Benford's law, also called the first-digit law, refers to the frequency distribution of digits in many (but not all) real-life sources of data. In this distribution, the number 1 occurs as the first digit about 30% of the time, while larger numbers occur in that position less frequently. Indeed, the number 9 occurs as the first digit less than 5% of the time. The probability for the distribution of the digit d shown above is very close to P(d)=log10(d+1)-log10(d).
This is a rather surprising result, as most people would guess that P(d) should be a uniform distribution. Remarkably, its origin is not completely understood. For more details, a good starting point can be found on Wikipedia.

Instructor Contact Information

Professor: Howard Haber

Office ISB 326
Phone 459-4228
Office Hours Mondays, 2:30--4:30 pm
e-mail haber@scipp.ucsc.edu

Teaching Assistant

Angelo Monteux

Office ISB 329
Phone 459-1962
Discussion Section Mondays 5:00--6:10 pm, Nat. Sci. Annex 101
e-mail amonteux@ucsc.edu

Learning Support Services Tutor

Khilesh Mistry

LSS tutoring sessions contact Khilesh for details
e-mail         kpmistry@ucsc.edu

Miscellaneous

Displayed above are graphs of the five Legendre polynomials, Pn(x) over the interval -1≤x≤1 for n=1,2,3,4 and 5. These functions are orthonormal with respect to the standard inner product. For further details see Wolfram MathWorld.

Displayed above are graphs of the first five Bessel functions of the first kind, Jn(x), for n=0,1,2,3 and 4 over the range 0 ≤ x ≤ 10. For further details see Wolfram MathWorld. In particular, the numerical values of the zeros of the Bessel functions and their first derivatives can be found here.

Displayed below are graphs of the first five Bessel functions of the second kind, Yn(x) [which is denoted by Nn(x) in Boas], for n=0,1,2,3 and 4 over the range 0 ≤ x ≤ 10.

For further details see Wolfram MathWorld.

If you have access to MATLAB [see Section III of this website], you can explore Sturm-Liouville eigenfunctions with a graphical user interface developed by Benjamin Dyhr. For more details check out the following website, courtesy of the University of Arizona mathematics department.

Temperature profile in a hot plate determined from the solution to Laplace's equation. Details on the boundary conditions can be found here. This webpage was created by Rudi Winter.

In June, 2012, when I was visiting the (newly restored) Palace of Versailles, located 10 miles west-southwest of Paris, I came across a statue of Pierre-Simon Laplace (1749--1827). He is among the few scientists honored at the Palace.

Mode of vibration of a circular membrane. The nodes are related to zeros of the Bessel function J3. More details and links can be found here. This webpage was created by Paul Nylander.

For a more interactive demonstration of the vibrational modes, check out this applet which simulates waves in a membrane (like a drum head), showing its various vibrational modes (which you access by clicking on the different squares).

Polar plots of some representative spherical harmonics Ym(θ,φ), produced by P.Wormer using MATLAB software. This figure is more easily viewed from here. The superscript c (or s) in the figure means that the real (or imaginary) part of the corresponding spherical harmonic has been taken. This notation is used since Re eimφ=cos φ and Im eimφ=sin φ. The overall normalization factors have been omitted as they do not alter the shapes of the polar plots.

There is a very nice demonstration project produced by Stephen Wolfram, the inventor of Mathematica, which allows the user to vary ℓ and m to see how the polar plots of the spherical harmonics change. Check out the Spherical Harmonics demonstration. You can download the relevant CDF file and view it on your computer with the freely available Wolfram CDF Player.

Suppose a large, thin copper plate is at some uniform temperature. Now we imagine a very tiny device that pours out heat: We embed the device at a point in the plate. The resulting plate temperature will spike upward toward infinity at the heat source, and drop off sharply in all directions. This is the Green function for Poisson's equation for heat flow, which is depicted in the figure above. For some interesting details on the life of George Green, check out this website.

This image was taken from the excellent Wikipedia article on the Monty Hall problem. For an entertaining book that describes the history and many variants of this problem, check out the book by Jason Rosenhouse entitled The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser.

You can play Let's Make a deal on the web. Click here to see if you win the brand new car! If you chose wrongly, press the button below:

The top figure exhibits the probability distribution for the sum of the spots on two dice after a roll. The bottom figure presents a histogram of an actual simulation involving 10,000 trials. By comparing the histogram with the predicted probability, one can instantly determine how well the sample distribution matches the predicted probability distribution. Note a typographical error in the actual count corresponding to an outcome of 8. Presumably, it should be something like 1390 and not 1670. For further details, check out the interesting website created by Cary Rhode.

Given a normal distribution, confidence levels corresponding to 1σ and 2σ are ilustrated above. This illustration is taken from the following webpage entitled Drawing conclusions from data: Statistics don't lie--or do they?. You may find some useful tips on this page.

SPECIAL ANNOUNCEMENTS

Final grades for Physics 116C have been assigned. The final exam grades and the final course grades are listed in this PDF file by student ID in numerical order. The distribution of all the final course grades is shown below:

Letter grades are based on the cumulative course average, which is weighted according to: homework (9 problem sets with the low score dropped)---40%; midterm exam---25%; and the final exam---35%. This cumulative course average is then converted into a letter grade according to the following approximate numerical ranges: A+ (92--100); A (83--92); A- (77--83); B+ (71--77); B (65--71); B- (59--65); C+ (53--59); C(47--53); D(41--47); F (0--41). Here is the statistical summary of the distribution of the cumulative course averages:

mean: 63.6               median: 61.7               standard deviation: 12.8               high: 96.9               low : 38.2

Students who did not take the final exam are not included in this statistical summary.

Solutions to the final exam (and a link to the final exam statistics) can be found in Section VI on this website.

Physics 116C: Mathematical Methods in Physics III

I. General Information and Syllabus

The General Information and Syllabus handout is available in either PDF or Postscript format     [PDF | Postscript]
Some of the information in this handout is reproduced below.

Class Hours

Lectures: Tuesdays and Thursdays, 12--1:45 pm, Physical Sciences Building, Room 110

Discussion Section

Mondays 5:00--6:10 pm, Natural Sciences Annex, Room 101

Instructor Office Hours

Mondays, 2:30--4:30 pm.
I will also be available for consultation after class on Tuesdays and Thursdays.

Required Textbook

Mathematical Methods in the Physical Sciences, by Mary L. Boas

The latest list of errata for the textbook is available in the following PDF file.

Course Grading and Requirements

40% Weekly Homework (9 problem sets)
25% Midterm Exam (Thursday, November 8, 2012, 12--1:45 pm)
35% Final Exam (Thursday, December 13, 2012, 12--3 pm)

Homework assignments will be posted on the course website on a weekly basis, and are due at the beginning of class on the due date specified on the assignment sheet. The homework problem sets are not optional. You are encouraged to discuss the class material and homework problems with your classmates and to work in groups, but all submitted problems should represent your own work and understanding. In order that homework can be graded efficiently and returned quickly, there will be a 50% penalty for late homework. This penalty may be waived in special circumstances if you see me before the original due date. Homework solutions will be posted to the course website one or two days after the official due date; no late homeworks will be accepted after that.

There will be one midterm exam and one final exam:

• Thursday November 8, 2012       12--1:45 pm       Midterm exam
• Thursday December 13, 2012         12--3 pm             Final exam
All exams will take place in Physical Sciences Building, Room 110. The midterm exam will be 1 hour and 45 minutes long and cover the first half of the course. The final exam will be three hours long and cover the complete course material. You must take the final exam to pass the course. You will be permitted to consult the class textbook, your own handwritten notes, and any class handout during the exams.

Course Syllabus for Physics 116C

A snapshot of the course lecture schedule appears below. The dates given below are approximate.

Brief Course Outline for Physics 116C

TOPIC Lecture dates Readings
Series solutions of differential equations Sept 27 Boas Chapter 12, sections 1, 11, 21
Legendre polynomials and functions Oct 2, 4, 9 Boas, Chapter 12, sections 2--10
Bessel functions Oct 9, 11, 16 Boas, Chapter 12, sections 12--18, 20
The Sturm-Liouville problem Oct 18 Boas, Chapter 12, section 23
Hermite and Laguerre functions and polynomials Oct 23 Boas, Chapter 12, section 22
Partial differential equations of mathematical physics Oct 25, 30 Boas, Chapter 13, sections 1--4
Problems with cylindrical and spherical symmetry Nov 1, 6 Boas, Chapter 13, sections 5--7
MIDTERM EXAM Nov 8 September/October lectures
Potential theory and Green function techniques Nov 13 Boas Chapter 13, section 8
Integral transform solutions of partial differential equations Nov 15 Boas Chapter 13, section 9
Theory of probability Nov 20, 27 Boas Chapter 15, sections 1--4
Random variables and probability distributions Nov 29, Dec 4 Boas Chapter 15, sections 5--9
Statistics and experimental measurements Dec 6 Boas Chapter 15, section 10
FIANL EXAM Dec 13 entire course material

A summary of the course syllabi for Physics 116A,B,C is available in either PDF or Postscript format     [PDF | Postscript]

II. Learning Support Services

Learning Support Services (LSS) offers course-specific tutoring that is available to all UCSC students. Students meet in small groups (up to 4 people per group) led by a tutor. Students are eligible for up to one-hour of tutoring per week per course, and may sign-up for tutoring beginning October 9th at 10:00am.

III. Computer algebra systems

Although the use of computer algebra is not mandatory in this class, it can be a very effective tool for pedagogy. In addition, if used correctly, it can be an invaluable problem solving tool. Two of the best computer algebra systems available are Mathematica and Maple. Another useful software system for applied mathematics is MATLAB, which provides a high-level language and interactive environment for numerical computation, visualization, and programming.

There are student versions of Mathematica, Maple and MATLAB available, which sell for the cost of a typical mathematical physics textbook. If you do not wish to invest any money at this time, you can use Mathematica and MATLAB for free at computer labs on campus. For further information click here. A complete list of all the software available in the Learning Technologies computer labs, along with the file path to find the programs on each workstation can be found at this link.

IV. Homework Problem Sets and Exams

Problem sets and exams are available in either PDF or Postscript format.

• Homework Set #1--due: Tuesday October 9, 2012           [PDF | Postscript]
• Homework Set #2--due: Tuesday October 16, 2012         [PDF | Postscript]
• Homework Set #3--due: Tuesday October 23, 2012         [PDF | Postscript]
• Homework Set #4--due: Tuesday October 30, 2012         [PDF | Postscript]
• Homework Set #5--due: Tuesday November 6, 2012         [PDF | Postscript]
• Midterm Exam---Thursday November 8, 2012                   [PDF | Postscript]
• Homework Set #6--due: Thursday November 15, 2012     [PDF | Postscript]
• Homework Set #7--due: Tuesday November 27, 2012       [PDF | Postscript]
• Homework Set #8--due: Tuesday December 4, 2012         [PDF | Postscript]
• Homework Set #9--due: Tuesday December 11, 2012       [PDF | Postscript]
• Final Exam---Thursday December 13, 2012                       [PDF | Postscript]

V. Practice Problems for the Midterm and Final Exams and their solutions

Practice midterm and final exams can be found here. These should give you some idea as to the format and level of difficulty of the exam. Also provided are additional practice problems to aid you in your preparation for the exams. Solutions to the practice exam and additional practice problems will be provided two days before the corresponding exam.

VI. Solutions to Homework Problem Sets and Exams

The homework set and exam solutions are available in either PDF or postscript format. Solutions will be posted one or two days after the homework is due and after the exams have been completed. Solutions to homework sets are provided by Angelo Monteux, with some further editing by Howard Haber. There is no password required to view the solutions.

VII. Other Class Handouts

1. The Wronskian is especially useful in the theory of ordinary linear differential equations. If y1(x), y2(x),..., yn(x) are linearly independent solutions to the differential equation, then their Wronskian is nonvanishing. Even if the solutions cannot be determined analytically, one can determine the Wronskian using Abel's formula. This formula is derived in this handout entitled Applications of the Wronskian to ordinary linear differential equations, and a number of other applications of the Wronskian are exhibited.   [PDF | Postscript].

2. Given a homogeneous second-order linear differential equation, one can determine whether a given point (such as the origin) is an ordinary point, a regular singular point or an irregular singular point. In the first case, one can determine a power series solution to the differential equation. In the second case, the Frobenius method can be applied (which yields a generalized power series solution). The irregular singular point is not amenable to either method.   [PDF | Postscript].

3. An optional handout entitled Series solutions to a second order linear differential equation with regular singular points provides details on how to evaluate the corresponding series and determine the two linearly independent solutions. In particular, if the roots of the indicial equation are equal, then one of the linearly independent solutions includes a logarithmic term. The logarithmic case can also arise if the roots of the indicial equation differ by an integer. After providing the general formulae for the various cases, a number of simple examples are given where the second linearly independent solution contains a logarithm. One appendix of this handout employs the Bessel differential equation to illustrate all the possible cases.   [PDF | Postscript].

4. The function (1-2xt+t2)-1/2 is the generating function for the Legendre polynomials. An explicit proof of this result is presented by constructing the Taylor series for (1-2xt+t2)-1/2 around t=0 and extracting the coefficient of tn. Some applications of the generating function are given.   [PDF | Postscript].

5. The Laplace and Poisson equations, subject to Dirichlet, Neumann, or mixed boundary conditions, are well-posed problems. In particular, the solution to these problems, if they exist, are unique (or in the Neumann case, unique up to an arbitrary additive constant). The proof of this latter statement is provided in the class handout entitled Uniqueness of solutions to the Laplace and Poisson equations.   [PDF | Postscript].

6. In solving linear second order partial differential equations subject to Dirichlet, Neumann or mixed boundary conditions, one can sometimes express the general solution to the partial differential equation as some kind of Fourier series prior to imposing the last boundary condition. The coefficients of that series are then determined uniquely once the final boundary condition is imposed. In a class handout entitled Various kinds of Fourier series, I have summarized the properties of five different kinds of Fourier series that can arise in practical problems. The Fourier series can involve only sines or only cosines, or it may contain both sines and cosines. Moreover, in certain cases only sums over odd integers appear in the Fourier sum.   [PDF | Postscript].

7. A handout entitled Spherical Harmonics examines the solutions to Laplace's equations in spherical coordinates. These solutions possess a radial and angular part; the latter are called the spherical harmonics, Ym(θ,φ), where ℓ=0,1,2,3,... and m=-ℓ,-ℓ+1,...,ℓ-1, ℓ. These notes provide a discussion of the properties of the spherical harmonics, along with the famous addition theorem. Two proofs of the addition theorem are given (one in an appendix), and the application to the expansion of the inverse distance function is provided.   [PDF | Postscript].

8. A handout entitled The Laplacian of the inverse distance and the Green function provides a derivation of ∇2(1/r)=-4πδ3(r). This result is then applied to solving Poisson's equation. These notes also introduce the concept of the inverse Laplacian and discuss its relation to the Green function. Finally, a derivation of the Green function using Fourier transforms is provided.   [PDF | Postscript].

9. A handout entitled The Exclusion-Inclusion principle is extremely useful for certain counting problems. Moreover, it provides a valuable generalization to eq. (3.6) on p. 732 of Boas. In this handout, the inclusion-exclusion principle is derived. This principle is then used to show how to count the number of derangements (i.e. permutations with no fixed points) of n objects.   [PDF | Postscript].

10. A handout entitled The normal approximation to the binomial distribution demonstrates that the binomial distribution for successes in n Bernoulli trials with probability for success of p and probability for failure of q=1-p, is very well approximated by the normal distribution when n, np and nq are all very large.   [PDF | Postscript].

11. A collection of independent and identically distributed (or iid for short) random variables are a set of random variables with the same probability distribution (or density). These are suitable for modeling independent trials. If x1, x2,..., xn are n iid random variables, then E(xi)=μ and Var(xi)=σ2, for any value of i=1,2,3,...,n. The average of the n random variables is defined by x=(x1+ x2+...+xn)/n. In class, I derived the results E(x)=μ and Var(x) =σ2/n. Details of the derivation are conveniently summarized in the following note by C. Alan Bester from the University of Chicago.   [PDF]   This note refers to slides that can be found here, although you should be able to confirm each step based on the results obtained in class.

12.The standard deviation of the mean (also called the standard error) provides an estimate of the uncertainty of the experimental determination of the expected value of a random variable. In the case of n measurements, standard error is a factor of n smaller than the standard deviation of the random variable. The latter corresponds to the uncertainty in a single measurement. This distinction is clarified in a class handout entitled The Standard Deviation of the Mean.     [PDF | Postscript]

VIII. Free online textbooks

1. A very good source for free mathematical textbooks can be found on FreeScience webpage. I can also provide another useful link to a list of free mathematics textbooks.

2. Sean Mauch (of Caltech) provides a free massive textbook (2321 pages) entitled Introduction to Methods of Applied Mathematics. However, many of the topics of Physics 116C, if treated by Mauch, are in a preliminary status.

3. James Nearing also provides a free textbook entitled Mathematical Tools for Physics. This book treats some of the topics of Physics 116C at a similar level of difficulty.

4. Russel L. Herman has provided notes for a second course in differential equations, covering linear and nonlinear systems and boundary value problems. His monograph is entitled A Second Course in Ordinary Differential Equations: Dynamical Systems and Boundary Value Problems. Of particular interest to Physics 116C are Chapter 6 on Sturm Liouville problems, Chapter 7 on special functions (including Bessel functions and a variety of orthogonal polynomials) and Chapter 8 on Green functions.

5. Kenneth Howell provides links to the chapters of his book entitled Applied Differential Equations, Parts 1--4. Of particular interest to Physics 116C students are the additional chapters that comprise Parts 5--7, which include six chapters on series solutions to differential equations, and four chapters on boundary value problems and Sturm Liouville problems. A consensed version of his chapter on Sturm Liouville problems can be found in his lecture notes for a course in mathematical physics.

6. R.S. Johnson, a professor of applied mathematics at the University of Newcastle upon Tyne has provided, free of charge, a series of mini-books (from the Notebook Series) that cover some of the topics of Physics 116C. These include

7. A textbook entitled Partial Differential Equations for Scientists and Engineers, by Geoffrey Stephenson is available from the following PDF link.

8. A textbook entitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell has been generously made available free of charge from the following PDF link, along with PDF solutions to all odd-numbered problems. I especially recommend Chapter 4 for its superb treatment of conditional probability, with applications to the Monty Hall problem, the two-child paradox, and other purported paradoxes that arise in probability theory.

9. Another free textbook on probability entitled Introduction to Probability by Dimitri P. Bertsekas and John N. Tsitsiklis is based on a course taught at MIT. The level of difficulty is appropriate for Physics 116C students.     [PDF]

IX. Articles and Books of Interest

1. All the examples of the indicial equations given in Boas possess real roots. But a quadratic equation with real coefficients can also have complex roots. Applying the Frobenius method in the case of complex indicial roots is treated pedagogically in a paper entitled The Frobenius method for complex roots of the indicial equation by Joseph L. Neuringer, International Journal of Mathematical Education in Science and Technology, Volume 9, Issue 1, 1978, pages 71--77.     [PDF]

2. The most readable account that I have come across on the Sturm Liouville problem, suitable for Physics 116C students, is Chapter 11 of R. Kent Nagle, Edward B. Saff and Arthur David Snider, Fundamentals of Differential Equations and Boundary Value Problems, 6th Edition (Pearson Education, Inc., Boston, MA, 2012). Unfortunately, this book is too expensive, and there are no previews available either from google or amazon. You may be able to find the most recent or previous editions of this book in the Science and Engineering Library.

3. If you were wondering whether the Laguerre differential equation can also be solved by factorizing the equation into a product of first order differential operators and constructing ladder (raising and lowering) operators, check out the article by H. Fakhri and A. Chenaghlou entitled Ladder operators for the associated Laguerre functions in J. Phys. A: Math. Gen. 37 (2004) pp. 7499--7507.     [PDF]

4. The factorization method for solving differential equations was made popular by physicists. The method was described in a classic review paper by L. Infeld and T.E. Hull entitled The Factorization Method in Reviews of Modern Physics 23 (1951) pp. 21--68.     [PDF]

5. One of the best elementary treatments of partial differential equations suitable for physicists can be found in a book by Gabriel Barton, Elements of Green's Functions and Propagation---Potentials, Diffusion and Waves, published by Oxford Science Publications in 1989. Among other things, there is a very clear discussion of which boundary conditions constitute a well-posed problem.

6. Spherical harmonics play a critical role in computer graphics. Colorful visual representations of the spherical harmonics provide added insight into their properties and significance. See for example, the following two articles:

• Volker Schönefeld, Spherical Harmionics     [PDF]
• Peter-Pike Sloan, Stupid Spherical Harmonics Tricks     [PDF]

7. In the Monty Hall problem, there are three doors. Behind one of them is a brand new car, whereas the other two doors conceal goats. Monty Hall asks you to choose a door without opening it. Then, he opens one of the other doors to reveal a goat. At this point, Monty asks you whether you wish to switch your choice of doors or to stand pat. Indeed, what is the probabilty of winning the car if you switch your choice of doors? When Marylin vos Savant answered this question in the September 9, 1990 issue of Parade magazine, the reader response was overwhelming, many of whom could not believe her proposed answer that switching resulted in a probability of 2/3 for winning the car. Wikipedia provides an excellent introduction to this problem. For a more detailed account, read about the history of the Monty Hall problem and its many variants in this enertaining book by Jason Rosenhouse entitled The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser (Oxford University Press, Oxford, UK, 2009).

8. Apparent paradoxes often arise in probability theory. One such paradox is the called the two-child paradox, which asks for the probability that a two-child family has two boys given that one of the children is known to be a boy. Once again, Wikipedia provides an excellent introduction to this problem. I also highly recommend Chapter 4 of the free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell previously cited in Section XIII of this website. In addition, check out a fascinating article by Peter Lynch, The Two-Child Paradox: Dichotomy and Ambiguity in Irish Mathematical Society Bulletin 67 (2011) pp. 67--73     [PDF].

9. The Monty Hall problem and the two-child paradox can be analyzed by using conditional probabilites. Once again, I highly recommend Chapter 4 of the free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell previously cited in Section XIII of this website. Another excellent treatment of these and other well-known puzzles in probability can be found in an article by Maya Bar-Hillel and Ruma Falk, Some teasers concerning conditional probabilities in Cognition 11 (1982) pp. 109--122     [ PDF].

10. Continuous sample spaces can lead to apparent paradoxical behavior. I highly recommend Chapter 2 of the free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell previously cited in Section XIII of this website. This chapter includes a discussion of Buffon's needle (and the relation of pi to the probability of where a dropped needle lies). This chapter also provides a treatment of Bertrand's paradox, which involves a randomly drawn chord in the unit circle.

11. Finally, I can resist in mentioning The Banach-Tarski paradox which exhibits some very unintuitive phenomena that can arise in dealing with sets with an uncountable infinite number of points. For a very readable account, check out the book by Leonard M. Wapner entitled The Pea and the Sun: A Mathematical Paradox (A.K. Peters, Ltd., Wellesley, MA, 2005). For a slightly more technical treatment (which is still accessible to advanced undergraduate students), have a look at R. French, The Banach-Tarski Theorem, The Mathematical Intelligencer 10 (1988) 21--28.

12. The law of large numbers and the central limit theorem are treated at an elementary level in Chapters 8 and 9, respectively of Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell, previously cited in Section XIII of this website.

13. David L. Streiner provides an article entitled Maintaining Standards: Differences between the Standard Deviation and Standard Error, and When to Use Each, which should help illuminate the distinction between the standard deviation and the standard error (also known as the standard deviation of the mean).     [PDF]

X. Other Web Pages of Interest

1. A superb resource for both the elementary functions and the special functions of mathematical physics is the Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun, which is freely available on-line. The home page for this resource can be found here. There, you will find links to a frames interface of the book. Another scan of the book can be found here. A third independent link to the book can be found here.

2. The NIST Handbook of Mathematical Functions (published by Cambridge University Press), together with its Web counterpart, the NIST Digital Library of Mathematical Functions (DLMF), is the culmination of a project that was conceived in 1996 at the National Institute of Standards and Technology (NIST). The project had two equally important goals: to develop an authoritative replacement for the highly successful Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards (M. Abramowitz and I. A. Stegun, editors); and to disseminate essentially the same information from a public Web site operated by NIST. The new Handbook and DLMF are the work of many hands: editors, associate editors, authors, validators, and numerous technical experts. The NIST Handbook covers the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas. A PDF copy of the handbook is provided here:   [PDF]

3. Another very useful reference for both the elementary functions and the special functions of mathematical physics is An Atlas of Functions (2nd edition) by Keith B. Oldham, Jan Myland and Jerome Spanier, published by Springer Science in 2009. This resource is freely available on-line to students at the University of California at this link.

4. Yet another excellent website for both the elementary functions and the special functions of mathematical physics is the Wolfram Functions site. This site was created with Mathematica and is developed and maintained by Wolfram Research with partial support from the National Science Foundation.

5. One of the classic references to special functions is a three volume set entitled Higher Transcendental Functions (edited by A. Erdelyi), which was compiled in 1953 and is based in part on notes left by Harry Bateman. This was the primary reference for a generation of physicists and applied mathematicians, which is colloquially referred to as the Bateman Manuscripts. Although much of the material goes considerably beyond the level of Physics 116C, this esteemed reference work continues to be a valuable resources for students and professionals. PDF versions of the three volumes are now available free of charge. Check out the three volumes by clicking on the relevant links here:     [Volume 1 | Volume 2 | Volume 3].

6. Google is invaluable for searching for mathematical information. For example, if I need the first few Bessel function zeros, I search with google for "Bessel function zeros." The first hit is Bessel function zeros on the Wolfram MathWorld site.

7. Wikipedia provides some good articles on various mathematical topics. For example, a simple exposition on the the law of large numbers can be found here. The Wikipedia treatment of central limit theorem is more technical. What I discussed in class was the classical CLT (or Lindeberg-Levy CLT), which is presented in the first paragraph of the Wikipedia article. However, Wikipedia has another page that illustrates the central limit theorem which is quite illuminating. Check it out at this link.

haber@scipp.ucsc.edu
Last Updated: February 7, 2013