Mathematics 528
METHODS OF APPLIED MATHEMATICS
SPRING 2007
NOTE: Solutions to homework assignments are no longer available
through this web page.
Final Exam:
 The final exam will be held on Thursday, May 3,
from 4:00 to 7:00 P.M., in Hill 124. The exam is cumulative.
 Office hours during reading period and exam week (all in Hill
520): Note change (5/1) in office hours for Thursday 5/3.
 Tuesday, May 1, 3:204:40 PM;
 Wednesday, May 2, 10:3011:30 AM;
 Thursday, May 3, 9:3010:30 AM, 3:103:50 PM;
 Monday, May 7, 1:302:30 PM;
 Tuesday, May 8, 3:204:40 PM.
 Here is a tentative formula sheet for the final
exam.
Exam 2 has been rescheduled to Thursday, April 12. It
will cover Chapters 23 and 24.
Exam 1 will be held in class on Tuesday, February 27. It
will cover through our work on conformal mapping.
Homework assignments.
 Assignment 13. Hildebrand, Chapter 2: 59, 60, 61, 62, 63. This
assignment will not be collected.
 Assignment 12. Hildebrand, Chapter 2: *8, *9, *13, *15. In
problem 13, show that the solutions are great circles.
 Assignment 11. Here is assignment 11
in pdf form. No problems from this assignment
will be collected.
 Section 24.5: 2 (b), (d), (h), (j); 3 (f), (i), (j); 4; 6 (a), (d), 7.
 Obviously, there are many more problems in Section 24.5 which you may
try; this seems a reasonable selection to practice on. We are omitting the
inverse Laplace and Fourier transforms.
 Assignment 10, starred problems due 4/03. Here is assignment 10
in pdf form.
 Assignment 9, starred problems due 3/27. Here is assignment 9
in pdf form.
 Solutions to Assignment 9.
 Section 24.2: 6 (c), *(d), *(f), (g), (h); 8 (a), *(c), (d);
*9; 11 (e), *(f), (j); 16 *(g).
 In 6(d), also identify explicitly the function which is the sum of the
series.
 For extra credit, identify explicitly the function which is the
sum of the series in 8(a).
 Assignment 8. This is due 3/20, and is here in pdf form.
 Assignment 7 was passed out in class Thursday, March 1. Its
due date has been revised to Thursday, March 8.
 Solutions to Assignment 7
 Section 23.2: 1. *(a), (b), (d), *(f), 3. (f), 5. *(a)
 Section 23.3: 1, 2, 4 (d), (f), (h), *(k), *(l), 5, *7
 Note: the problems 23.3:1,2 are just thought problems to help you
understand Cauchy's theorem. After that, there are a good many problems,
none very hard, just so that you can practice doing these contour
integrals. Use Cauchy's Theorem whenever possible in (Section 23.3)! For
23.3:4(f), see Example 3 in that section. Cauchy's Theorem could help you
do 22.3.2:1(d) but of course you don't know it at that point.
 Assignment 6. Further problems on conformal mapping. These
problems will not be collected.
 Assignment 5, due 2/20. Turn in only starred problems. Here is assignment 5 in pdf form.
 Solutions to Assignment 5.
 Section 22.3: 10 (a), (e), *(g), 11 (a), *(c), 14(a), *(d), *(e)
 Supplementary exercise 5.1: In each case below, find a conformal
mapping w=f(z) carrying the given region D onto the upper
half plane v>0 (here w=u+iv and in describing D we
always write z=x+iy). Give a brief explanation of your answer, but
not a full proof.
 (a) D is the right half plane x>0.
 (b) D is the second quadrant x<0, y>0. Hint: think about z^2.
 *(c) D is the intersection of the right half
plane with the unit disk: x>0, x^2+y^2<1. Hint: start with a
bilinear transfomation, then use the idea of (b).
 *(d) D is the strip 1>x>0.
 (e) D is the half strip 1>x>0, y>0. Hint: modify
a homework problem.
 Assignment 4, due 2/13. Turn in only starred problems. Here is assignment 4 in pdf form.
 Solutions to Assignment 4.
 Section 21.5: 15 *(a), (b), *(d).
For (d), work in polar coordinates. In
these coordinates you can use equation (1) of section 20.3 to determine if
a given function is harmonic, and the appropriate CauchyRiemann equations
(equations (30) of Section 21.5) to construct the conjugate harmonic
function of a given harmonic function.
 Section 21.5: *16 (a), (c).
 Section 22.2: *1.
 Assignment 3, due 2/06. All problems are from Section
21.5. Turn in only starred problems. Here is assignment 3 in pdf form.
 Solutions to Assignment 3.
 2(a), *2(b),
5(a).
 *5(b) and *10(e). Think about these two problems together. Show
that if you calculate the derivative of 1/z (i) by the rule for
powers (or equivalently the quotient rule), (ii) directly as a limit, from
(8), as in Example 2, or (iii)by any of the formulas (19), the answer is
the same. NOTE: There was an error on the copy of assignment 3 handed
out in class: problem 10(c) was assigned. The correct assignment is 10(e),
as above.

*9. Hint: this is a bit tricky. To calculate any partial derivatives of
u(x,y) or v(x,y) at x=y=0, or to try to calculate the
derivative of f at z=0, you must use the direct
definition of the derivative as a limit, since there is a special formula
for f(0). Calculate all the partial derivatives this way, then try
to calculate f'(0) as z approaches 0 along the direction
x=y.
As an additional exercise (not to be turned in): From the results of
the problem, and Theorem 21.5.1 (or more properly from the version of this
theorem stated in class) we know that not all of the partial derivatives
u_{x}, u_{y}, v_{x}, and
v_{y} can be continuous at x=y=0. Be sure you
understand why this is true, then show it by direct calculation.

*10(a), 10(g), *12(b), 14(a). Hint for 14(a): use the CauchyRiemann
equations.
 Assignment 2, due 1/30. Turn in only starred problems.
 Section 21.3: 2(b), *2(f), 9(e), 12, *13, 16(c), *18(c).
Note: In problem 21.3.13, the hint referred to
is in problem 12, not problem 11.
 Section 21.4: 11(c), *11(e), 11(f), *13.
 Solutions to assignment 2 will be distributed in class.
 Assignment 1, due 1/23. Greenberg 21.2: 9(d,h), 11(b,d); 21.4:
4(f), 5(c,f), 8(c,f).
Course handouts: