Case III. Complex conjugate roots are of minor practical importance, and we discuss the derivation of real solutions from complex ones just in terms of a typical example
Step 3. Solution of the Entire Problem
18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem
Recall from Sec. 10.8 that harmonic functions are solutions to Laplace’s equation and their second-order partial derivatives are continuous. In this section we explore how general properties of harmonic functions often can be obtained from properties of analytic functions. This can frequently be done in a simple fashion. Specifically, important mean value properties of harmonic functions follow readily from those of analytic functions.
The details are as follows.
T H E O R E M 1 Mean Value Property of Analytic Functions
Let be analytic in a simply connected domain D. Then the value of at a point in D is equal to the mean value of F(z)on any circle in D with center at z0. z0
F(z) F(z)
P R O O F In Cauchy’s integral formula (Sec. 14.3) (1)
we choose for C the circle in D. Then and
(1) becomes (2)
The right side is the mean value of Fon the circle ( value of the integral divided by the length of the interval of integration). This proves the theorem.
For harmonic functions, Theorem 1 implies
T H E O R E M 2 Two Mean Value Properties of Harmonic Functions
Let be harmonic in a simply connected domain D. Then the value of at a point in D is equal to the mean value of on any circle in D with center at This value is also equal to the mean value of on any circular disk in D with center [See footnote 1 in Sec. 18.4.]
P R O O F The first part of the theorem follows from (2) by taking the real parts on both sides,
The second part of the theorem follows by integrating this formula over rfrom 0 to (the radius of the disk) and dividing by
(3)
The right side is the indicated mean value (integral divided by the area of the region of integration).
Returning to analytic functions, we state and prove another famous consequence of Cauchy’s integral formula. The proof is indirect and shows quite a nice idea of applying the ML- inequality. (A bounded regionis a region that lies entirely in some circle about the origin.) T H E O R E M 3 Maximum Modulus Theorem for Analytic Functions
Let be analytic and nonconstant in a domain containing a bounded region R and its boundary. Then the absolute value cannot have a maximum at an interior point of R. Consequently, the maximum of is taken on the boundary of R. If F(z)⫽0in R, the same is true with respect to the minimum of ƒF(z)ƒ.
ƒF(z)ƒ ƒF(z)ƒ F(z)
䊏
£(x0, y0)⫽ 1
pr02 冮0r0冮02p£(x0⫹r cos a, y0⫹r sin a)r da dr.
r02>2,
r0
£(x0, y0)⫽Re F(x0⫹iy0)⫽ 1
2p冮02p£(x0⫹r cos a, y0⫹r sin a) da.
(x0, y0).
£(x, y) (x0, y0).
£(x, y) (x0, y0)
£(x, y)
£(x, y)
䊏 2p
⫽ F(z0)⫽ 1
2p 冮02pF(z0⫹reia) da.
z⫺z0⫽reia, dz⫽ireia da, z⫽z0⫹reia
F(z0)⫽ 1
2pi 冯C zF(z)⫺z0 dz
P R O O F We assume that has a maximum at an interior point of R and show that this leads to a contradiction. Let be this maximum. Since is not constant, is not constant, as follows from Example 3 in Sec. 13.4. Consequently, we can find a circle Cof radius rwith center at such that the interior of C is in R and is smaller than Mat some point P of C. Since is continuous, it will be smaller than Mon an arc of Cthat contains P(see Fig. 425), say,
for all zon
Let have the length Then the complementary arc of Chas the length
We now apply the ML-inequality (Sec. 14.1) to (1) and note that We then obtain (using straightforward calculation in the second line of the formula)
that is, which is impossible. Hence our assumption is false and the first statement is proved.
Next we prove the second statement. If in R, then is analytic in R.
From the statement already proved it follows that the maximum of lies on the boundary of R. But this maximum corresponds to the minimum of This completes
the proof. 䊏
ƒF(z)ƒ. 1>ƒF(z)ƒ 1>F(z) F(z)⫽0
M⬍M,
⬉ 1 2pa
M⫺k
r b L1⫹ 1 2pa
M
r b (2pr⫺L1)⫽M⫺ kL1 2pr ⬍M M⫽ ƒF(z0)ƒ ⬉ 1
2p `冮C
1
F(z)
z⫺z0 dz` ⫹ 1 2p `冮C
2
F(z) z⫺z0 dz`
ƒz⫺z0ƒ ⫽r.
2pr⫺L1. C2
L1. C1
C1. ƒF(z)ƒ ⬉M⫺k (k⬎0)
C1
ƒF(z)ƒ
ƒF(z)ƒ z0
ƒF(z)ƒ
F(z) ƒF(z0)ƒ ⫽M
z0
ƒF(z)ƒ
Fig. 425. Proof of Theorem 3
C1
z0
P C2
This theorem has several fundamental consequences for harmonic functions, as follows.
T H E O R E M 4 Harmonic Functions
Let be harmonic in a domain containing a simply connected bounded region R and its boundary curve C. Then:
(I) (Maximum principle)If is not constant, it has neither a maximum nor a minimum in R. Consequently, the maximum and the minimum are taken on the boundary of R.
(II) If is constant on C, then is a constant.
(III) If is harmonic in R and on C and if on C, then everywhere in R.
h(x, y)⫽£(x, y)
h(x, y)⫽ £(x, y) h(x, y)
£(x, y)
£(x, y)
£(x, y)
£(x, y)
P R O O F (I) Let be a conjugate harmonic function of in R. Then the complex
function is analytic in R, and so is Its absolute
value is
From Theorem 3 it follows that cannot have a maximum at an interior point of R.
Since is a monotone increasing function of the real variable the statement about the maximum of follows. From this, the statement about the minimum follows by replacing by
(II) By (I) the function takes its maximum and its minimum on C. Thus, if is constant on C, its minimum must equal its maximum, so that must be a constant.
(III) If hand are harmonic in Rand on C, then is also harmonic in Rand on C, and by assumption, everywhere on C. By (II) we thus have
everywhere in R, and (III) is proved.
The last statement of Theorem 4 is very important. It means that a harmonic function is uniquely determined in R by its values on the boundary of R.Usually, is required to be harmonic in Rand continuous on the boundary of R, that is,
where is on the boundary and is in R.
Under these assumptions the maximum principle (I) is still applicable. The problem of determining when the boundary values are given is called the Dirichlet problem for the Laplace equation in two variables, as we know. From (III) we thus have, as a highlight of our discussion,
T H E O R E M 5 Uniqueness Theorem for the Dirichlet Problem
If for a given region and given boundary values the Dirichlet problem for the Laplace equation in two variables has a solution, the solution is unique.
£(x, y)
(x, y) (x0, y0)
x:xlim0
y:y0
£(x, y)⫽ £(x0, y0),
£(x, y)
䊏 h⫺£⫽0 h⫺ £⫽0
h⫺ £
£
£(x, y)
£(x, y)
£(x, y)
⫺£.
£
£
£, e£
ƒG(z)ƒ
ƒG(z)ƒ ⫽eRe F(z)⫽e£(x, y).
G(z)⫽eF(z). F(z)⫽ £(x, y)⫹i°(x, y)
£(x, y)
°(x, y)
PROBLEMS RELATED TO THEOREMS 1 AND 2 1–4 Verify Theorem 1 for the given and circle of radius 1.
1.
2.
3.
4.
5. Integrate around the unit circle. Does the result contradict Theorem 1?
6. Derive the first statement in Theorem 2 from Poisson’s integral formula.
ƒzƒ
(z⫺1)ⴚ2, z0⫽ ⫺1 (3z⫺2)2, z0⫽4 2z4, z0⫽ ⫺2 (z⫹1)3, z0⫽52
F(z), z0,
7–9 Verify (3) in Theorem 2 for the given and circle of radius 1.
7.
8.
9.
10. Verify the calculations involving the inequalities in the proof of Theorem 3.
11. CAS EXPERIMENT. Graphing Potentials. Graph the potentials in Probs. 7 and 9 and for two other functions of your choice as surfaces over a rectangle in the xy-plane. Find the locations of the maxima and minima by inspecting these graphs.
x⫹y⫹xy, (1, 1) x2⫺y2, (3, 8) (x⫺1)(y⫺1), (2, ⫺2) (x0, y0),
£(x, y),
P R O B L E M S E T 1 8 . 6
12. TEAM PROJECT. Maximum Modulus of Analytic Functions. (a) Verify Theorem 3 for (i) and
the rectangle (ii)
and the unit disk, and (iii) and any bounded domain.
(b) is not zero in the disk and
has a minimum at an interior point. Does this contradict Theorem 3?
(c) (x real) has a maximum 1 at Why can this not be a maximum of in a domain containing ?
(d) If is analytic and not constant in the closed
unit disk D: and on the unit
circle, show that must have a zero in D.
13–17 MAXIMUM MODULUS
Find the location and size of the maximum of in the unit disk
13. F(z)⫽cos z ƒzƒ⬉1.
ƒF(z)ƒ F(z)
ƒF(z)ƒ ⫽c⫽const ƒzƒ⬉1
F(z)
z⫽p>2
ƒF(z)ƒ ⫽ ƒsin zƒ p>2.
F(x)⫽sin x
ƒzƒ ⬉2 F(z)⫽1⫹ ƒzƒ
F(z)⫽ez
F(z)⫽sin z 1⬉x⬉5, 2⬉y⬉4,
F(z)⫽z2
1. Why can potential problems be modeled and solved by methods of complex analysis? For what dimensions?
2. What parts of complex analysis are mainly of interest to the engineer and physicist?
3. What is a harmonic function? A harmonic conjugate?
4. What areas of physics did we consider? Could you think of others?
5. Give some examples of potential problems considered in this chapter. Make a list of corresponding functions.
6. What does the complexpotential give physically?
7. Write a short essay on the various assumptions made in fluid flow in this chapter.
8. Explain the use of conformal mapping in potential theory.
9. State the maximum modulus theorem and mean value theorems for harmonic functions.
10. State Poisson’s integral formula. Derive it from Cauchy’s formula.
11. Find the potential and the complex potential between the plates and kept at 10 V and 110 V, respectively.
12. Find the potential and complex potential between the coaxial cylinders of axis 0 (hence the vertical axis
in space) and radii kept at
potential and respectively.
13. Do the task in Prob. 12 if and the outer cylinder is grounded, U2⫽0.
U1⫽220 V U2⫽2 kV, U1⫽200 V
r1⫽1 cm, r2⫽10 cm, y⫽x⫹10
y⫽x
14. If plates at and are kept at potentials is the potential at larger or smaller than the potential at in Prob. 12?
No calculation. Give reason.
15. Make a list of important potential functions, with applications, from memory.
16. Find the equipotential lines of
17. Find the potential in the first quadrant of the xy-plane if the x-axis has potential 2 kV and the y-axis is grounded.
18. Find the potential in the angular region between the plates kept at 800 V and
kept at 600 V.
19. Find the temperature Tin the upper half-plane if, on
the x-axis, for and for
20. Interpret Prob. 18 as an electrostatic problem. What are the lines of electric force?
21. Find the streamlines and the velocity for the complex potential Describe the flow.
22. Describe the streamlines for
23. Show that the isotherms of are
hyperbolas.
24. State the theorem on the behavior of harmonic functions under conformal mapping. Verify it for
and
25. Find V in Prob. 22 and verify that it gives vectors tangent to the streamlines.
w⫽u⫹iv⫽z2.
£*⫽eu sin v
F(z)⫽ ⫺iz2⫹z F(z)⫽12z2⫹z.
F(z)⫽(1⫹i)z.
x⬍1.
⫺30°C x⬎1
T⫽30°C
Arg z⫽p>3 Arg z⫽p>6
F(z)⫽i Ln z.
r⫽5
x⫽5 U1⫽200 V, U2⫽2 kV,
x2⫽10 x1⫽1
C H A P T E R 1 8 R E V I E W Q U E S T I O N S A N D P R O B L E M S
14.
15.
16. (a, bcomplex,
17.
18. Verify the maximum principle for and the rectangle
19. Harmonic conjugate. Do and a harmonic conjugate in a region Rhave their maximum at the same point of R?
20. Conformal mapping. Find the location of the
maximum of in
where Find the region Rthat is mapped
onto by Find the potential in R
resulting from and the location of the maximum. Is the image of ? If so, is this just by chance?
(x1, y1) (u1, v1)
(x1, y1)
£* w⫽f(z)⫽z2. R*
w⫽u⫹iv.
R*: ƒwƒ⬉1, v⭌0,
£*⫽eu cos v
(u1, v1)
°
£
a⬉x⬉b, 0⬉y⬉2p.
£(x, y)⫽ex sin y F(z)⫽2z2⫺2
a⫽0) F(z)⫽az⫹b
F(z)⫽sinh 2z F(z)⫽exp z2
Potential theoryis the theory of solutions of Laplace’s equation (1)
Solutions whose second partial derivatives are continuous are called harmonic functions. Equation (1) is the most important PDE in physics, where it is of interest in two and three dimensions. It appears in electrostatics (Sec. 18.1), steady-state heat problems (Sec. 18.3), fluid flow (Sec. 18.4), gravity, etc. Whereas the three- dimensional case requires other methods (see Chap. 12), two-dimensional potential theory can be handled by complex analysis, since the real and imaginary parts of an analytic function are harmonic (Sec. 13.4). They remain harmonic under conformal mapping (Sec. 18.2), so that conformal mapping becomes a powerful tool in solving boundary value problems for (1), as is illustrated in this chapter.
With a real potential in (1) we can associate a complex potential
(2) (Sec. 18.1).
Then both families of curves and have a physical meaning.
In electrostatics, they are equipotential lines and lines of electrical force (Sec. 18.1).
In heat problems, they are isotherms (curves of constant temperature) and lines of heat flow (Sec. 18.3). In fluid flow, they are equipotential lines of the velocity potential and streamlines (Sec. 18.4).
For the disk, the solution of the Dirichlet problem is given by the Poisson formula (Sec. 18.5) or by a series that on the boundary circle becomes the Fourier series of the given boundary values (Sec. 18.5).
Harmonic functions, like analytic functions, have a number of general properties;
particularly important are the mean value property and the maximum modulus property(Sec. 18.6), which implies the uniqueness of the solution of the Dirichlet problem (Theorem 5 in Sec. 18.6).
° ⫽const
£⫽const
F(z)⫽£⫹i°
£
ⵜ2£⫽0.
S U M M A R Y O F C H A P T E R 1 8
Complex Analysis and Potential Theory
S o f t w a r e ( p . 7 8 8 – 7 8 9 )
C H A P T E R 1 9 Numerics in General C H A P T E R 2 0 Numeric Linear Algebra C H A P T E R 2 1 Numerics for ODEs and PDEs
787
P A R T E
Numeric Analysis
Numeric analysisor briefly numerics continues to be one of the fastest growing areas of engineering mathematics. This is a natural trend with the ever greater availability of computing power and global Internet use. Indeed, good software implementation of numerical methods are readily available. Take a look at the updated list of Software starting on p. 788. It contains software for purchase (commercial software) and software for free download (public-domain software). For convenience, we provide Internet addresses and phone numbers. The software list includes computer algebra systems (CASs), such as Maple andMathematica, along with the Maple Computer Guide, 10th ed., and Mathematica Computer Guide, 10th ed., by E. Kreyszig and E. J. Norminton related to this text that teach you stepwise how to use these computer algebra systems and with complete engineering examples drawn from the text. Furthermore, there is scientific software, such as IMSL,LAPACK (free download), and scientific calculators with graphic capabilities such as TI-Nspire. Note that, although we have listed frequently used quality software, this list is by no means complete.
In your career as an engineer, appplied mathematician, or scientist you are likely to use commercially available software or proprietary software, owned by the company you work for, that uses numeric methods to solve engineering problems, such as modeling chemical or biological processes, planning ecologically sound heating systems, or computing trajectories of spacecraft or satellites. For example, one of the collaborators of this book (Herbert Kreyszig) used proprietary software to determine the value of bonds, which amounted to solving higher degree polynomial equations, using numeric methods discussed in Sec. 19.2.
However, the availability of quality software does not alleviate your effort and responsibility to first understand these numerical methods. Your effort will pay off because, with your mathematical expertise in numerics, you will be able to plan your solution approach, judiciously select and use the appropriate software, judge the quality of software, and, perhaps, even write your own numerics software.
Numerics extends your ability to solve problems that are either difficult or impossible to solve analytically. For example, certain integrals such as error function [see App. 3, formula (35)] or large eigenvalue problems that generate high-degree characteristic polynomials cannot be solved analytically. Numerics is also used to construct approximating polynomials through data points that were obtained from some experiments.
Part E is designed to give you a solid background in numerics. We present many numeric methods as algorithms, which give these methods in detailed steps suitable for software implementation on your computer, CAS, or programmable calculator. The first chapter, Chap. 19, covers three main areas. These are general numerics (floating point, rounding errors, etc.), solving equations of the form (using Newton’s method and other methods), interpolation along with methods of numeric integration that make use of it, and differentiation.
Chapter 20 covers the essentials of numeric linear algebra. The chapter breaks into two parts: solving linear systems of equations by methods of Gauss, Doolittle, Cholesky, etc.
and solving eigenvalue problems numerically. Chapter 21 again has two themes: solving ordinary differential equations and systems of ordinary differential equations as well as solving partial differential equations.
Numerics is a very active area of research as new methods are invented, existing methods improved and adapted, and old methods—impractical in precomputer times—are rediscovered. A main goal in these activities is the development of well-structured software. And in large-scale work—millions of equations or steps of iterations—even small algorithmic improvements may have a large significant effect on computing time, storage demand, accuracy, and stability.
Remark on Software Use.Part E is designed in such a way as to allow compelete flexibility on the use of CASs, software, or graphing calculators.The computational requirements range from very little use to heavy use. The choice of computer use is at the discretion of the professor. The material and problem sets (except where clearly indicated such as in CAS Projects, CAS Problems, or CAS Experiments, which can be omitted without loss of continuity) do not require the use of a CAS or software. A scientific calculator perhaps with graphing capabilities is all that is required.
Software
See also http://www.wiley.com/college/kreyszig/
The following list will help you if you wish to find software. You may also obtain information on known and new software from websites such as Dr. Dobb’s Portal, from articles published by the American Mathematical Society(see also its website at www.ams.org), the Society for Industrial and Applied Mathematics (SIAM, at www.siam.org), the Association for Computing Machinery(ACM, at www.acm.org), or the Institute of Electrical and Electronics Engineers(IEEE, at www.ieee.org). Consult also your library, computer science department, or mathematics department.
f(x)⫽0
TI-Nspire.Includes TI-Nspire CAS and programmable graphic calculators. Texas Instru- ments, Inc., Dallas, TX. Telephone: 1-800-842-2737 or (972) 917-8324; website at www.education.ti.com.
EISPACK.See LAPACK.
GAMS (Guide to Available Mathematical Software). Website at http://gams.nist.gov.
Online cross-index of software development by NIST.
IMSL (International Mathematical and Statistical Library). Visual Numerics, Inc., Houston, TX. Telephone: 1-800-222-4675 or (713) 784-3131; website at www.vni.com.
Mathematical and statistical FORTRAN routines with graphics.
LAPACK.FORTRAN 77 routines for linear algebra. This software package supersedes LINPACK and EISPACK. You can download the routines from www.netlib.org/lapack.
The LAPACK User’s Guide is available at www.netlib.org.
LINPACKsee LAPACK
Maple. Waterloo Maple, Inc., Waterloo, ON, Canada. Telephone: 1-800-267-6583 or (519) 747-2373; website at www.maplesoft.com.
Maple Computer Guide. For Advanced Engineering Mathematics, 10th edition. By E. Kreyszig and E. J. Norminton. John Wiley and Sons, Inc., Hoboken, NJ. Telephone:
1-800-225-5945 or (201) 748-6000.
Mathcad. Parametric Technology Corp. (PTC), Needham, MA. Website at www.ptc.com.
Mathematica.Wolfram Research, Inc., Champaign, IL. Telephone: 1-800-965-3726 or (217) 398-0700; website at www.wolfram.com.
Mathematica Computer Guide.For Advanced Engineering Mathematics, 10th edition.
By E. Kreyszig and E. J. Norminton. John Wiley and Sons, Inc., Hoboken, NJ. Telephone:
1-800-225-5945 or (201) 748-6000.
Matlab. The MathWorks, Inc., Natick, MA. Telephone: (508) 647-7000; website at www.mathworks.com.
NAG.Numerical Algorithms Group, Inc., Lisle, IL. Telephone: (630) 971-2337; website at www.nag.com. Numeric routines in FORTRAN 77, FORTRAN 90, and C.
NETLIB.Extensive library of public-domain software. See at www.netlib.org.
NIST. National Institute of Standards and Technology, Gaithersburg, MD. Telephone:
(301) 975-6478; website at www.nist.gov. For Mathematical and Computational Science Division telephone: (301) 975-3800. See also http://math.nist.gov.
Numerical Recipes.Cambridge University Press, New York, NY. Telephone: 1-800-221- 4512 or (212) 924-3900; website at www.cambridge.org/us. Book, 3rd ed. (in see App. 1, Ref. [E25]; source code on CD ROM in which also contains old source code (but not text) for (out of print) 2nd ed. C, FORTRAN 77, FORTRAN 90 as well as source code for (out of print) 1st ed. To order, call office at West Nyack, NY, at 1-800-872-7423 or (845) 353-7500 or online at www.nr.com.
FURTHER SOFTWARE IN STATISTICS.See Part G.
C⫹⫹,
C⫹⫹)
790
C H A P T E R 1 9
Numerics in General
Numeric analysis or briefly numerics has a distinct flavor that is different from basic calculus, from solving ODEs algebraically, or from other (nonnumeric) areas. Whereas in calculus and in ODEs there were very few choices on how to solve the problem and your answer was an algebraic answer, in numerics you have many more choices and your answers are given as tables of values (numbers) or graphs. You have to make judicous choices as to what numeric method or algorithm you want to use, how accurate you need your result to be, with what value (starting value) do you want to begin your computation, and others. This chapter is designed to provide a good transition from the algebraic type of mathematics to the numeric type of mathematics.
We begin with the general concepts such as floating point, roundoff errors, and general numeric errors and their propagation. This is followed in Sec. 19.2 by the important topic of solving equations of the type by various numeric methods, including the famous Newton method. Section 19.3 introduces interpolation methods. These are methods that construct new (unknown) function values from known function values. The knowledge gained in Sec. 19.3 is applied to spline interpolation (Sec. 19.4) and is useful for under- standing numeric integration and differentiation covered in the last section.
Numerics provides an invaluable extension to the knowledge base of the problem- solving engineer. Many problems have no solution formula (think of a complicated integral or a polynomial of high degree or the interpolation of values obtained by measurements).
In other cases a complicated solution formula may exist but may be practically useless.
It is for these kinds of problems that a numerical method may generate a good answer.
Thus, it is very important that the applied mathematician, engineer, physicist, or scientist becomes familiar with the essentials of numerics and its ideas, such as estimation of errors, order of convergence, numerical methods expressed in algorithms, and is also informed about the important numeric methods.
Prerequisite:Elementary calculus.
References and Answers to Problems:App. 1 Part E, App. 2.