 Suppose you made a different basis choice at each point in space. Then, your landscape might look like the famous lithograph by M.C. Escher above entitled Print Gallery. Escher was also fond of mirror reflecions (see below). For more information on Escher and his work, check out the official Escher website; many of his works can be viewed from the "Picture Gallery" link on that webpage. Another website with many Escher prints for viewing can be found here.

Instructor Contact Information

Professor: Howard Haber

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

Teaching Assistant

Jonathan Cornell

Office ISB 314
Phone 459-4138
Discussion Section Mondays, 4:15--5:45 pm, in ISB 235
e-mail jcornell@ucsc.edu

Learning Support Services Tutor

Matteo Crismani

e-mail         mcrisman@ucsc.edu

Required textbook Click here for the errata.

Miscellaneous If you stack n bricks on a table, how far can you make them extend over the edge without toppling? For bricks of unit length, the answer is given by one-half the nth harmonic number, i.e. the sum of the series 1/2 + 1/4 + 1/6 + 1/8 + ... + 1/2n. For more details, click here and scroll down to the section entitled Hung over. This problem is also discussed in this note written by David Bressoud. A proof that (1/2)+(1/4)+(1/8)+...=1 by geometric construction.  This is an excerpt from a 1996 Foxtrot cartoon created by Bill Amend. On the first midterm exam, I asked you to prove that this series was conditionally convergent. The actual summation of the series is more difficult. A solution to this problem (along with the complete Foxtrot cartoon) can be found here. Mathematical posters such as this one from RJ:Design can be found at this link. The complex plane can be mapped onto the surface of the sphere and vice versa. The north pole of the sphere maps onto the "point of infinity". That is, all of complex infinity is mapped onto a single point! For further details, check out the Wikipedia article on the Riemann sphere. This image is a plot of the complex function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). The hue represents the argument (Arg) of the complex function, while the brightness represents the magnitude. Further details can be found at this link. This image also appears on Wikipedia. Above, a plot of the gamma function Γ(x) as a function of the real variable x. Below, a plot of the absolute magnitude of the gamma function |Γ(x+iy)| plotted in the complex plane. Note the spikes occur on the negative real axis for x=0,-1,-2,-3,... where the abolute value of the gamma function approaches infinity. A great source for graphs of mathematical functions, both real-valued and complex-valued, can be found in the Gallery of mathematical functions. This is an artistic rendition of a four-dimensional hypersphere. For further details on how we can view objects in four dimensions, check out this link, which is maintained by Rebecca Frankel at MIT. Above is a plot of the absolute value of the Riemann zeta function |ζ(z)| plotted in the complex plane (taken from this source). One of the most famous unproven mathematical conjectures states that all non-trivial zeros of ζ(z) lie along the line Re z=1/2. The matrix has you. Click here for more information. Hollywood seems to have a thing for linear algebra. The figure above is a convenient mnemonic device for computing the determinant of a 3x3 matrix. In the active transformation (left), point P moves relative to the coordinate frame to location P', while the coordinate frame remains unchanged (in this case, the position of P has rotated clockwise by angle θ), while in a passive transformation (right), point P is observed in two different coordinate frames (pictured rotated relative to one another by angle θ) The coordinates of P' in the active case are the same as the coordinates of P in the rotated frame in the passive case if the rotation of P is clockwise and the rotation of the axes is counterclockwise. (Reference: Wikipedia) A line passing through the origin (denoted by the thick blue line) is a one-dimensional subspace of R3. It is the intersection of two planes (colored green and yellow), each of which is a two-dimensional subspace R3. This figure is taken from Wikipedia, which has a rather nice comprehensive description of linear vector spaces. The distribution of eigenvalues of a symmetric random matrix with entries chosen from a standard normal distribution is illustrated above for a random 5000 x 5000 matrix. This is an illustration of Wigner's Semicircle Law. The figure above is a visualization of a symmetric tensor in three dimensions. The object in the figure is the sum of a spear, a plate and a sphere. The spear describes the principal direction of the tensor, where the length is proportional to the largest eigenvalue. The plate describes the plane spanned by the eigenvectors corresponding to the two largest eigenvalues. The sphere, with a radius proportional to the smallest eigenvalue, shows how isotropic the tensor is. For further details, check out the following link.

SPECIAL ANNOUNCEMENTS Nearly eight months after the end of this class, we have been treated to a wonderful article by Brian Hayes in the November--December 2011 issue of American Scientist on the incredibly shrinking N-dimensional hypersphere. Check out the link in the tenth item listed in Section IX of this website.

Final grades for Physics 116A have been assigned. You can pick up your final exam and any uncollected homeworks from my office.

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 (10 problem sets with the low score dropped)---40%; first midterm exam---15%; second first midterm exam---15%; and the final exam---30%. This cumulative course average is then converted into a letter grade according to the following approximate numerical ranges: A+ (88--100); A (82--88); A- (76--82); B+ (70--76); B (64--70); B- (58--64); C+ (52--58); C(46--52); D(40--46); F (0--40). Here is the statistical summary of the distribution of the cumulative course averages:

mean: 61.6               median: 63.1               high: 88.0               low : 11.1

Solutions to the final exam (and a link to the final exam statistics) can be found in Section VI on this website. In addition to the solutions to the exam problems, three supplementary appendices have been included which provide additional material connected to two of the exam questions. I have also posted one last class handout to Section VII of this website, which clarifies an assertion made in Appendix C of the exam solutions.

This website will be kept live for the remainder of the calendar year 2011. Feel free to consult it while you are taking Physics 116B and C.

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 (led by Jonathan Cornell)

Mondays, 4:15--5:45 pm, ISB 235
Due to campus closures on Monday January 17 and February 21, the discussion section normally scheduled for those two Mondays will be held on Tuesday, 5:30--7:00 pm, ISB 213 (note the room change).
Homework assignments that are due on Tuesday January 18 and February 22 can be handed in no later than 5 pm on the following day to my ISB mailbox without penalty.

Instructor Office Hours

Mondays, 2--4 pm.
In addition, I will be available in my office from 5:00--6:30 pm on most Mondays, Tuesdays and Thursdays.

Required Textbook

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

The author keeps track of the latest list of errata for her textbook in the following PDF file.

40% Weekly Homework (10 problem sets)
15% First Midterm Exam (Thursday, January 27, 2011)
15% Second Midterm Exam (Thursday, February 24, 2011)
30% Final Exam (Wednesday, March 16, 2011, 4--7 pm)

Homework assignments will be posted on the course website on a weekly basis, and are due each Tuesday at the beginning of class. (The final homework set of the course will be due the day before the final exam.) 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 made available one or two days after the official due date; no late homeworks will be accepted after that.

The two midterm exams and final exams will be held in the same classroom as the lectures. Each midterm will be a one hour exams, and will be followed by a shortened lecture of 45 minutes. 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

The course syllabus is available in either PDF or Postscript format     [PDF | Postscript]
A snapshot of the course lecture schedule appears below. The dates given below are approximate.

Brief Course Outline for Physics 116A

Infinite series, power series and asymptotic series Jan 4, 6, 11 Boas Chapter 1
Complex numbers and complex functions Jan 13, 18, 20 Boas, Chapter 2
Special functions defined by integrals Jan 25, 27, Feb 1, 3 Boas, Chapter 11
Matrices, linear algebra and vector spaces Feb 8, 10, 15, 17, 22 Boas, Chapter 3
Eigenvalue problems and matrix diagonalization Feb 24, March 1, 3 Boas, Chapter 3
Tensor Algebra March 8, 10 Boas, Chapter 10

There will be two midterm exams and one final exam:

• Thursday January 27, 12--1 pm       Midterm exam #1
• Thursday February 24, 12--1 pm     Midterm exam #2
• Wednesday March 16, 4--7 pm             Final exam
All exams will take place in Physical Sciences Building, Room 110.

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 by clicking on this link beginning January 12 at 10:00 am. For further information, consult the following FAQ sheet.

The LSS tutor assigned to Physics 116A is Matteo Crismani (e-mail: mcrisman@ucsc.edu)

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.

There are student versions of both Mathematica and Maple 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 for free at computer labs on campus. For further information click here.

IV. Homework Problem Sets and Exams

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

• Homework Set #1--due: Tuesday January 11, 2011       [PDF | Postscript]
• Homework Set #2--due: Tuesday January 18, 2011       [PDF | Postscript]
• Homework Set #3--due: Tuesday January 25, 2011       [PDF | Postscript]
• Midterm Exam #1---Thursday January 27, 2011             [PDF | Postscript]
• Homework Set #4--due: Tuesday February 1, 2011       [PDF | Postscript]
• Homework Set #5--due: Tuesday February 8, 2011       [PDF | Postscript]
• Homework Set #6--due: Tuesday February 15, 2011       [PDF | Postscript]
• Homework Set #7--due: Tuesday February 22, 2011       [PDF | Postscript]
• Midterm Exam #2---Thursday February 24, 2011           [PDF | Postscript]
• Homework Set #8--due: Tuesday March 1, 2011           [PDF | Postscript]
• Homework Set #9--due: Tuesday March 8, 2011           [PDF | Postscript]
• Homework Set #10--due: Tuesday March 15, 2011        [PDF | Postscript]
• Final Exam---Wednesday March 16, 2011                     [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. Solutions 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 PDF format. Solutions will be posted one or two days after the homework is due and after the exams have been completed. There is no password required to view the solutions.

• Solution Set #1     [PDF | Postscript]         mean score: 88 (out of 120 available points)
• Solution Set #2     [PDF | Postscript]         mean score: 111 (out of 140 available points)
• Solution Set #3     [PDF | Postscript]         mean score: 82 (out of 100 available points)
• Solutions to Midterm Exam #1         [PDF | Postscript]     [exam statistics]
• Solution Set #4     [PDF | Postscript]         mean score: 57 (out of 70 available points)
• Solution Set #5     [PDF | Postscript]         mean score: 70 (out of 110 available points)
• Solution Set #6     [PDF | Postscript]         mean score: 86 (out of 100 available points)
• Solution Set #7     [PDF | Postscript]         mean score: 90 (out of 110 available points)
• Solutions to Midterm Exam #2         [PDF | Postscript]     [exam statistics]
• Solution Set #8     [PDF | Postscript]         mean score: 58 (out of 70 available points)
• Solution Set #9     [PDF | Postscript]         mean score: 70 (out of 90 available points)
• Solution Set #10   [PDF | Postscript]         mean score: 63 (out of 80 available points)
• Solutions to Final Exam         [PDF | Postscript]     [exam statistics]

VII. Other Class Handouts

1. This is a handout on the alternating series test, with a clarification on the behavior of series that do not meet all the conditions of the test.   [PDF | Postscript].

2. Section 11 in Chapter 1 of Boas mentions a number of theorems about power series, although little detail is provided. In this handout, I have elaborated on these theorems. In particular, I state an important result, called Abel's Theorem, that asserts that if a series is convergent at an endpoint of the interval of convergence, then the series must be continuous at this point.   [PDF | Postscript].

3. Applying Abel's theorem incorrectly can lead to apparent paradoxes in the evaluation of power series at the endpoint of its interval of convergence. This handout describes the apparent paradox and its resolution.   [PDF | Postscript].

4. (Optional handout) The sum of an infinite series of functions is pointwise convergent over an interval defined by a < x < b if the sum is convergent at each point in the interval. However, the derivative of an infinite pointwise convergent sum of functions is not always equal to the sum of the derivatives. Likewise, the integral of an infinite pointwise convergent sum of functions is not always equal to the sum of the integrals. It is useful to introduce a stricter notion of convergence called uniform convergence. Uniformly convergent sums are much better behaved. All power series with a radius of convergence R is uniformly convergent over any closed interval [-r,r], where 0 < r < R. For further details, see the following class handout , which distinguishes between these two concepts of a convergent sum of functions:   [PDF | Postscript].

5. Taylor series can be used to approximate functions. By summing over a finite number of terms, it is often possible to obtain a bound on the error made by replacing the function with the sum of the first N terms of the Taylor series. (The error will depend on the point at which the function is evaluated.) One such example is provided by Theorem 14.4 on p. 35 of Boas. A proof of this theorem and a simple example are provided in this class handout.   [PDF | Postscript].

6. A list of Taylor series for some well known functions is provided in this handout. The coefficients of the Maclaurin series for tan(x) and sec(x) depend on the Bernoulli and Euler numbers, respectively. Some properties of these numbers are presented.   [PDF | Postscript].

7. In this handout, asymptotic power series are defined and contrasted with convergent series. The properties of asymptotic series are briefly summarized. A case example is studied to illuminate the choice of N such that the sum of the first N terms of the asymptotic series provides the optimal approximation to the function.   [PDF | Postscript].

8. Suppose we wish to study the behavior of a function f(x) for values of x in the neighborhood of x=a, where a is a real (or complex) number. We would like to know how f(x) approaches f(a) as x  →  a. This handout provides an introduction to the Big Oh (order) symbol and how to use it in expressing the behavior of a function.   [PDF | Postscript].

9. A complex number can be written in polar form as z=ei θ, where θ is called the principal value of the argument of z and is denoted by Arg z. One can also define a multi-valued argument function as arg z=Arg z+2πn (where n is any integer). Details of the properties of arg z and Arg z can be found in this handout.   [PDF | Postscript].

10. A complex logarithm, exponential and power functions do not possess many of the simple properties associated with the corresponding real-valued functions. Ignoring this simple fact can lead to apparent paradoxes. In these highly detailed (and sometime dense) notes, I carefully define the complex logarithm, exponential and power functions, paying careful attention to distinguish between the multi-valued functions and their single-valued principal value counterparts.   [PDF | Postscript].

11. (Optional handout) The inverse trigonometric and the inverse hyperbolic functions are multi-valued functions that can be written in terms of the complex logarithm. In these comprehensive notes, I provide a derivation of these expressions. I also carefully define the (single-valued) principal values of the inverse trigonometric and the inverse hyperbolic functions and their branch cuts following the conventions of Mathematica 8 computer algebra software, and compare with the standard conventions for the real-valued inverse functions.   [PDF | Postscript].

12. This handout provides a brief discussion of the properties of the logarithmic derivative of the Gamma function. In the process, we learn some fundamental facts about Euler's constant. [PDF | Postscript].

13. In this handout, I show you how to compute the volume and surface "area" of an n-dimensional hypersphere. The results depend on the Gamma function. [PDF | Postscript].

14. This handout provides an introduction to the Riemann zeta function, with details of the derivations presented in class.     [PDF | Postscript].

15. (Optional handout) There is a remarkable connection between the Bernoulli numbers and the Riemann zeta function evaluated at the even integers. This handout provides a "proof" of the formula that exhibits this relationship. A little handwaving motivation is used in one step where the mathematical rigor is lacking. (Euler glossed over the same subtlety!)     [PDF | Postscript].
Further details and derivations (at a slightly more sophisticated level) can be found in the following article:     [PDF]

16. Numerous problems in linear algebra can be solved by employing elementary row operations to convert a matrix into its reduced row echelon form. In this handout, explicit forms for the elementary matrices that correspond to the elementary row operations are given. The reduced row echelon form is defined and a procedure for achieving this form is outlined, called Gauss-Jordan elimination. Using these methods, I provide a proof that the rank of a matrix is equal to the rank of its transpose. I also provide details on how to compute the inverse (if it exists) using Gauss-Jordan elimination.     [PDF | Postscript].

17. This handout discusses vector components (also called coordinates), matrix elements and changes of basis. Applications to the matrix diagonalization problem and the properties of three-dimensional rotation matrices are provided.     [PDF | Postscript].

18. This handout demonstrates a number of basic properties of the characteristic polynomial of a matrix. These notes discuss how the coefficients of the characteristic polynomial are related to the eigenvalues, and provides a general formula for the coefficients in terms of traces of powers of the matrix. A proof is given of the Cayley-Hamilton theorem, which states that any matrix satisfies its own characteristic equation.     [PDF | Postscript].

19. This handout examines the properties and derives an explicit form for the most general 3x3 proper and improper rotation matrices. Using these results one can easily determine the rotation axis and the angle of rotation given an arbitrary orthogonal matrix of determinant +1. Likewise, one can easily determine the reflection plane and the rotation angle around an axis perpendicular to the plane given an arbitrary orthogonal matrix of determinant -1.     [PDF | Postscript].

20. (Optional handout) Any three-dimensional rotation R(n,θ) can be expressed as a product of simpler rotations. In this handout, we explicitly show how to write a three-dimensional rotation matrix as a product of rotation matrices whose rotation axes are along either the x, y or z axis. The most famous representation of a three-dimensional rotation of this type is the Euler angle parameterization. The Euler angles are used often in classical mechanics and quantum mechanics. The relation between the Euler angles and (n,θ) is derived in these notes. [PDF | Postscript].

21. The diagonalization of a 2x2 real symmetric matrix appears often in physics problems. So, it is useful to solve this problem once and for all. In this handout, I compute the eigenvalues and eigenvectors of the most general 2x2 real symmetric matrix A, and then find the real orthogonal matrix that diagonalizes A.     [PDF | Postscript].

22. (Optional handout) This handout treats the diagonalization of hermitian and normal matrices by a unitary similarity transformation. It also provides details of the proof that two commuting hermitian matrices are simultaneously diagonalizable by a common unitary similarity transformation. The latter implies that two commuting hermitian matrices possess simultaneous eigenvectors.     [PDF | Postscript]

23. A new derivation of the explicit forms for the most general proper and improper rotation matrices, which are parameterized by a rotation angle θ and a unit vector n, is presented. This elegant derivation (which is significantly simpler than the one presented in the previous handout 19) is based on the methods of tensor algebra. [PDF | Postscript].

24. (Optional Handout) This handout provides a derivation of the full asymptotic expansion for Γ(z+x) as z→∞ where |Arg z|<π and x is a fixed real number. By setting z=iy, we obtain the corresponding asymptotic expansion for Γ(x+iy) as |y|→∞ and explain why all corrections of O(1/|y|p) [for any positive power p] are absent if and only if x=0, ½ or 1. This result addresses an assertion made in Appendix C of the exam solutions. [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 here.

2. One of the most useful undergraduate textbooks on mathematical physics is: Essential Mathematical Methods for Physicists, by Hans J. Weber and George B. Arfken, which is available for online viewing here. It takes a while to load, so be patient.

3. Lecture notes on infinite series written by Vince Vatter (who is a professor of mathematics at the University of Florida) provide an elementary treatment of sequences, infinite series and power series. These notes, which are available here in PDF format, are very readable and provide more details than Chapter 1 of Boas. You will also find additional problems along with some solutions.

4. Infinite sequences and sums are some of the initial topics studied in a mathematical real analysis course. Such a course provides a sophisticated treatment of concepts first learned in calculus. Here, I provide a free textbook by Brian S. Thomson, Judith B. Bruckner and Andrew M. Bruckner, Elementary Real Analysis, taken from http://www.classicalrealanalysis.com. For this course, I recommend that you peruse Chapters 2, 3, 14 and 16, if you wish to read more about infinite series and power series. These chapters are quite readable for Physics 116A students (even if the rest of the book is somewhat advanced).

5. Sean Mauch (of Caltech) provides a free massive textbook (2321 pages) entitled Introduction to Methods of Applied Mathematics. You may be particularly interested in his treatment of complex numbers. He takes 180 pages to cover the material of Chapter 2 of Boas. I have isolated the two relevant chapters on complex numbers and complex functions to this PDF link. These chapters also provide useful worked out examples.

6. James Nearing also provides a free textbook entitled Mathematical Tools for Physics. This book treats most of the topics of Physics 116A at a similar level of difficulty. Although not as comprehensive as the one by Sean Mauch, you may find some of the presentations enlightening.

7. David A. Santos provides a free textbook on linear algebra which I have uploaded to this website in PDF format. I believe that you will find this quite useful. It covers all the topics of chapters 3 of Boas in greater depth, but at about the same level of sophistication. In addition, it provides many worked out examples.

8. William Chen has a extensive set of lecture notes on linear algebra consisting of twelve chapters on topics, many of which have been treated in Physics 116A. Click on the individual chapter headings to download the pdf versions of each chapter.

9. For a more comprehensive (and somewhat more advanced, although still very readable) treatment of matrix analysis and linear algebra, check out Matrix Analysis and Applied Linear Algebra by Carl D. Meyer. This modern textbook and solution manual, published by SIAM, have been generously made available for free viewing. Please respect the restrictions to printing and distribution of this material.

10. If you need more practice on problems in matrices and linear algebra, I highly recommend a free textbook by Jim Hefferon entitled Linear Algebra available as a free PDF download along with the answers to exercises. Note that if you save the two files in the same directory, then clicking on an exercise will send you to its answer and clicking on an answer will send you to its exercise.

11. Yet Another free text book on linear algebra by Sergei Treil entitled Linear Algebra Done Wrong can be found here in PDF format. This book is advertised as a first linear algebra course for mathematically advanced students. It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more rigorous mathematics than is presented in a "cookbook style" calculus type course. You will have to check out the preface of this book to see why the author chose such a strange title for his opus.

IX. Articles of Interest

1. If the rearrangement of conditionally convergent series strikes your fancy, check out this link for a more in depth discussion of rearranging the alternating harmonic series.

2. There are many proofs that the infinite harmonic series is divergent. Click here for twenty different proofs of this well known theorem. This paper is very elementary and clearly written. Applications of the harmonic series are ubiquitous. For example, see this note written by David Bressoud.

3. The history of pi is fascinating. Check out this review paper by Peter Borwein, one of the giants in the field of pi research, which is available in PDF format. A slightly older and shorter review by the same author can be found here: [PDF | Postscript]. Almost anything you wanted to know about pi can be found in this amazing book Pi Unleashed by Jorg Arndt and Christoph Haenel. For a more sophisticated account of pi, I can recommend The Number Pi by Pierre Eymard and Jean-Pierre Lafon.

4. Gregory's series for pi is very slowly converging. Nevertheless, if you sum over the first 500,000 terms, although the 6th digit after the decimal point is wrong, the next ten digits are correct! In fact, only four digits of the first forty are incorrect! The explanation of this curious phenomenon can be understood as a consequence of a certain asymptotic expansion. More details can be found here in an article by J.M. Borwein, P.B. Borwein and K. Dilcher in [PDF] format.

5. Did you ever wonder how to prove that ?     Click here for fourteen different proofs of this result. If fourteen proofs are too much to handle, try clicking here for a more modest presentation that presents six different ways to sum the series. The latter presentation is slightly more elementary in style as compared to the former.

6. Rudolfo Rosales produced a pedagogical set of notes on branch points and branch cuts for his complex variables course at MIT. You can find these notes here in your favorite format: [PDF | Postscript]. A shorter and very readable treatment entitled Branches of Functions is excerpted from Complex Analysis for Mathematics and Engineering by John H. Mathews and Russell W. Howell and is available in PDF format. You will also find here a graphical representation of the Riemann surface of the complex square root function.

7. Click on the following links to learn more about the gamma function, Euler's constant, the Riemann zeta function and the Bernoulli numbers.

Introduction to the Gamma function       [PDF | Postscript]
The Euler constant      [PDF | Postscript]
Collection of formulae for Euler's constant      [PDF | Postscript]
The Riemann zeta function       [PDF | Postscript]
Introduction to the Bernoulli numbers       [PDF | Postscript]

These articles were found on a web site entitled Numbers, constants and computation, which contains quite a lot of fascinating material. The article on the gamma function covers a number of topics that we were not able to cover in class, including the theorem on the conditions on the uniqueness of the gamma function as the continuous function extrapolation of the factorials of the positive integers.

8. If you would like to learn more about Euler's constant and its connections to the harmonic series and other areas of mathematics, look no further than the highly entertaining book by Julian Havil entitled Gamma: Exploring Euler's Constant. The level of this book is suitable for any calculus student!

9. A beautifully typeset article (with a pleasing use of colors and photographs) entitled Values of the Riemann zeta function at integers by R.J. Dwilewicz and Jan Minac provides another introduction to some of the mysteries of the Riemann zeta function. In this article, the gamma function, its logarithmic derivative, Euler's constant, and the Bernoulli numbers all play a role in the story. The article is quite readable for the most part (you can ignore the more advanced material that occasionally appears) and definitely worth perusing. It can be found here:     [PDF].

10. In problem 7 on Homework Set 5, you learned that the the volume of a hypersphere of radius 1 in d-dimensions reaches a maximum between d=5 and d=6 and then approaches zero as d → 0. These results are the subject of a beautifully written article by Brian Hayes in the November--December 2011 issue of American Scientist, entitled An Adventure in the Nth Dimension. This article can be found here:     [PDF].

11. Boas quotes the following important theorem at the bottom of page 154. A matrix can be diagonalized by a unitary similarity transformation if and only if it is normal, i.e. if the matrix commutes with its hermitian conjugate. For a proof of this result along with a pedagogical treatment of normal matrices written especially for physics students, see Philip A. Macklin, Normal Matrices for Physicists, American Journal of Physics 52, 513--515 (1984). If you are logged onto a campus computer, then you can directly download the article here:     [PDF].

X. Other Web Pages of Interest

1. An apocryphal story relates how a nine-year old Carl Friedrich Gauss managed to compute the sum 1+2+3+...+100 in a few seconds. But is the story true? For an attempt to answer this question, check out an article in the May-June, 2006 issue of the American Scientist by Brian Hayes entitled Gauss's Day of Reckoning. A collection of versions of the Gauss anecdote can be found here.

2. 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.

3. 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]

4. 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.

5. 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.

6. A great source for graphs of mathematical functions, both real-valued and complex-valued, can be found in the Gallery of mathematical functions.

7. An encyclopedic treatment of linear algebra and the theory and applications of matrices can be found in the Handbook of Linear Algebra, edited by Leslie Hogben (Chapman and Hall/CRC, Boca Raton, FL, 2007). An electronic version of this book is available for viewing and downloading if you are using a computer on the UCSC network.

8. Problems 1 and 4 on the practice final exam provide two independent techniques for solving homogeneous linear difference equations with constant coefficients. The function F(x) in problem 1 is called a generating function for the sequence of integers 0,1,1,3,5,11,21,43,.... To learn more about this technique, I can recommend an amazing book by Herbert S. Wilf, entitled "Generatingfunctionology." The second edition of this book is available for free in pdf format.

These same techniques can also be used to derive a closed-form expression for the nth integer of the Fibonacci sequence (0,1,1,2,3,5,8,13,21,...). The Fibonacci numbers are also intimately connected to the golden ratio (sometimes called the golden mean) which is often denoted by φ=(1+√5)/2. To learn more about the golden ratio, check out this Wikipedia link.

haber@scipp.ucsc.edu
Last Updated: October 26, 2011