## METHODS OF APPLIED MATHEMATICS

### 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, 3:20-4:40 PM;
• Wednesday, May 2, 10:30-11:30 AM;
• Thursday, May 3, 9:30-10:30 AM, 3:10-3:50 PM;
• Monday, May 7, 1:30-2:30 PM;
• Tuesday, May 8, 3:20-4:40 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 Cauchy-Riemann 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 ux, uy, vx, and vy 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 Cauchy-Riemann 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).