quantity k is determined by theboundary conditions, as we will see in upcoming chapters.

A Helmholtz equation can always be produced from second-order PDEs in whichthe space and time derivatives occur separately. Consider the homogeneous wave

1Your inner mathematician may be asking: ‘What if =RT 0?’ We are explicitly not looking for the trivialsolution.


2-2

equation, equation (1.12). By separating variables with ψ =r rt R T t( , ) ( ) ( ), weobtain the analog of equation (2.4):

∇ = − =R

R kc T

Tt

1 1 1 dd

. (2.7)2 22

2

2

The time-dependent part T t( ) is readily obtained (check it!):

= + −T t A B( ) e e . (2.8)kct kcti i

The spatial part rR( ) is obtained as the solution of the Helmholtz equation (2.6).The homogeneous PDEs in table 1.1 can, in each case, be reduced to the Helmholtz

equation through the method of separation of variables. The Laplace equation is aspecial case with =k 02 . We thus turn to the Helmholtz equation in the majorcoordinate systems utilized in scientific work.

2.2 Helmholtz equation in rectangular coordinatesThe Helmholtz equation in rectangular coordinates is, from equation (2.6),

⎛⎝⎜

⎞⎠⎟

∂∂

+ ∂∂

+ ∂∂

+ =x y z

k R x y z( , , ) 0. (2.9)2

2

2

2

2

22

To solve equation (2.9) try as a solution a product of three unknown functions,

=R x y z X x Y y Z z( , , ) ( ) ( ) ( ). (2.10)

Substitute equation (2.10) into equation (2.9) and divide by =R XYZ to obtain

″ + ″ + ″ + =X

XY

YZ

Z k1 1 1

0, (2.11)2

where the notation ″X indicates a second derivative of X with respect to itsargument, ″ ≡X X xd /d2 2. Once more we make the separation-constant argument;rewrite equation (2.11) in the form

″ = − + ″ + ″− − −X X k Y Y Z Z( ) .x y zdepends only on depends only on and

1 2 1 1��

The term on the left is independent of y and z, the terms on the right are independentof x; the only way we can have an equality for all x, y, z, is if each side is equal to aconstant. The argument can be repeated for each term in equation (2.11). Weconclude that each term in equation (2.11) is equal to a constant:

″ = − ″ = − ″ = −X

X kY

Y kZ

Z k1 1 1

, (2.12)12

22

32

where + + =k k k k12

22

32 2. We have taken the separation constants in equation

(2.12) to each be negative; that might need to be revised depending on the particularsof the system.


2-3

Thus, we have reduced the Helmholtz equation, a PDE in three variables, to threeODEs, a step that comes at the expense of introducing three unknown constants.Actually, only two are independent because of the constraint + + =k k k k1

222

32 2

(if k2 is known); alternatively, k1, k2, and k3 could be found through the impositionof boundary conditions, with k2 then defined as = + +k k k k2

12

22

32. The ODEs

in equation (2.12) have simple solutions:

= + = += +

− −

−

X x A B Y y A B

Z z A B

( ) e e ( ) e e

( ) e e .(2.13)

k x k x k y k y

k z k z

1i

1i

2i

2i

3i

3i

1 1 2 2

3 3

Example. Laplace equation in two dimensionsConsider the two-dimensional steady-state heat equation, ∇ =T 02 , or

∂∂

= −∂∂

Tx

Ty

.2

2

2

2

Writing =T x y X x Y y( , ) ( ) ( ) implies, from the PDE,

″ = − ″X

XY

Y1 1

.

From now on, we will skip the argument that such an equality can be achievedonly if each side is equal to a constant (did you get that?). Taking theseparation constant α− 2 to be negative, X and Y must satisfy

α α″ + = ″ − =X X Y Y0 0.2 2

In this case, the solutions for X and Y can be written

α α α α= + = +X x A x B x Y y C y D y( ) cos sin ( ) cosh sinh . (2.14)

The solution to the PDE (∇ =T 02 ) is incomplete until the constants α andA, B,C, D are known; these will be determined once boundary conditions have beenspecified.

The method of separation of variables is useful because it reduces a difficult problem(a PDE) to one with which we are more comfortable (a set of ODEs). But the methoddoes not always work: a PDE expressed in a coordinate frame is said to be separablewhen it can be reduced to separate ODEs in those coordinates; otherwise, it is said tobe non-separable. Evidently, the Helmholtz equation in rectangular coordinates isseparable. It can be shown that the Helmholtz equation is separable in 11 differentcoordinate systems [1]—only three of which will be important in the following.

2.3 Helmholtz equation in cylindrical coordinatesThe Helmholtz equation in cylindrical coordinates is:


2-4

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟ρ ρ

ρρ ρ ϕ

ψ ρ ϕ∂∂

∂∂

+ ∂∂

+ ∂∂

+ =z

k z1 1

( , , ) 0, (2.15)2

2

2

2

22

where we have changed notation again ( ψ↔R ); we will use R for another purpose.Try as a solution, a product of three unknown functions:

ψ ρ ϕ ρ ϕ= Φz R Z z( , , ) ( ) ( ) ( ). (2.16)

Substitute equation (2.16) into equation (2.15), and divide by ψ:

⎛⎝⎜

⎞⎠⎟ρ ρ

ρρ ρ

+Φ

Φ″ + ″ + =R

RZ

Z k1 d

ddd

1 1 10. (2.17)

22

In the usual way (i.e. proceeding by an argument that we hope is familiar by now),let us introduce a separation constant. Let

ΦΦ″ = −m

1, (2.18)2

the solution of which is

ϕΦ = ϕ( ) e . (2.19)mi

We will not bother with integration constants here; that will come later when weassemble the solution ψ ρ ϕ ρ ϕ= Φz R Z z( , , ) ( ) ( ) ( ). We also do not bother to split outseparate solutions for ±m; that is taken care of by letting m have positive andnegative values in equation (2.19). The quantity m is particularly simple to ascertain.The solution for Φ must be single-valued: if the azimuthal coordinate ϕ is permittedto ‘wrap around’ the z-axis, we demand that

ϕ π ϕΦ + = Φ( 2 ) ( ). (2.20)

The only way equation (2.20) can be satisfied by ϕΦ = ϕ( ) e mi is if m is an integer.Equation (2.20) can be considered a type of boundary condition, one imposed by theinternal consistency of the solution. With m determined, equation (2.17) reduces to

⎛⎝⎜

⎞⎠⎟ρ ρ

ρρ ρ

− + ″ + = =R

R mZ

Z k m1 d

ddd

10 ( integer). (2.21)

2

22

Time for another separation constant. Let

α″ = −Z

Z k1

. (2.22)2 2

The right side of equation (2.22) could be positive or negative depending on the system,i.e. depending on boundary conditions. If α > k2 2 the solution of equation (2.22) is

α α= − + − −( ) ( )Z z A k z B k z( ) exp exp . (2.23)2 2 2 2

For α < k2 2, the form of Z z( ) would be in terms of complex exponentials. Theimportant point is that the form of Z z( ) is known.


2-5

Having introduced the separation constants m and α, equation (2.21) reduces to

⎛⎝⎜

⎞⎠⎟ρ

αρ

″ + ′ + − =R Rm

R1

0. (2.24)22

2

The differential equation for ρR( ) is called the radial equation. Equation (2.24) canbe simplified through a change of variables. Let αρ≡x , where we assume thatα ≠ 0. It can then be shown (do this) that equation (2.24) is the same as

⎛⎝⎜

⎞⎠⎟

″ + ′ + − =R xx

R xmx

R x( )1

( ) 1 ( ) 0, (2.25)m m m

2

2

where we have labeled the solution of equation (2.25) by the value of m. Because forevery m there is a new ODE to solve, it is convenient to label the solutions by m.

In some sense we are done, or at least we will be done when we know how tosolve equation (2.25). We started out with the Helmholtz equation in cylindricalcoordinates, equation (2.15), for ψ ρ ϕ z( , , ), a function of three variables. By‘guessing’ that ψ = ΦR Z can be written as the product of three unknown functions,we have been able, by introducing two separation constants, to obtain the formof ϕΦ( ), equation (2.19), and that for Z z( ), equation (2.23). It remains to solveequation (2.25), which is likely to be unfamiliar because it is an ODE withnon-constant coefficients. Equation (2.25) is Bessel’s differential equation, and itssolutions are Bessel functions. We will learn all about Bessel functions in chapter 7.There are two linearly independent solutions of equation (2.25): J x( )m , the Besselfunction, and N x( )m , the Neumann function. The solution of equation (2.25) cantherefore be written (without knowing the form of Jm and Nm):

= +R x A J x B N x( ) ( ) ( ), (2.26)m m m m m

where Am and Bm are constants.For the special case of α = 0, equation (2.24) reduces to

ρ ρ″ + ′ − =R R

mR

10. (2.27)

2

2

The solution to equation (2.27) has the form (check it!)

⎧⎨⎩ρρ ρ

ρ=

+ ≠+ =

−R

a b ma b m

( )0

ln 0.(2.28)

m m

2.4 Helmholtz equation in spherical coordinatesLet us perform this procedure in spherical coordinates. From equation (2.6),

⎜ ⎟ ⎜ ⎟⎛⎝⎜

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎞⎠⎟θ θ

θθ θ ϕ

ψ θ ϕ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

+ =r r

rr r r

k r1 1

sinsin

1sin

( , , ) 0. (2.29)2

22 2 2

2

22

We are going to implement separation of variables here in two stages. Write

ψ θ ϕ θ ϕ=r R r Y( , , ) ( ) ( , ), (2.30)


2-6

essential mathematics for the physical sciences7 bessel functions and friends 7-1 7.1 small-argument...

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