Fabian Waleffe -- Math 321 -- U. of Wisconsin

Math 321 homework : Spring 2004

Textbook: Mathematical Methods for Physicists, 5th Edition, by Arfken and Weber.

Read the Notes on vectors FIRST and try out as many of the implicit and explicit exercises given there as possible. Also read sections 1.1-1.5 in A&W (you may skip pages 8-11 for now). Don't miss `Vectors and Vector Space' on page 12.

Arfken and Weber Exercises:
(problem numbers in parentheses are interesting but may be slightly more advanced or involve concepts (e.g. physics concepts) that you may not be familiar with).

Section 1.4: 4, 5, 7, 9, 10, 11, 14, 15, 16
Section 1.5: 1, 5, 6, 7, 8, 9, 11, (15), (17), (18)
Sections 1.1 and 1.3: 1.1.9, 1.1.11 (Hubble's law), 1.3.3 (position vector exercises).

See the notes and exercises on tensors .

Vector Calculus READ Arfken & Weber sections 1.6-1.14, and don't be afraid to look back at your favorite Math 222 and Math 234 Calculus book.
Exercises:

Section 1.6 1,2,3,5
Section 1.7 1,5,6
Section 1.8 2,3,4,6,11,13,14,16
Section 1.9 1,2,3,4,5,7,8,10
Section 1.10 2,3,4,5
Section 1.11 1,2,3,4
Section 1.12 1,2,3,5,7,8,9
Section 1.13 1, 7, 8
Section 1.14 1,2

Chapter 5 Key ideas assumed to be known from math 221 and 222: what is a series? a geometric series? what is the partial sum of a geometric series? [5.1], ratio test [5.2], Taylor Series (including "binomial theorem") [5.6], Taylor series of exp(x), cos(x), sin(x), exp(i theta) where i²=-1 and theta is real.

Section 5.1. write the number 0.9999999... using a geometric series and show that it equals 1; write 0.123123123123... as the ratio of 2 integers.
Section 5.6. 1, 5, 8; FInd Taylor series about x=0 for 1/(1-x), 1/(1-x^2), 1/(1+x^2), ln(1+x) and discuss their domain of convergence using the ratio test. [Hint: using geometric series bypasses a lot of derivatives.] What is the Taylor series of 1/(1-x) about x=a? what is its domain of convergence? [Hint:1-x = (1-a) - (x-a)]

Chapter 6 Functions of a Complex Variable

Section 6.1. 1, 3, 5-11, 13, 15, 16, 21
Section 6.2. 1, 2, 5, 10
Section 6.3. 1,2, 3
Extras:
1. Compute the integral of 1/z over a square of sides 2 centered at z=0 explicitly (i.e. by using the definition of the integral, parametrizing the curve etc.. not by Cauchy's formula). If you think before computing (usually a good thing to do) there is no integral to compute!
2. Compute the integrals of z² and e^z over the path (a) 0 -> 1 -> 1+i, (b) the circular arc from 4 to 4i, (c) the circle of radius 3 centered at 4+4i, by explicit calculation.
Section 6.4. 1, 3, 4, 5,
Extras
1. Complete in full details the calculation of the integral of 1/(a²+x²) over the real axis using complex integration, as done in-class for a=1.
2. Calculate the integral of e^z/(z(z²+9)) over the circle of center z=1+i and radius 2 by (1) the "cut-out method" and explicit parametrization of small circles, and (2) by Cauchy's integral formula.
3. Calculate the integral of z² from z=1 to z=1+i. Discuss.
4. Calculate the integral of z^(-1) from z=1 to z=1+i. Discuss!
5. Calculate the integrals of cos(z)/z and sin(z)/z over a circle of radius 2 centered at the origin.
Section 6.5. 1, 2, 8
Extras
1. Laurent vs Taylor: Expand 1/z into a Laurent series about z=0 (hint: this is really easy...). Can you expand 1/z in a Taylor series about z=0? What about a Taylor series about z=1? Here's another nice Laurent vs. Taylor: Expand 1/(1+z²) in a Taylor series about z=0 (we've done it in class). Show that it converges only for |z|<1. Now expand in a Laurent series about z=0 (Hint: let z=1/t, t complex and expand the t function in a Taylor series about t=0, i.e. z=infinity(!), the Taylor series in t is your Laurent series in z. Show that the Laurent series converges only for |z|>1. Hence the Taylor and Laurent series are complementary. Cool!)
2. Find the first 3 terms in the Laurent series expansion of 1/sin(z) about z=0. Calculate the integral of 1/sin(z) about a circle of radius 1 centered at the origin.
3. Calculate the integral of exp(1/z) about an ellipse of major axis 7 in the x direction and minor axis 2 in the y direction, centered at the origin.
4. Calculate the integral of exp(1/z) and exp(1/z^3) about the unit circle centered at the origin.
5. Calculate the integral of e^z/[(z²+1)(z+1)²] about a countour that encloses z=-1 and z=i.
6. Same but for a countour that encloses z=-1, z=i and z=-i.
7. Calculate the integral of g(z)= z² sin (1/z) over the unit circle. (How about z⁴ sin(1/z) and sin(z²) sin(1/z)?)
8. Calculate the integral of z⁵ exp(1/z) about the unit circle.
  Hints and explicit solved examples for the following types of integrals are given in section 7.2.
9. Calculate the integral from -infinity to + infinity of 1/(ax²+ bx + c) where a,b,c and x are real and a>0, b²-4ac < 0. [Hint: same approach as for 1/(1+x²) done in class].
10. Calculate the integral from -infinity to + infinity of 1/(1+x⁴).
11. Calculate the integral from -infinity to infinity of cos(x)/(1+x²) and of cos(x)/(1+x⁴)
12. Given the Poisson integral [integral from -infinity to + infinity of exp(-x²) = sqrt(Pi)], calculate the Fresnel integrals: integral from 0 to infinity of cos(x²) and of sin(x²). Justify!
Section 7.2 7, 8, 9, 10, 11, 12, 14, 15, 20, 21, 22, 24, 25
Singularities, Branch points, branch cuts: Read section 7.1. Plot exp(1/z) for z=x (real), -1< x < 1 and for z=i y with -1< y < 1. Same exercise for cos(1/z). Try exercises 6.6.4, (example 7.1.1), 7.2.18 (NEED -1< a <0 ! so 7.2.18 and 7.2.19 are in fact identical!).